111Department of Mathematics, University of California, Los Angeles

# A Hele-Shaw Limit With A Variable Upper Bound and Drift

Raymond Chu<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

We investigate a generalized Hele-Shaw equation with a source and drift terms
where the density is constrained by an upper-bound density constraint that
varies in space and time. By using a generalized porous medium equation
approximation, we are able to construct a weak solution to the generalized
Hele-Shaw equations under mild assumptions. Then we establish uniqueness of
weak solutions to the generalized Hele-Shaw equations. Our next main result is
a pointwise characterization of the density variable in the generalized Hele-
Shaw equations when the system is in the congestion case. To obtain such a
characterization for the congestion case, we derive a new uniform lower bounds
on the time derivative pressure of the generalized porous medium equation via
a refined Aronson-Bénilan estimate that implies monotonicity properties of the
density and pressure.

##  1\. Introduction

We consider solutions $\varrho_{k}$ for $k>1$ of the modified Porous Medium
Equation (mPME)

(1.1) $\partial_{t}\varrho_{k}=\nabla\cdot(\varrho_{k}(\nabla
p_{k}+\vec{b}))+f\varrho_{k}\text{ in }Q_{T}:={\mathbb{R}}^{d}\times(0,T),$

where the pressure
$p_{k}:=\frac{k}{k-1}\left(\frac{\varrho_{k}}{m}\right)^{k-1}$. In this
equation $m,\vec{b}$, and $f$ are given smooth functions (see Section 2 for
the exact assumptions on these functions), while the unknown is the density
$\varrho_{k}$. Here $k>1$ is a constant that reduces congestion by having
strong diffusion when the density is larger than the constraint $m$. We can
also rewrite this equation as

(1.2) $\partial_{t}\varrho_{k}=\nabla\cdot(m\nabla
v_{k}^{k}+\varrho_{k}\vec{b})+f\varrho_{k}\text{ in }Q_{T}$

because $\varrho_{k}\nabla p_{k}=m\nabla v_{k}^{k}$ where
$v_{k}:=\varrho_{k}/m$ is the normalized density. For the special case of $m$
being a constant, this equation reduces to the Porous Medium Equation (PME)
with a given drift $-\vec{b}$ and source term $f$. This anti-congestion effect
has led to the PME being used to model many physical phenomena such as
population dynamics [3, 4, 25] and density based tumor growth [22, 8]. Due to
the physical importance of the pressure, we obtain from the constitutive law
$p_{k}=\frac{k}{k-1}\left(\frac{\varrho_{k}}{m}\right)^{k-1}$ that

(1.3) $\partial_{t}p_{k}=|\nabla p_{k}|^{2}+\nabla
p_{k}\cdot\vec{b}+(k-1)p_{k}\left[\frac{1}{m}\nabla\cdot(m\nabla
p_{k})+\frac{\nabla\cdot(m\vec{b})+mf-\partial_{t}m}{m}\right].$

We are interested in the $\mathit{Hele}$-$\mathit{Shaw}$ limit of
$\eqref{PME_Density}$ and $\eqref{PME_pressure1}$, that is the limit as
$k\rightarrow\infty$. Formally if we let $(\varrho_{\infty},p_{\infty})$
denote the limit of $(\varrho_{k},p_{k})$, then we have the following free
boundary problem

(1.4) $\begin{cases}\partial_{t}\varrho_{\infty}=\nabla\cdot(m\nabla
p_{\infty}+\varrho_{\infty}\vec{b})+f\varrho_{\infty}\text{ in }Q_{T}\\\
\nabla\cdot(m\nabla p_{\infty})+\nabla\cdot(m\vec{b})+mf-\partial_{t}m=0\text{
on }\\{p_{\infty}>0\\}\\\ \varrho_{\infty}(x,t)\leq m(x,t)\text{ in }Q_{T}\\\
p_{\infty}(m-\varrho_{\infty})=0\text{ in }Q_{T}\\\ \end{cases}.$

The case of $m$ being a constant is well studied and in this case (1.4) is
known as the Hele-Shaw equations. The Hele-Shaw equations are used to model
physical phenomena with a density constraint such as congested crowd motion
[20] and tumor growth [22, 8]. In the tumor growth models, $m$ represents the
maximum packaging density of cells [23], while for the congested crowd motion
models, $m$ represents the maximum pedestrian population. By allowing $m$ to
vary in space and time, instead of being a constant, it allows these models to
account for spatial and time changes to the maximum population or packaging
density of cells.

Figure 1. Numerical Simulations of the mPME $\eqref{PME_Density}$ with
$f=\vec{b}=0$. In this figure, we have $k=40$ and
$m(x,t)=\exp(-t/6-x^{2}/10)$, which corresponds to the case of congestion, and
we see the solution $\varrho_{40}$ becomes saturated on $m$.

By combining the techniques in the papers [22, 12, 8], we were able to take
the Hele-Shaw limit in $L^{1}(Q_{T})$ of $\eqref{PME_Density}$ and
$\eqref{PME_pressure1}$ under mild assumption of $m=m(x,t)$, $\vec{b}$, $f$,
and the initial data. In addition, by using methods from [22], we were able to
derive that the limiting density and pressure are the unique weak solutions to
$(\ref{Hele-Shaw})$. Then by using the arguments in [12], we derive the normal
velocity law for $\partial\Sigma(t)$ where
$\Sigma(t)=\\{x:p_{\infty}(x,t)>0\\}$ in the viscosity sense. The velocity law
$V$ is given by

(1.5) $V=\left[-\frac{m\nabla
p_{\infty}}{(m-\varrho^{E})}-\vec{b}\right]\cdot\vec{\nu},$

where $\vec{\nu}$ is the outward normal of $\partial\Sigma(t)$ and
$\varrho^{E}$ is the external density (the limit of $\varrho_{\infty}$ from
outside $\Sigma(t)$). The velocity law $(\ref{velocity_law})$ shows that a
typical feature of the $\partial\Sigma(t)$ is that it can have infinite normal
velocity.

Then we focus on the case of when

(1.6) $m_{t}<mf+\nabla\cdot(m\vec{b})\text{ in }Q_{T},$

which we call the congestion case. This assumption has been used when $m$ is a
constant in [15] and corresponds to when $m$ grows strictly slower than the
external density. In the congestion case, we were able to derive an explicit
pointwise formula for the limiting density up to $\partial\Sigma(t)$ that is
consistent with the identification in [15] when $m$ is constant. This
identification formula implies that when we are the congestion case, if the
initial density is a patch, then it is as a patch for all time (that is if
$\varrho_{\infty}(x,0)=m(x,0)\chi_{A(0)}$, then
$\varrho_{\infty}(x,t)=m(x,t)\chi_{A(t)}$).

We were able to explicitly identify the limiting density by deriving a novel
uniform Aronson-Bénilan estimate in the congestion case. This estimate appears
to be new even for the case of $m$ being constant with non-zero $\vec{b}$. An
important consquence of the estimate is that it leads to a uniform time
derivative estimate of the pressure $p_{k}$ along the streamlines for all
$k>1$ is large

(1.7)
$\frac{d}{dt}p_{k}(X(t,x_{0}),t)\geq-\left(C+\frac{1}{t}\right)p_{k}(X(t,x_{0}),t),$

where the constant $C>0$ is independent of $k$. For each
$x_{0}\in{\mathbb{R}}^{d}$, the streamline $X(t,x_{0})$ is defined as the
(unique) solution of

(1.8) $\begin{cases}\frac{d}{dt}X(t,x_{0})=-\vec{b}(X(t,x_{0}),t)\text{ in
}(0,\infty)\\\ X(0,x_{0})=x_{0}\end{cases}.$

Note in the special case of $\vec{b}\equiv 0$ that the streamlines reduce to
the identity map.

We use (1.7) to show that the normalized density
$v_{\infty}:=\varrho_{\infty}/m$ is non-decreasing with respect to the
streamlines. This means that if $t_{1}<t_{2}$, then
$v_{\infty}(X(t_{1},x_{0}),t_{1})\leq v_{\infty}(X(t_{2},x_{0}),t_{2})$. This
estimate also implies the geometric property that $\Sigma(t)$ is non-
decreasing with respect to the streamlines. That is if $x_{0}\in\Sigma(t)$,
then the streamline containing the point $(x_{0},t)$ is in $\Sigma(s)$ for all
$s\geq t$. The last monotonicity property is enough to allow us to
characterize the limiting density up to $\partial\Sigma(t)$ pointwise.

Similar uniform time derivative estimates of the pressure are only known in
the literature when $m$ is constant in the congestion case with $\vec{b}=0$ or
$m$ is constant with $f=\vec{b}=0$. Uniform lower bounds on
$\partial_{t}p_{k}$ similar to $\eqref{new_time_derivative}$ played an
essential role in taking the Hele-Shaw limit for weak solutions with no drift
in the congestion case [5, 2, 9, 10, 22, 14].

The congestion assumption (1.6) is essentially necessary to have estimate such
as $(\ref{new_time_derivative})$. Indeed, the the congestion assumption is
formally equivalent to having $p_{\infty}$ satisfy $\nabla\cdot(m\nabla
p_{\infty})<0$ inside $\Sigma(t)$ for all $t\geq 0$. If instead there was some
time $t_{0}$ such that $\nabla\cdot(m\nabla p_{\infty})(\cdot,t_{0})\geq 0$ in
$\Sigma(t_{0})$ where $\Sigma(t_{0})$ open, connected, and smooth, then the
maximum principle for uniformly elliptic operators implies that
$p_{\infty}\equiv 0$. That is the set $\Sigma(t)$ instantaneously collapsed at
time $t_{0}$. However, if the time derivative estimate
$(\ref{new_time_derivative})$ was true for this case, then the estimate
implies for positive time that $\Sigma(t)$ is non-decreasing with respect to
the streamlines, which is a contradiction. So uniform time derivative estimate
such as $\eqref{new_time_derivative}$ can not hold in general when there is a
$t_{0}>0$ such that
$[\partial_{t}m-mf-\nabla\cdot(m\vec{b})](\cdot,t_{0})=\nabla\cdot(m\nabla
p_{\infty}))(\cdot,t_{0})\geq 0$ on $\Sigma(t_{0})$.

We also mention that it is necessary to use some form of weak theory
(viscosity solutions, gradient flow in Wasserstein space, or weak solutions)
to understand $(\ref{Hele-Shaw})$. It is well known even for $m$ being a
constant that if we start with smooth data, the pressure can become
discontinuous in finite time due to topological changes (see [18, 22] for
examples of how the pressure can become discontinuous in finite time). In the
recent paper [16], the authors produced some regularity results of
$\partial\Sigma(t)$ for constant $m$ under the congestion assumption for
continuous viscosity solutions.

Now we review the literature on similar Hele-Shaw limits. These Hele-Shaw
limits are well studied when $m$ is a constant and until recently, these
limits were only investigated when the system is under congestion with $m$
constant or when $f=\vec{b}=0$ with $m(x,t)$ either being constant or $1/t$.
The weak limit of $\eqref{PME_Density}$ when $m\equiv 1$, $f=0$, and
$\vec{b}=0$ was studied in [5] and was again studied in [2] with more general
initial data. This investigation was continued in [9] and [10] when the
equations $(\ref{PME_Density})$ were posed on a domain
$\Omega\times(0,\infty)$ with general initial data and pressure boundary data.
In [10] the authors considered the case of $m(t)=1/t$ of $(\ref{Hele-Shaw})$
in their analysis. Then in [18], the author showed uniqueness and existence of
(1.4) when $m$ is constant with $f$ and $\vec{b}=0$ for viscosity solutions.
In addition, in [22] and [14] the authors dealt with the Hele-Shaw limits with
the condition of $f>0$, $\vec{b}=0$, and $m\equiv 1$ in a weak and viscosity
sense respectively. In [1] the authors were able to incorporate a non-zero
drift when $\vec{b}(x,t)=\nabla\Phi(x)$ such that $\Delta\Phi>0$ with $f=0$
and $m\equiv 1$ into their analysis by using viscosity solutions and gradient
flows in Wasserstein space. Then in [15] the authors were able to deal with
the case of $f+\nabla\cdot\vec{b}>0$ by using viscosity solutions.

The above literature primarily focused on the case when $m$ is constant with
congestion or $f=\vec{b}=0$. These assumptions were used when $\vec{b}=0$ to
obtain similar time derivatives as the one in $\eqref{new_time_derivative}$ to
obtain the weak limit. In [15], which dealt with $m\equiv 1$ when there is
congestion with drift, $\Sigma(t)$ non-decreasing along streamlines played an
essential part in their analysis. As mentioned earlier, these estimates and
monotonicity property of $\Sigma(t)$ are no longer generally true when there
is a general source term and drift.

In [12], the authors were able to remove the monotonicity conditions when
there is a source term $f$ with $\vec{b}=0$ in the exterior of a bounded
smooth domain with pressure boundary data and $m$ is constant. In addition,
the authors of [8] were also able to take the Hele-Shaw limit with $m$
constant without monotonicity conditions on their source and drift terms.

The authors of [26] investigated (1.1) and its Hele-Shaw limit using the
theory of gradient flow in Wasserstein space (see [21] for the gradient flow
structure of the PME) when $f=0$, $\vec{b}=0$, $\partial_{t}m(x,t)<0$
(congestion case), and $m(x,t)$ is approximately linear. The approximately
linear constraint on $m$ was used in [26] to ensure a well defined limit from
their JKO approximation scheme and was conjectured to be necessary to take the
Hele-Shaw limit of (1.1) and (1.3). In comparison, our assumptions are rather
mild (see Section 2) and we do not need the approximately linear condition of
$m(x,t)$ used in [26]. Our methods appear to be the first to be able to take
the Hele-Shaw limit for a wide range of $m$ that varies in space and time.

##  2\. Notation, Assumptions, and Summary of Results

Throughout this paper, we denote the pair $(\varrho_{k},p_{k})$ as weak
solutions of (1.1) and (1.3) respectively with initial conditions
$\varrho_{k}^{(0)}$ and $p_{k}^{(0)}$.

######  Definition 2.1 (Weak Solutions to mPME).

We say that $\varrho_{k}$ is a weak sub solution to (1.1) with initial data
$\varrho_{k}^{0}\in L^{2}({\mathbb{R}}^{d})$ if $\varrho_{k}\in L^{2}(Q_{T})$
and $v_{k}:=\varrho_{k}/m$ satisfies $v_{k}^{k}\in H^{1}(Q_{T})$ such that for
all non-negative test functions $\varphi\in
C^{\infty}_{c}({\mathbb{R}}^{d}\times[0,\infty))$

(2.1)
$\iint_{Q_{T}}-\varrho_{k}\partial_{t}\varphi\leq\int_{{\mathbb{R}}^{d}}\varphi(t=0)\varrho_{k}^{0}(x)+\iint_{Q_{T}}-m\nabla\varphi\cdot\nabla
v_{k}^{k}-\varrho_{k}(\nabla\varphi\cdot\vec{b})+f\varphi\varrho_{k},$

where $Q_{T}:={\mathbb{R}}^{d}\times(0,T)$.

Similarly, we say that $\varrho_{k}$ is a weak super solution if the
inequality in (2.1) is reversed. Then we say that $\varrho_{k}$ is a weak
solution if it is both a weak sub and super solution to (1.1). Now if we
denote $p_{k}:=\frac{k}{k-1}\left(\frac{\varrho_{k}}{m}\right)^{k-1}$, then we
say that $p_{k}$ is a weak solution (respectively super/sub) solution of (1.3)
if $\varrho_{k}$ is a weak (respectively super/sub) solution of (1.1).

We refer the reader to Section 9 where we discuss properties of weak solutions
of (1.1) such as existence, comparison principle, and smooth approximation. As
standard in the PME literature [24], when we obtain a priori estimates, we
treat $\varrho_{k}$ as if it is smooth because we can instead use our smooth
approximations and take limits to obtain the estimate for our non-smooth
$\varrho_{k}$.

To simplify our notation we let
$F:=(\nabla\cdot(m\vec{b})+mf-\partial_{t}m)/m$, then (1.3) simplifies to

(2.2) $\partial_{t}p_{k}=|\nabla p_{k}|^{2}+\nabla
p_{k}\cdot\vec{b}+(k-1)p_{k}\left(\frac{1}{m}\nabla\cdot(m\nabla
p_{k})+F\right).$

We also let $w_{k}:=\frac{1}{m}\nabla\cdot(m\nabla p_{k})$ because this
quantity frequently shows up in our calculation. In addition, for many
computations, it will be easier to work with the normalized density
$v_{k}:=\varrho_{k}/m$ compared to the density $\varrho_{k}$. The normalized
density solves

(2.3) $\partial_{t}v_{k}=\frac{1}{m}\nabla\cdot(m\nabla v_{k}^{k})+\nabla
v_{k}\cdot\vec{b}+Fv_{k}\text{ in }Q_{T}.$

We will refer to the first equation in (1.4) as the density equation and the
second equation as the pressure equation. And we will use the letter $C$ to
denote a constant that is independent of $k$ that may change from line to
line. And we denote $\vec{b}=(b_{1},...,b_{d})^{T}$ as the negative drift, $f$
as the source term, and $m(x,t)$ as the hard constraint. In addition, because
many computations will involve derivatives of $m$ being divided by $m$, we
will use the notation $\lambda:=\log(m)$ to simplify our expressions. Also we
will also denote for $\tau,t>0$ the sets $Q_{t}:={\mathbb{R}}^{d}\times(0,t)$
and $Q_{\tau,t}:={\mathbb{R}}^{d}\times(\tau,t)$ and we fix a $T>0$. Finally
the operators $\nabla,\nabla\cdot$, and $\Delta$ will only be taken in the
space variables.

Now we list some assumptions that will be used throughout this paper.

######  Assumption 1 (Hard Constraint, Drift, and Source Term Assumptions).

We assume that $m(x,t)$, $\vec{b}(x,t)$, and $f(x,t)$ are in
$C^{\infty}(Q_{T})$ with bounded derivatives of all order such that there is a
$\delta>0$ with $m(x,t)\geq\delta.$

######  Remark 2.2.

If $m$ was allowed to go down to zero, the operator $\nabla\cdot(m\nabla
p_{\infty})$ in $\eqref{Hele-Shaw}$ would become degenerate elliptic on the
zero level set of $m$. But if a Hele-Shaw limit exists when $m$ is allowed to
go down to zero, it seems that the zero level set of $m$ will lead to an
additional obstacle problem for $\varrho_{\infty}$ on
$\partial\\{x:m(x,t)>0\\}$.

######  Assumption 2 (Initial Density and Pressure).

We will always assume that the initial density
$\varrho_{k}^{(0)}(x)=\varrho_{k}(x,0)\in W^{1,1}({\mathbb{R}}^{d})$ is such
that $\varrho_{k}^{0}\geq 0$ almost everywhere. Also we will assume that there
is a compact set $K\subset{\mathbb{R}}^{d}$ such that for all $k>1$ we have

$\text{supp}(\varrho_{k}^{(0)})\subset K$

and

$\sup_{k>1}||\varrho_{k}^{(0)}||_{L^{\infty}({\mathbb{R}}^{d})}\leq C\text{ ,
}\sup_{k>1}||\varrho_{k}^{(0)}||_{W^{1,1}({\mathbb{R}}^{d})}\leq C\text{ , and
}\sup_{k>1}||p_{k}^{(0)}(x)||_{L^{\infty}({\mathbb{R}}^{d})}\leq C.$

Finally we assume there is a limiting density $\varrho_{\infty}^{(0)}(x)$ such
that $\varrho_{k}^{(0)}\rightarrow\varrho_{\infty}^{(0)}$ in
$L^{1}({\mathbb{R}}^{d})$.

######  Remark 2.3.

The purpose of the condition of
$\varrho_{k}^{(0)}\rightarrow\varrho_{\infty}^{(0)}$ in
$L^{1}({\mathbb{R}}^{d})$ in Assumption 2 is to specify the initial data for
the density in (1.4).

######  Assumption 3 (Hard Constraint Restrictions).

We assume that either $m(x,t)=m(|x|,t)$ ($m$ is radial in space), $d=1$ (one
space dimension), or there is an $R>0$ such that for
$\varepsilon=\varepsilon(\delta)>0$ sufficiently small, we have that $|\nabla
m(x,t)|\leq\varepsilon/|x|$ for $|x|\geq R$ .

######  Remark 2.4.

The only purpose of Assumption 3 is to construct a super solution of the
pressure equation $(\ref{PME_pressure})$ to derive uniform $L^{\infty}$ and
compact support bounds on the pressure.

With these assumptions, we have the following compactness theorem:

######  Theorem 2.1.

If Assumptions 1, 2, and 3 are met, then the sequence of pairs
$\\{(\varrho_{k},p_{k})\\}_{k>1}$ has a unique limit in $L^{1}(Q_{T})$ as
$k\rightarrow\infty$. Also the limiting densities and pressures are in the
regularity class $(\varrho_{\infty},p_{\infty})\in
L^{\infty}(0,T;L^{1}({\mathbb{R}}^{d})\cap L^{\infty}({\mathbb{R}}^{d}))\times
L^{2}(0,T;H^{1}({\mathbb{R}}^{d})\cap L^{\infty}({\mathbb{R}}^{d}))$ with
compact support. Also the limiting density and pressure
$(\varrho_{\infty},p_{\infty})$ are a weak solution to the density equation in
$(\ref{Hele-Shaw})$ with initial data $\varrho_{\infty}^{(0)}$.

######  Definition 2.5 (Weak Solutions of Hele-Shaw Density).

We say that the pair $(\varrho_{\infty},p_{\infty})$ is a solution to the
density equation with initial data $\varrho_{\infty}^{(0)}$ for $\eqref{Hele-
Shaw}$ if for all test functions $\varphi\in C^{\infty}_{c}(Q_{T})$

$\int_{{\mathbb{R}}^{d}}\varphi(t=0)\varrho_{\infty}^{(0)}+\iint_{Q_{T}}-(\partial_{t}\varphi)\varrho_{\infty}=\iint_{Q_{T}}-\nabla\varphi\cdot(m\nabla
p_{\infty}+\varrho_{\infty}\vec{b})+\varphi f\varrho_{\infty}$

and $\varrho_{\infty}(x,t)\leq m(x,t)$ with $p_{\infty}(m-\varrho_{\infty})=0$
almost everywhere.

######  Remark 2.6.

For our analysis, we need an extra assumption to be able to show $p_{\infty}$
is a weak solution to the pressure equation in $(\ref{Hele-Shaw})$.

We will refer to showing that $p_{\infty}$ is a weak solution to the pressure
equation in $(\ref{Hele-Shaw})$ as the complimentary relationship.

######  Assumption 4 (Complimentary Relationship Condition).

Assume there is a constant $C>0$ independent of $k$ such that

$\int_{{\mathbb{R}}^{d}}|\nabla p_{k}(x,0)|^{2}dx\leq C.$

As in [22] the complimentary relationship is equivalent to $\nabla
p_{k}\rightarrow\nabla p_{\infty}$ in $L^{2}(Q_{T})$. So this control of the
$||\nabla p_{k}(x,0)||_{L^{2}({\mathbb{R}}^{d})}$ seems rather natural.
However, in [12] the authors obtained the complimentary relationship without
this assumption with no drift and $m$ constant by obtaining an obstacle
problem formulation of the pressure. When drift is added into the system, it
is not clear if the obstacle problem formulation in [12] still holds.

Assumption 4 was also used in [8] to obtain the complimentary relationship
with $m$ constant with source and drift terms. We will modify the proof in [8]
to obtain the optimal $L^{4}(Q_{T})$ bounds on $\nabla p_{k}$ (see [7] for an
example of the optimality of the bounds), which will give us enough
compactness to obtain the complimentary relationship.

######  Theorem 2.2 (Complimentary Relationship).

Assume that Assumptions 1, 2, 3, and 4 are met, then for any $\varphi\in
C^{\infty}_{c}(Q_{T})$ we have that

$\iint_{Q_{T}}-mp_{\infty}[\nabla\varphi\cdot\nabla
p_{\infty}]-m\varphi|\nabla p_{\infty}|^{2}+\varphi p_{\infty}mF=0.$

That is the limiting pressure satisfies the complimentary relationship

$p_{\infty}\left[\nabla\cdot(m\nabla p_{\infty})+mF\right]=0\text{ in }Q_{T}$

in the sense of distributions.

Then with the existence and uniqueness of the generalized Hele-Shaw problem,
we focus on geometric properties of $\Sigma(t):=\\{x:p_{\infty}(x,t)>0\\}$.

######  Theorem 2.3.

The normal velocity $V$ of the free-boundary $\partial\Sigma(t)$ satisfies

$V=\left[-\frac{m\nabla
p_{\infty}}{(m-\varrho^{E})}-\vec{b}\right]\cdot\vec{\nu}$

in the sense of comparison with classical barriers where $\varrho^{E}$ is the
external density. The external density is the limit of the density
$\varrho_{\infty}$ from outside the saturated region $\Sigma(t)$. See Section
6.1 for a definition of classical barriers.

Then we focus on the case of congestion (1.6). This constraint is a
generalization of the one used in [15] used for $m\equiv 1$. And in this case,
we can identify the limiting pressure up to $\partial\Sigma(t)$ when we have
control of the initial limiting pressure:

######  Theorem 2.4 (Explicit Identification of $\varrho_{\infty}$ in The
Congestion Case).

Assume that Assumptions 1, 2, and 3, are met such that $p_{k}(x,0)\rightarrow
p_{\infty}(x,0)$ uniformly such that $\Sigma(0)$ is open and we are in the
congestion case $(\ref{congestion})$. Then we have the pointwise formula

$\varrho_{\infty}(x,t)=\begin{cases}\varrho^{E}(x,t)\text{ for a.e.
}(x,t)\in\text{Int}(\\{p(x,t)=0\\})\\\ m(x,t)\text{ for a.e.
}(x,t)\in\\{p(x,t)>0\\}\end{cases},$

where the external density $\varrho^{E}(x,t)$ solves the continuity equation

$\begin{cases}\partial_{t}\varrho^{E}=\nabla\cdot(\varrho^{E}\vec{b})+f\varrho^{E}\text{
in }Q_{T}\\\ \varrho^{E}(x,0)=\varrho_{\infty}(x,0)\end{cases}.$

######  Remark 2.7.

Note that this identification Theorem 2.4 is consistent with the one in [15],
which dealt with the case of $m\equiv 1$ using viscosity solutions.

We were able to obtain this identification by refining the new Aronson-Bénilan
estimate for the PME with drift found in [17]. We showed that under the
congestion case for all $k>1$ there exists a $C>0$ independent of $k$ such
that

$\frac{1}{m}\nabla\cdot(m\nabla p_{k})\geq-F-\frac{C}{k-1}-\frac{1}{(k-1)t}.$

Now our Aronson-Bénilan estimate implies that we have the following uniform
estimate of the pressure $p_{k}$ along the streamlines $X(t,x_{0})$ for all
$k>1$ for some $C>0$ independent of $k$

$\frac{d}{dt}p_{k}(X(t,x_{0}),t)\geq-(C+\frac{1}{t})p_{k}(X(t,x_{0}),t).$

In particular, we deduce that $\Sigma(t)$ is non-decreasing with respect to
streamlines for positive time. That is if $x\in\Sigma(t)$ for $t>0$ then the
streamline containing $x$ is in $\Sigma(s)$ for all $T\geq s\geq t$. Then by
using the uniform convergence of the initial limiting pressure and $\Sigma(0)$
is open, we can use the barrier constructed in Lemma 8.4 for the congestion
case to conclude that $\Sigma(t)$ is non-decreasing with respect to the
streamline for all $0\leq t\leq T$. This is enough for us to conclude the
identification of $\varrho_{\infty}$.

###  2.1. Open Problems

Let us denote $(\varrho_{\infty},p_{\infty})$ as the limiting density and
pressure respectively.

  * $\circ$

$\mathit{Obstacle}$ $\mathit{Problem}$ $\mathit{Formulation}$ $\mathit{of}$
$\mathit{the}$ $\mathit{Pressure}$ In [12] the authors derived an obstacle
problem formulation of $p_{\infty}$ when $\vec{b}\equiv 0$ and $m\equiv 1$.
However, when there is a non-zero drift $\vec{b}$, the proof used in [12] does
not carry over.

  * $\circ$

$\mathit{When}$ $\mathit{m}$ $\mathit{is}$ $\mathit{allowed}$ $\mathit{to}$
$\mathit{go}$ $\mathit{to}$ $\mathit{zero}$ When $m$ is allowed to reach zero,
the limiting PDE $(\ref{Hele-Shaw})$ seems to be modified into a new obstacle
problem that includes the zero level set of $m$.

###  2.2. Acknowledgements

The author was partially supported by NSF grant DMS-1900804 and the Summer
Mentored Research Fellowship at the University of California, Los Angeles
during this project. The author is also very grateful to his advisor Inwon Kim
for suggesting this project, for her guidance and mentorship throughout this
project, and for reading the manuscript. The author is also very grateful to
the two anonymous referees for their very detailed comments.

##  3\. Existence of the Hele-Shaw Limit

###  3.1. $L^{\infty}(Q_{T})$ Bounds and Finite Speed of Propagation

We will derive enough estimates to be able to pass the limit in
$(\ref{PME_Density})$ and $\eqref{PME_pressure}$ to $(\ref{Hele-Shaw})$ as
$k\rightarrow\infty$ in the weak sense. Our first estimate will be used to
obtain $L^{\infty}(Q_{T})$ and control the support of the pressure $p_{k}$.

First notice as $p_{k}(x,0)\geq 0$ and $0$ is a solution to the pressure
equation, we may apply the comparison principle to deduce that $p_{k}(x,t)\geq
0$ almost everywhere. Now we will show the following bounds:

######  Lemma 3.1 ($L^{\infty}(Q_{T})$ Bounds and Finite Speed of
Propagation).

If Assumptions 1, 2, and 3 are met then there is a constant $C=C(T)$ and a
compact set $\Omega\subset\mathbb{R}^{d}$ independent of $k>1$ such that

$0\leq p_{k}\leq C\text{ a.e. in
}Q_{T},\quad\text{supp}(p_{k}(t))\subset\Omega\text{ for a.e. }0\leq t\leq T.$

###### Proof.

To prove this lemma, we use the viscosity super solution constructed in Lemma
8.2 and 8.3. We choose the initial conditions of this barrier so that the
barrier at time $0$ is larger than
$\sup_{k>1}||p_{k}^{(0)}||_{L^{\infty}({\mathbb{R}}^{d})}$ inside $K$. So by
the comparison principle, $p_{k}$ is dominated by the barrier. Then as the
barrier is compactly supported and bounded for all times, we deduce the same
is true for $p_{k}$. ∎

This immediately implies compact support and $L^{\infty}(Q_{T})$ bounds on the
density and normalized density.

######  Corollary 3.2.

The density $\varrho_{k}$ and normalized density $v_{k}$ are also uniformly
bounded in $L^{\infty}(Q_{T})$, uniformly compactly supported in $Q_{T}$, and
non-negative almost everywhere.

###  3.2. BV Estimates on the Pressure and Density

Now we will derive $L^{2}(Q_{T})$ estimates on $|\nabla p_{k}|$, which follows
from just integrating (2.2). Then we obtain $L^{1}({\mathbb{R}}^{d})$
estimates for $|\nabla v_{k}(t)|$ for a.e. $t\in[0,T]$ via an integral
Grönwall’s inequality with Kato’s inequality argument.

######  Lemma 3.3 ($L^{2}(Q_{T})$ bounds for $|\nabla p_{k}|$).

If Assumptions 1, 2, and 3 are met then for all $k\geq 6$ there is a constant
$C=C(T)$ such that

$\iint_{Q_{T}}|\nabla p_{k}|^{2}\leq C(T).$

###### Proof.

First we use $\frac{1}{m}\nabla\cdot(m\nabla p_{k})=\Delta
p_{k}+\nabla\lambda\cdot\nabla p_{k}$ and integrate the pressure equation over
$Q_{T}$ to see that

$(k-2)\iint_{Q_{T}}|\nabla
p_{k}|^{2}=\int_{{\mathbb{R}}^{d}}p_{k}(x,0)-p_{k}(x,T)+\iint_{Q_{T}}\nabla
p_{k}\cdot(\vec{b}+(k-1)p_{k}\nabla\lambda)+(k-1)p_{k}F.$

Now we recall that Young’s Inequality with $\varepsilon>0$ says that
$|\vec{v}_{1}\cdot\vec{v}_{2}|\leq\varepsilon|\vec{v}_{1}|^{2}+|\vec{v}_{2}|^{2}/(4\varepsilon)$.
Using Young’s Inequality with $\varepsilon$ combined with our uniform sup
bounds and uniform support control on $p_{k}$ (Lemma 3.1) gives us that there
is $C=C(T)$ such that

$(k-2)\iint_{Q_{T}}|\nabla p_{k}|^{2}\leq
kC(T)+\frac{k+1}{2}\iint_{Q_{T}}|\nabla p_{k}|^{2}.$

Now because $k\geq 6$, we have that $(k-2)-\frac{k+1}{2}\geq 1/2$ and
$\sup_{k\geq 6}$ $((k-2)-\frac{k+1}{2})/k\leq\tilde{C}$, which implies the
desired bound.

∎

Note that this $L^{2}(Q_{T})$ estimate combined with Hölder’s inequality and
our uniform compact support of $p_{k}$ (see Lemma 3.1) gives us $L^{1}(Q_{T})$
control of $\nabla p_{k}$. Thus to obtain $L^{1}(Q_{T})$ convergence of the
pressure along a sub-sequence as $k\rightarrow\infty$, it remains to show time
derivative bounds on the pressure. Uniform time derivative bounds on $p_{k}$
is one of the most delicate parts of our analysis. This is unsurprising
because in [22] the authors constructed examples of $p_{\infty}$ that have
strong jump discontinuities in time.

Our first step in obtaining time derivative control of $p_{k}$ is to obtain BV
bounds for the normalized density $v_{k}$. To derive our BV bounds, we use the
following lemma that shows $|\nabla v_{k}^{k}|\approx|\nabla p_{k}|$.

######  Lemma 3.4.

If Assumptions 1, 2, and 3 are met with $k\geq 6$ then there is a constant
$C=C(T)$ such that for the normalized density $v_{k}$ we have that

$\iint_{Q_{T}}|\nabla v_{k}^{k}|^{2}\leq C(T).$

###### Proof.

Note that from Corollary 3.2 we have that $0\leq v_{k}\leq C$ almost
everywhere, so

$|\nabla v_{k}^{k}|\leq Ckv_{k}^{k-2}|\nabla v_{k}|=C|\nabla p_{k}|.$

Hence, by integrating this inequality and using Lemma 3.3, we conclude the
desired inequality. ∎

Now we have enough estimates to obtain $L^{1}({\mathbb{R}}^{d})$ bounds on
$|\nabla v_{k}(t)|$ for a.e. $t\in[0,T]$.

######  Lemma 3.5 (BV bounds for the Normalized Density).

Assume that Assumptions 1, 2, and 3 are met with $k\geq 6$. Then there is a
constant $C=C(T)$ such that for almost every $t\in[0,T]$

$\int_{{\mathbb{R}}^{d}}|\nabla v_{k}(t)|\leq C(T).$

###### Proof.

We differentiate (2.3) in $x_{i}$ using that $\frac{1}{m}\nabla\cdot(m\nabla
v_{k}^{k})=\Delta(v_{k}^{k})+\nabla\lambda\cdot\nabla v_{k}^{k}$ to obtain for
$\omega_{i}:=\partial_{x_{i}}v_{k}$ that

(3.1)
$\partial_{t}\omega_{i}=\Delta(kv_{k}^{k-1}\omega_{i})+\nabla(\partial_{x_{i}}\lambda)\cdot\nabla
v_{k}^{k}+\nabla\lambda\cdot\nabla(kv_{k}^{k-1}\omega_{i})+\nabla(\omega_{i})\cdot\vec{b}+\nabla
v_{k}\cdot(\partial_{x_{i}}\vec{b})+v_{k}[\partial_{x_{i}}F]+\omega_{i}F.$

Note that our assumptions tells us that $F$ is a smooth bounded function with
bounded derivatives. To deal with the term
$\nabla\lambda\cdot\nabla(kv_{k}^{k-1}\omega_{i})$, we observe that

$\nabla\lambda\cdot\nabla(kv_{k}^{k-1}\omega_{i})=\nabla\cdot(kv_{k}^{k-1}\omega_{i}\nabla\lambda)-kv_{k}^{k-1}\omega_{i}\Delta\lambda=\nabla\cdot(kv_{k}^{k-1}\omega_{i}\nabla\lambda)-(\partial_{x_{i}}v_{k}^{k})\Delta\lambda.$

Now we multiply (3.1) by $\text{sign}(\omega_{i})$ and use Kato’s Inequality,
Cauchy-Schwarz, and $v_{k}\leq C$ (see Corollary 3.2) to see that

$\partial_{t}|\omega_{i}|\leq\Delta(kv_{k}^{k-1}|\omega_{i}|)+C|\nabla
v_{k}^{k}|+\nabla\cdot(kv_{k}^{k-1}|\omega_{i}|\nabla\lambda)+C|\partial_{x_{i}}v_{k}^{k}|+\nabla|\omega_{i}|\cdot\vec{b}+C|\nabla
v_{k}|+C|\omega_{i}|+C.$

Hence, integrating this using that $\omega_{i}$ is compactly supported and
that $|\nabla v_{k}^{k}|\in L^{1}(Q_{t})$ uniformly in $k$ (see Lemma 3.4)
gives

$\frac{d}{dt}\iint_{Q_{t}}|\omega_{i}|\leq
C+\int_{{\mathbb{R}}^{d}}|\omega_{i}|(t=0)+\iint_{Q_{t}}-|\omega_{i}|(\nabla\cdot\vec{b})+C|\omega_{i}|+C\left(\sum_{j=1}^{d}|\omega_{j}|^{2}\right)^{1/2}.$

Summing this over all $1\leq i\leq d$ with
$\omega:=\sum_{j=1}^{d}|\omega_{j}|$ along with our initial BV bounds
(Assumption 2) gives

$\frac{d}{dt}\iint_{Q_{t}}\omega\leq C+C\iint_{Q_{t}}\omega.$

Now Grönwall’s Inequality implies with our initial BV bounds (Assumption 2)
the desired bounds. ∎

###  3.3. Generalized Semi-Convexity Estimate

Now we aim to control the time derivatives of the normalized density and
pressure. To do this, we need a semi-convexity estimate in a similar vein to
the Aronson-Bénilan estimate. The famous Aronson-Bénilan (see [24]) estimate
says that for the case of $m(x,t)\equiv 1,f=0$, and $\vec{b}=\vec{0}$ one has
the following inequality

$\Delta p_{k}\geq-\frac{1}{(k-1)t}.$

We will derive for our case the following similar estimate

$w_{k}=\frac{1}{m}\nabla\cdot(m\nabla p_{k})\geq-\frac{2}{(k-1)t}-K_{1},$

where $K_{1}>0$ is a constant independent of $k$. This generalized semi-
convexity estimate will be used to show for any $\tau>0$ and a.e.
$t\in[\tau,T]$ that $||\partial_{t}v_{k}(t)||_{L^{1}({\mathbb{R}}^{d})}\leq
C(\tau,T)$. For this section, we fix a $k\geq 6$ and will drop the subscript
$k$ on the $p_{k}$ and $w_{k}$.

The main idea of the generalized semi-convexity estimate is to derive that $w$
solves a parabolic PDE and modify the barrier introduced in [17]. In Section
7, we modify the barrier used in this section under the congestion assumption
to derive a refined Aronson-Bénilan estimate.

We begin deriving the PDE that $w$ solves. Observe that

(3.2) $w_{t}=\nabla\lambda_{t}\cdot\nabla p+\nabla\lambda\cdot\nabla
p_{t}+\Delta p_{t}.$

Now we will derive expressions for $\nabla p_{t}$ and $\Delta p_{t}$. By
differentiating (2.2) in $x_{i}$ we see that

(3.3)
$\partial_{x_{i}}p_{t}=\sum_{j=1}^{d}\left[2(\partial_{x_{j}}p)(\partial^{2}_{x_{j}x_{i}}p)+(\partial^{2}_{x_{i}x_{j}}p)b_{j}+(\partial_{x_{j}}p)(\partial_{x_{i}}b_{j})\right]+(k-1)\left[(\partial_{x_{i}}p)(w+F)+p(\partial_{x_{i}}w+\partial_{x_{i}}F)\right].$

Differentiating in $x_{i}$ again and summing gives

(3.4) $\Delta p_{t}=\sum_{j=1}^{d}2(\partial^{2}_{x_{i}x_{j}}p)^{2}+2\nabla
p\cdot\nabla(\Delta p)+\nabla(\Delta p)\cdot\vec{b}+\nabla
p\cdot(\Delta\vec{b})+2\sum_{i,j=1}^{d}(\partial^{2}_{x_{i}x_{j}}p)(\partial_{x_{i}}b_{j})+$
(3.5) $+(k-1)\left\\{\Delta p(w+F)+2\nabla p\cdot(\nabla w+\nabla F)+p(\Delta
w+\Delta F)\right\\}.$

Using (3.3), (3.4), and (3.5) in (3.2), we obtain the following PDE for $w$

(3.6) $w_{t}=\nabla\lambda_{t}\cdot\nabla p+(k-1)\left[w(w+F)+(\nabla w+\nabla
F)\cdot(p\nabla\lambda+2\nabla p)+p(\Delta w+\Delta F)\right]+$ (3.7)
$+\sum_{i,j=1}^{d}2(\partial^{2}_{x_{i}x_{j}}p)^{2}+2(\nabla w\cdot\nabla
p-(\nabla p)^{T}D^{2}\lambda(\nabla p))+\nabla(w-\nabla
p\cdot\nabla\lambda)\cdot\vec{b}+\nabla p\cdot(\Delta\vec{b})+$ (3.8)
$+\sum_{i,j=1}^{d}(\partial^{2}_{x_{i}x_{j}}p)(b_{j})(\partial_{x_{i}}\lambda)+\sum_{i,j=1}^{d}(\partial_{x_{j}}p)(\partial_{x_{i}}b_{j})(\partial_{x_{i}}\lambda)+2\sum_{i,j=1}^{d}(\partial^{2}_{x_{i}x_{j}}p)(\partial_{x_{i}}b_{j}).$

To derive this equation we used the identity

$\sum_{i,j=1}^{d}2(\partial_{x_{j}}p)(\partial^{2}_{x_{i}x_{j}}p)(\partial_{x_{i}}\lambda)+2\nabla
p\cdot\nabla(\Delta p)=2\\{\nabla w\cdot\nabla p-(\nabla
p)^{T}D^{2}\lambda(\nabla p)\\},$

which follows from expanding the right hand side.

Now we will use (3.6),(3.7), and (3.8) to derive that $w$ is a super solution
to a simpler parabolic differential equation. Indeed, by Cauchy-Schwarz and
Young’s Inequality with $\varepsilon$ we have that

(3.9) $\nabla\lambda_{t}\cdot\nabla
p\geq-\frac{1}{2\varepsilon}||\nabla\lambda_{t}||^{2}_{\infty}-\frac{\varepsilon}{2}|\nabla
p|^{2}.$

In addition, we have the following bound

(3.10) $\nabla(\nabla
p\cdot\nabla\lambda)\cdot\vec{b}=\sum_{i,j=1}^{d}(\partial^{2}_{x_{i}x_{j}}p)(\partial_{x_{i}}\lambda)b_{j}+(\partial^{2}_{x_{i}x_{j}}\lambda)(\partial_{x_{i}}p)b_{j}$
(3.11)
$\geq\sum_{i,j=1}^{d}-\frac{\varepsilon}{2}|\partial^{2}_{x_{i}x_{j}}p|^{2}-\frac{1}{2\varepsilon}\left(||\partial_{x_{i}}\lambda||^{2}_{\infty}||b_{j}||^{2}_{\infty}\right)-\frac{\varepsilon}{2}|\nabla
p|^{2}-\frac{1}{2\varepsilon}\sum_{i,j=1}^{d}||\partial^{2}_{x_{i}x_{j}}\lambda||^{2}_{\infty}||b_{j}||^{2}_{\infty}.$

Also we observe that

(3.12) $\nabla p\cdot(\Delta\vec{b})\geq-\frac{\varepsilon}{2}|\nabla
p|^{2}-\frac{1}{2\varepsilon}||\Delta\vec{b}||^{2}_{\infty}.$

Then by using Young’s Inequality with $\varepsilon$, we see that

(3.13)
$\sum_{i,j=1}^{d}(\partial_{x_{j}}p)(\partial_{x_{i}}b_{j})(\partial_{x_{i}}\lambda)+2\sum_{i,j=1}^{d}(\partial^{2}_{x_{i}x_{j}}p)(\partial_{x_{i}}b_{j})\geq-C(\varepsilon)-\varepsilon|\nabla
p|^{2}-\varepsilon\sum_{i,j=1}^{d}|\partial^{2}_{x_{i}x_{j}}p|^{2}.$

Now by using that $p\geq-C$ (see Lemma 3.1)

(3.14) $w(w+F)\geq\frac{w^{2}}{2}-C,\quad\nabla F\cdot(p\nabla\lambda+2\nabla
p)\geq-C-\frac{|\nabla p|^{2}}{2},\text{ and }p\Delta F\geq-C.$

Combining these inequalities along with making $\varepsilon$ sufficiently
small (independent of $k$) gives

$\partial_{t}w\geq-|\nabla p|^{2}+\nabla
w\cdot\vec{b}+(k-1)\left\\{\frac{w^{2}}{2}+p\nabla w\cdot\nabla\lambda+2\nabla
w\cdot\nabla p+p\Delta w-\frac{|\nabla
p|^{2}}{2}-C\right\\}+\sum_{i,j=1}^{d}(\partial^{2}_{x_{i}x_{j}}p)^{2}+$
$+2(\nabla w\cdot\nabla p-(\nabla p)^{T}D^{2}\lambda(\nabla p)).$

Now we treat $p$ as a known function and define the parabolic differential
operator

(3.15) $L[\phi]:=\partial_{t}\phi-\nabla w\cdot\vec{b}+|\nabla
p|^{2}-2(\nabla\phi\cdot\nabla p-(\nabla p)^{T}D^{2}\lambda(\nabla p))+$
(3.16)
$-(k-1)\left\\{\frac{\phi^{2}}{2}+\nabla\phi\cdot(p\nabla\lambda+2\nabla
p)+p\Delta\phi-\frac{|\nabla p|^{2}}{2}-C\right\\},$

then we have by construction that $L[w]\geq 0$. Now we construct a sub-
solution of this operator using the barrier introduced in [17].

######  Lemma 3.6 (Subsolution for semi-convexity estimate).

We denote $p=p_{k}$ and $w=w_{k}$ for ease of notation. Then there exists a
sufficiently large $K_{1}>0$ such that for any $\tau>0$

$\psi(x,t;\tau):=-\frac{2}{(k-1)(t+\tau)}+p-K_{1}$

satisfies $L[\psi]\leq 0$ whenever $k>1$ is sufficiently large where $L$ is
the parabolic differential operator in (3.15) and (3.16).

######  Remark 3.1.

This barrier $\psi(x,t;\tau)$ is inspired by the one in [17].

###### Proof.

We plug in $\psi$ to $L$ to see that

$L[\psi]=\frac{2}{(k-1)(t+\tau)^{2}}+2(\nabla p)^{T}D^{2}\lambda(\nabla
p)+(k-1)(pF-\frac{\psi^{2}}{2}-\frac{3}{2}|\nabla p|^{2}+C).$

Now we use that

$\psi^{2}=\frac{4}{(k-1)^{2}(t+\tau)^{2}}-2(p-K_{1})\left(\frac{2}{(k-1)(t+\tau)}\right)+(p-K_{1})^{2}$
$\geq\frac{4}{(k-1)^{2}(t+\tau)^{2}}+(p-K_{1})^{2},$

where we chose $K_{1}>\sup_{k>1}||p_{k}||_{L^{\infty}(Q_{T})}$ (which is
finite thanks to Lemma 3.1), so that $p-K_{1}\leq 0$. Now from $k\geq 2$, we
have that
$(k-1)\frac{-2}{(k-1)^{2}(t+\tau)^{2}}\leq\frac{-2}{(k-1)^{2}(t+\tau)^{2}}$,
which implies

$L[\psi]\leq\underbrace{\frac{1}{(k-1)(t+\tau)^{2}}\left[2-2\right]}_{0}+\underbrace{2(\nabla
p)^{T}D^{2}\lambda(\nabla p)+(k-1)\left[-\frac{3}{2}|\nabla
p|^{2}\right]}_{\mathcal{I}_{1}}+$
$+\underbrace{(k-1)(-\frac{(p-K_{1})^{2}}{2}+C)}_{\mathcal{I}_{2}}.$

Note that we used the $L^{\infty}(Q_{T})$ pressure bounds (Lemma 3.1) to have
$pF\leq C$ for some $C>0$. Now for all $(x,t)\in Q_{T}$, let
$\lambda_{1}(x,t)\geq\lambda_{2}(x,t)\geq...\geq\lambda_{n}(x,t)$ be the
eigenvalues of $D^{2}\lambda(x,t)$. Then let $\Lambda:=\sup_{(x,t)\in
Q_{T}}\\{\lambda_{1}(x,t)\\}$ be the biggest eigenvalue of $D^{2}\lambda$ over
$Q_{T}$. This implies

$\mathcal{I}_{1}\leq 2\Lambda|\nabla p|^{2}-(k-1)\frac{3}{2}|\nabla p|^{2}\leq
0\text{ for }k\geq\frac{4}{3}\Lambda+1.$

Then for $\mathcal{I}_{2}$, by making $K_{1}$ larger if necessary, we can also
assume that $K_{1}\geq\sup_{k>1}||p_{k}||_{L^{\infty}(Q_{T})}+\sqrt{2C}$ ,
which implies $\mathcal{I}_{2}\leq 0$. So we conclude that $L[\psi]\leq 0$. ∎

Now we reintroduce the subscript $k$.

######  Lemma 3.7 (Semi-Convexity Estimate).

There exists a $K_{1}$ independent of $k$ such that we have

$w_{k}:=\frac{1}{m}\nabla\cdot(m\nabla p_{k})\geq-\frac{2}{(k-1)t}-K_{1}$

for sufficiently large $k$ in the distribution sense.

###### Proof.

Using the $\psi(x,t;\tau)$ from Lemma 3.6, we see that

$\lim_{\tau\downarrow 0}\psi(x,0;\tau)=-\infty.$

This implies for small enough $\tau=\tau(k)>0$ we can compare
$\psi(\cdot,\cdot,\tau)$ to $w_{k}$ (because $L[w_{k}]\geq 0$ and
$L[\psi(\cdot,\cdot;\tau)]\leq 0$) to obtain the claim. ∎

This gives us enough control to derive an $L^{1}(Q_{\tau,T})$ estimate on
$w_{k}$.

######  Lemma 3.8 ($L^{1}(Q_{\tau,T})$ bound on the semi-convexity term).

Assume that Assumptions 1, 2, and 3 are met then if $\tau>0$, there is a
constant $C=C(\tau,T)$ such that for all sufficiently large $k$

$\iint_{Q_{\tau,T}}|w_{k}|\leq C(\tau,T).$

###### Proof.

Recall that $|w_{k}|=w_{k}+2|w_{k}|_{-}$ where $|x|_{-}:=-\min(x,0)$ for
$x\in{\mathbb{R}}$. Then our semi-convexity estimate Lemma 3.7 implies that
$|w_{k}|_{-}\leq\frac{2}{(k-1)\tau}+K_{1}$ in $Q_{\tau,T}$. Now using that
$w_{k}$ is compactly supported thanks to Lemma 3.1, we conclude the claim by
integrating. ∎

###  3.4. Time Derivative Estimates

Now equipped with our semi-convexity estimate we can derive
$L^{1}({\mathbb{R}}^{d})$ estimates on $|\partial_{t}v_{k}(t)|$ for a.e.
$t\in[\tau,T]$ and $L^{1}(Q_{\tau,T})$ estimates on $|\partial_{t}p_{k}|$.
Note that we cannot expect to go down to $t=0$ without stronger conditions on
our initial data.

To begin we first observe the semi-convexity estimate Lemma 3.7 that

$\partial_{t}p_{k}\geq\vec{b}\cdot\nabla
p_{k}+(k-1)p_{k}\left[-\frac{2}{(k-1)t}-K_{1}\right].$

Then by using $p_{k}=\frac{k}{k-1}(v_{k}^{k-1})$ we have that

$\partial_{t}v_{k}\geq\vec{b}\cdot\nabla
v_{k}-v_{k}\left[K_{1}+\frac{2}{(k-1)t}\right].$

Cauchy-Schwarz implies that

$\partial_{t}v_{k}\geq-C|\nabla
v_{k}|-v_{k}\left[K_{1}+\frac{2}{(k-1)t}\right].$

This estimate implies the following bound on the negative part of
$\partial_{t}v_{k}$,

$|\partial_{t}v_{k}|_{-}\leq C|\nabla
v_{k}|+|v_{k}|\left[K_{1}+\frac{2}{(k-1)t}\right].$

Thus our BV bounds on $v_{k}$ (Lemma 3.5) and $L^{\infty}(Q_{T})$ bounds on
$v_{k}$ (Corollary 3.2), implies for a.e. $t\in[\tau,T]$ that

$\int_{{\mathbb{R}}^{d}}|\partial_{t}v_{k}|_{-}(t)dx\leq C(\tau,T).$

Now we integrate in space (2.3) to see that

$\left|\int_{{\mathbb{R}}^{d}}\partial_{t}v_{k}\right|\leq C.$

So using that
$|\partial_{t}v_{k}|=\partial_{t}v_{k}+2|\partial_{t}v_{k}|_{-}$, we see for
a.e. $t\in[\tau,T]$ that

$\int_{{\mathbb{R}}^{d}}|\partial_{t}v_{k}|(t)\leq C(\tau,T).$

That is we have the following lemma:

######  Lemma 3.9 (Time Derivative Control of The Normalized Density).

Assume that Assumptions 1, 2, and 3 are met, then for any $\tau>0$ we have the
bound for almost every $t\in[\tau,T]$

$\int_{{\mathbb{R}}^{d}}|\partial_{t}v_{k}|(t)\leq C(\tau,T)$

for sufficiently large $k$.

Now we will obtain $L^{1}(Q_{\tau,T})$ estimates of $\partial_{t}p_{k}$. But
we first need the following lemma based on an estimate in [12].

######  Lemma 3.10.

Assume that Assumptions 1, 2, and 3 are met and let $\tau>0$. Then if we write
$q:=\partial_{t}v$ (we drop the subscript $k$ in this lemma). Then we have for
sufficiently large $k$ and almost every $0<\tau<T$ that

$\iint_{Q_{\tau,T}}kv^{k-1}|q|\leq C(\tau,T).$

###### Proof.

Observe by differentiating the normalized density equation $(\ref{normalized-
density})$ in time that we have

$\partial_{t}q=\Delta(kv^{k-1}q)+\nabla\lambda_{t}\cdot\nabla
v^{k}+\nabla\lambda\cdot\nabla(kv^{k-1}q)+\nabla q\cdot\vec{b}+\nabla
v\cdot(\partial_{t}\vec{b})+qF+v(\partial_{t}F).$

Multiplying by $\text{sign}(q)$ and using Kato’s inequality implies that

$\partial_{t}|q|\leq\Delta(kv^{k-1}|q|)+|\nabla\lambda_{t}\cdot\nabla
v^{k}|+\nabla\lambda\cdot\nabla(kv^{k-1}|q|)+\nabla\cdot(|q|\vec{b})-|q|\nabla\cdot\vec{b}$
$+|\nabla v\cdot(\partial_{t}\vec{b})|+C|q|+C|v|.$

Now we observe that
$\Delta(kv^{k-1}|q|)+\nabla\lambda\cdot\nabla(kv^{k-1}|q|)=\nabla\cdot(m\nabla(kv^{k-1}|q|))/m$.
Hence, we have from Young’s Inequality and Cauchy-Schwarz for any smooth
$\phi=\phi(x)\geq 0$ that is compactly supported

$\phi(\partial_{t}|q|)\leq\frac{\phi}{m}\nabla\cdot(m\nabla(kv^{k-1}|q|))+\phi(\frac{|\nabla
v^{k}|^{2}}{2}+C)+\phi\left[\nabla\cdot(|q|\vec{b})-|q|\nabla\cdot\vec{b}\right]+C\phi|\nabla
v|+C\phi|q|+C\phi|v|.$

So by integration by parts and using our $L^{2}(Q_{T})$ bounds on $|\nabla
v^{k}|$ (Lemma 3.4), BV bounds on $|\nabla v|$ (Lemma 3.5),
$L^{\infty}(Q_{T})$ bounds on $v$ (Corollary 3.2), and $L^{1}(Q_{\tau,T})$
bounds on $q$ (Lemma 3.9) we have that

$\iint_{Q_{\tau,T}}\phi(\partial_{t}|q|)\leq\iint_{Q_{\tau,T}}(kv^{k-1}|q|)[\nabla\cdot(m\nabla(\phi/m))]+C(\phi,\tau,T).$

We now let $R>0$ be so large such that

$\text{supp}(v_{k})\subset B_{R}(0)\text{ for all }k>1\text{ for }t\in[0,T].$

Then we define $\phi=m\varphi$ where $\varphi$ is the unique solution of the
following uniformly elliptic PDE

$\begin{cases}\nabla\cdot(m\nabla\varphi)=-1\text{ in }B_{2R}(0)\\\
\phi=0\text{ on }\partial B_{2R}(0)\end{cases}$

and defined to be 0 outside of this ball. Hence, we deduce that

$\iint_{Q_{\tau,T}}(kv^{k-1}|q|)\leq\int_{{\mathbb{R}}^{d}}\phi(x)\left[|q|(x,\tau)-|q|(x,T)\right]dx+C(\tau,T)\leq
C(\tau,T),$

where we used Lemma 3.9 in the final inequality. ∎

Now this gives enough estimates to prove our $L^{1}(Q_{\tau,T})$ bounds on
$\partial_{t}p$. Indeed we proceed as in [22] and use that
$|\partial_{t}p|=kv^{k-2}|q|$ to see

$\iint_{Q_{\tau,T}}|\partial_{t}p|\leq\iint_{\\{v<1/2\\}\cap
Q_{\tau,T}}kv^{k-2}|q|+2\iint_{\\{v\geq 1/2\\}\cap Q_{\tau,T}}kv^{k-1}|q|$
$\leq\iint_{Q_{\tau,T}}k\frac{1}{2^{k-2}}|q|+2kv^{k-1}|q|\leq C(\tau,T),$

where in the final bound we used Lemma 3.9 and 3.10. That is we deduced

######  Lemma 3.11.

Assume that Assumptions 1, 2, and 3 are met, then for almost every $\tau>0$ we
have the bound for sufficiently large $k>1$

$\iint_{Q_{\tau,T}}|\partial_{t}p_{k}|\leq C(\tau,T).$

###  3.5. Passage To The Limit For The Density

Our bounds imply $v_{k}$ and $p_{k}$ is a bounded family in
$W^{1,1}(Q_{\tau,T})$ for almost every $0<\tau<T$ (so the set of such $\tau$
is dense). Hence, along a subsequence (which we still denote by $k$), we can
find a $v_{\infty},p_{\infty}\in L^{1}(Q_{\tau,T})$ such that

$v_{k}\rightarrow v_{\infty}\text{ in }L^{1}(Q_{\tau,T})\text{ and
}p_{k}\rightarrow p_{\infty}\text{ in }L^{1}(Q_{\tau,T}).$

Then by using a standard diagonal argument combined with the
$L^{\infty}(Q_{T})$ and support control of the pressure and density Lemma 3.1,
we see that we can extract a sub-sequence (which we denote by $k)$ such that
$(v_{k},p_{k})$ has a limit in $L^{1}(Q_{T})\times L^{1}(Q_{T})$ . We denote
the limit (along this sub-sequence) as $v_{\infty}$ and $p_{\infty}$. Now if
we denote by $\varrho_{\infty}:=mv_{\infty}$, then it implies
$\varrho_{k}\rightarrow\varrho_{\infty}$ in $L^{1}(Q_{T})$ since
$m\geq\delta>0$. Next observe from $v_{k}p_{k}=C_{k}p_{k}^{\frac{k}{k-1}}$
where $C_{k}\rightarrow 1$ as $k\rightarrow\infty$ that from the pointwise
convergence a.e. (we can use a further sub-sequence if needed) that

$p_{\infty}(v_{\infty}-1)=0\text{ a.e. }\Rightarrow
p_{\infty}(m-u_{\infty})=0\text{ a.e. }.$

Finally by considering the weak form of the density equation in
$(\ref{PME_Density})$ and letting $k\rightarrow\infty$ with our estimates,
shows that for any test function $\varphi$ we have

$\begin{cases}\iint_{Q_{T}}\varphi_{t}\varrho_{\infty}-\nabla\varphi\cdot(m\nabla
p_{\infty}+\varrho_{\infty}\vec{b})+\varphi
f\varrho_{\infty}+\int_{{\mathbb{R}}^{d}}\varphi(t=0)\varrho_{\infty}^{0}=0\\\
p_{\infty}(m-\varrho_{\infty})=0\text{ a.e. in }Q_{T}\end{cases}.$

That is $(\varrho_{\infty},p_{\infty})$ is a weak solution to the density
equation in $(\ref{Hele-Shaw})$ with initial data $\varrho_{\infty}^{0}$. This
gives us the existence portion of Theorem 2.1.

##  4\. Uniqueness and Comparison Principle of the Generalized Hele-Shaw Flow

In this section, we consider weak solutions $(\varrho,p)$ to the density
equation $(\ref{Hele-Shaw})$ such that $\varrho\in
L^{\infty}([0,T];L^{1}({\mathbb{R}}^{d})\cap L^{\infty}({\mathbb{R}}^{d}))$
and $p\in L^{2}([0,T];H^{1}({\mathbb{R}}^{d})\cap
L^{\infty}({\mathbb{R}}^{d}))$ with compact support for any fixed time and we
will show uniqueness among this class of functions with the same initial
density. We proceed via the Hilbert’s duality method used in [22, 12, 8]. Let
$(\varrho_{1},p_{1})$ and $(\varrho_{2},p_{2})$ be two weak solutions of
$(\ref{Hele-Shaw})$ with the same initial data, then we see that for any
$\varphi\in C^{\infty}_{c}({\mathbb{R}}^{d}\times[0,T))$ we have

(4.1)
$\iint_{Q_{T}}((\varrho_{1}-\varrho_{2})+(p_{1}-p_{2}))\left[A(\partial_{t}\varphi)+B\nabla\cdot(m\nabla\varphi)-A(\nabla\varphi\cdot\vec{b}-\varphi
f)\right]=0,$

where

(4.2) $0\leq
A:=\frac{\varrho_{1}-\varrho_{2}}{(\varrho_{1}-\varrho_{2})+(p_{1}-p_{2})}\leq
1\text{ and }0\leq
B:=\frac{p_{1}-p_{2}}{(\varrho_{1}-\varrho_{2})+(p_{1}-p_{2})}\leq 1.$

Note that these bounds on $A$ and $B$ follow from $p_{i}(m-\varrho_{i})=0$.
for $i=1,2$ (where $A:=0$ when $\varrho_{1}=\varrho_{2}$ and $B:=0$ when
$p_{1}=p_{2}$). Now let $R>0$ be so large such that $\varrho_{i}(t)$ and
$p_{i}(t)$ are supported in $B_{R}(0)\subset{\mathbb{R}}^{d}$ for $t\in[0,T]$.
This leads us to be interested in solving the dual problem; that is if
$\psi\in C_{c}^{\infty}(Q_{T})$ we want to find a $\varphi$ such that

(4.3)
$\begin{cases}A(\partial_{t}\varphi)+B\nabla\cdot(m\nabla\varphi)-A(\nabla\varphi\cdot\vec{b}-\varphi
f)=A\psi\text{ in }Q_{T}\\\ \varphi(t=T)=0\text{ and }\varphi(x,t)=0\text{ on
}\partial B_{R}(0)\times(0,T)\end{cases},$

where if necessary, we take $R$ larger to ensure the support of $\psi$ is
contained in $B_{R}(0)\times[0,T]$. We also define $\varphi=0$ for $|x|>R$.

Observe that $(\ref{dual_problem_1})$ is a backwards degenerate parabolic
operator due to the possible degeneracy of $A=0$ and $B=0$ at some points. So
to solve $(\ref{dual_problem_1})$ we use an approximation scheme. Indeed,
consider compactly supported smooth approximations $A_{n}$ and $B_{n}$ such
that

$\begin{cases}||A_{n}-A||_{L^{2}(Q_{T})}\leq\frac{1}{n}\text{ and
}\frac{1}{n}\leq A_{n}\leq 1\\\
||B_{n}-B||_{L^{2}(Q_{T})}\leq\frac{1}{n}\text{ and }\frac{1}{n}\leq B_{n}\leq
1\end{cases}.$

Now we note that the approximate problem for $\psi\in C^{\infty}_{c}(Q_{T})$

(4.4)
$\begin{cases}\partial_{t}\varphi_{n}+\frac{B_{n}}{A_{n}}\left[\nabla\cdot(m\nabla\varphi_{n})\right]-(\nabla\varphi_{n}\cdot\vec{b}-\varphi_{n}f)=\psi\text{
in }Q_{T}\\\ \varphi_{n}(t=T)=0\text{ and }\varphi_{n}(x,t)=0\text{ on
}\partial B_{R}(0)\times(0,T)\end{cases}.$

is a uniform backwards parabolic equation with a terminal time condition, so
the equation is classically solvable. In addition, since the coefficients are
smooth, the solution is smooth with a smooth solution (see [24]). Then using
$\varphi_{n}$ as a test function in $(\ref{Dual-Problem})$ gives us that

$\iint_{Q_{T}}(\varrho_{1}-\varrho_{2})\psi=\underbrace{\iint_{Q_{T}}\left[(\varrho_{1}-\varrho_{2})+(p_{1}-p_{2})\right](A\frac{B_{n}}{A_{n}}-B_{n}+B_{n}-B)[\nabla\cdot(m\nabla\varphi_{n})]}_{\mathcal{I}_{n}}.$

For notational simplicity, we denote
$Z:=\left[(\varrho_{1}-\varrho_{2})-(p_{1}-p_{2})\right]$ and we observe that

(4.5)
$\mathcal{I}_{n}=\underbrace{\iint_{Q_{T}}Z\frac{B_{n}}{A_{n}}(A-A_{n})[\nabla\cdot(m\nabla\varphi_{n})]}_{\mathcal{I}_{n,1}}+\underbrace{\iint_{Q_{T}}Z(B_{n}-B)[\nabla\cdot(m\nabla\varphi_{n})]}_{\mathcal{I}_{n,2}}.$

Now we collect similar estimates on the dual problem to the ones in [22]:

######  Lemma 4.1.

There are constant $C=C(T,\psi)>0$ independent of $n$ such that for all
$t\in[0,T]$

$||\varphi_{n}(t)||_{L^{\infty}({\mathbb{R}}^{d})}\leq
C,\quad||\nabla\varphi_{n}(t)||_{L^{2}({\mathbb{R}}^{d})}\leq
C,\quad||(B_{n}/A_{n})^{1/2}\nabla\cdot(m\nabla\varphi_{n})||_{L^{2}(Q_{T})}\leq
C.$

###### Proof.

For the first inequality, observe that the equation for $\varphi_{n}$ is a
backwards uniformly parabolic equation, and we can compare $\varphi_{n}$ to
$e^{\pm Ct}-1$ for large $C>0$ to obtain the bounds with $C$ independent of
$n$.

For the second bound, we multiply $(\ref{reg_dual})$ by
$\nabla\cdot(m\nabla\varphi_{n})$ and integrate over
$Q_{t,T}={\mathbb{R}}^{d}\times(t,T)$ for $0\leq t<T$ to see that

(4.6)
$\underbrace{\iint_{Q_{t,T}}-\frac{m}{2}\partial_{t}|\nabla\varphi_{n}|^{2}}_{\mathcal{I}_{1}}+\frac{B_{n}}{A_{n}}|\nabla\cdot(m\nabla\varphi_{n})|^{2}-\underbrace{(\nabla\cdot(m\nabla\varphi_{n}))(\nabla\varphi_{n}\cdot\vec{b}-\varphi_{n}f)}_{\mathcal{I}_{2}}=\underbrace{\iint_{Q_{t,T}}\psi(\nabla\cdot(m\nabla\varphi_{n}))}_{\mathcal{I}_{3}}.$

Now observe that from integration by parts that

(4.7)
$\mathcal{I}_{1}=-\frac{1}{2}\iint_{Q_{t,T}}\partial_{t}(m|\nabla\varphi_{n}|^{2})-|\nabla\varphi_{n}|^{2}(\partial_{t}m)=\frac{1}{2}\int_{{\mathbb{R}}^{d}}m(t)|\nabla\varphi_{n}(t)|^{2}dx+\frac{1}{2}\iint_{Q_{t,T}}(\partial_{t}m)|\nabla\varphi_{n}|^{2}$
(4.8)
$\geq\int_{{\mathbb{R}}^{d}}\frac{\delta}{2}|\nabla\varphi_{n}(t)|^{2}-C\iint_{Q_{t,T}}|\nabla\varphi_{n}|^{2}.$

And integration by parts along with Young’s Inequality combined with $m$ being
bounded gives

(4.9)
$|\mathcal{I}_{3}|\leq\iint_{Q_{t,T}}|m(\nabla\psi\cdot\nabla\varphi_{n})|\leq
C(\psi)+C\iint_{Q_{t,T}}|\nabla\varphi_{n}|^{2}.$

For the final term, we split it

$\mathcal{I}_{2}=\underbrace{\iint_{Q_{t,T}}(\nabla\cdot(m\nabla\varphi_{n}))(\nabla\varphi_{n}\cdot\vec{b})}_{\mathcal{I}_{2,1}}-\underbrace{\iint_{Q_{t,T}}(\nabla\cdot(m\nabla\varphi_{n}))(\varphi_{n}f)}_{\mathcal{I}_{2,2}}.$

Then we have from integration by parts and Young’s Inequality that combined
with our first estimate of
$||\varphi_{n}(t)||_{L^{\infty}({\mathbb{R}}^{d})}\leq C$ that

(4.10)
$|\mathcal{I}_{2,2}|\leq\iint_{Q_{t,T}}|m\nabla\varphi_{n}\cdot(f\nabla\varphi_{n}+\varphi_{n}\nabla
f)|\leq C+C\iint_{Q_{t,T}}|\nabla\varphi_{n}|^{2}.$

Now to deal with $\mathcal{I}_{2,1}$ observe

$-\mathcal{I}_{2,1}=\iint_{Q_{t,T}}(m\nabla\varphi_{n})\cdot\nabla(\nabla\varphi_{n}\cdot\vec{b})=\iint_{Q_{t,T}}m\left(\sum_{i,j=1}^{d}\frac{\partial\varphi_{n}}{\partial
x_{i}}\frac{\partial^{2}\varphi_{n}}{\partial x_{i}\partial
x_{j}}b_{j}+\sum_{i,j=1}^{d}\frac{\partial\varphi_{n}}{\partial
x_{i}}\frac{\partial\varphi_{n}}{\partial x_{j}}\frac{\partial b_{j}}{\partial
x_{i}}\right).$

Then notice that

$\iint_{Q_{t,T}}m\sum_{i,j=1}^{d}\left|\frac{\partial\varphi_{n}}{\partial
x_{i}}\frac{\partial\varphi_{n}}{\partial x_{j}}\frac{\partial b_{j}}{\partial
x_{i}}\right|\leq
C\iint_{Q_{t,T}}\sum_{i,j=1}^{d}\left|\frac{\partial\varphi_{n}}{\partial
x_{i}}\frac{\partial\varphi_{n}}{\partial x_{j}}\right|\leq
C\iint_{Q_{t,T}}|\nabla\varphi_{n}|^{2},$

where for the final inequality we used Young’s Inequality. And we also have
from integration by parts on $x_{j}$ that

$\iint_{Q_{t,T}}\sum_{i,j=1}^{d}\frac{\partial\varphi_{n}}{\partial
x_{i}}\frac{\partial^{2}\varphi_{n}}{\partial x_{i}\partial
x_{j}}(mb_{j})=\iint_{Q_{t,T}}-\sum_{i,j=1}^{d}\frac{\partial^{2}\varphi_{n}}{\partial
x_{i}\partial x_{j}}\frac{\partial\varphi_{n}}{\partial
x_{i}}(mb_{j})-\sum_{i,j=1}^{d}|\frac{\partial\varphi_{n}}{\partial
x_{i}}|^{2}\frac{\partial(mb_{j})}{\partial x_{j}}.$

Hence, we deduce that

$\iint_{Q_{t,T}}\sum_{i,j=1}^{d}\frac{\partial\varphi_{n}}{\partial
x_{i}}\frac{\partial^{2}\varphi_{n}}{\partial x_{i}\partial
x_{j}}(mb_{j})=-\frac{1}{2}\iint_{Q_{t,T}}\sum_{i,j=1}^{d}|\frac{\partial\varphi_{n}}{\partial
x_{i}}|^{2}\frac{\partial(mb_{j})}{\partial x_{j}}.$

This implies that we have from Young’s Inequality that

(4.11) $|\mathcal{I}_{2,1}|\leq C+C\iint_{Q_{t,T}}|\nabla\varphi_{n}|^{2}.$

Using our bounds of $\mathcal{I}_{1},\mathcal{I}_{2}$, and $\mathcal{I}_{3}$
(see (4.7), (4.11), (4.10), (4.9)) in (4.6) gives

(4.12)
$\int_{{\mathbb{R}}^{d}}\frac{\delta}{2}|\nabla\varphi_{n}(t)|^{2}+\iint_{Q_{t,T}}\frac{B_{n}}{A_{n}}|\nabla\cdot(m\nabla\varphi_{n})|^{2}\leq
C(\psi)+C(\psi)\iint_{Q_{t,T}}|\nabla\varphi_{n}(t)|^{2},$

which by $B_{n}/A_{n}\geq 0$, lets us use the integral form of Grönwall’s
inequality to deduce that

(4.13) $\int_{{\mathbb{R}}^{d}}|\nabla\varphi_{n}(t)|^{2}\leq C(\psi,T).$

To obtain the 3rd inequality in the statement of Lemma 4.1, we use (4.13) and
(4.12) to see that

$||(B_{n}/A_{n})^{1/2}\nabla\cdot(m\nabla\varphi_{n})||_{L^{2}(Q_{t,T})}\leq
C(\psi,T)$

for all $0\leq t\leq T$, so by letting $t\rightarrow 0$ we conclude the third
inequality. ∎

Now these inequalities will be used to bound $\mathcal{I}_{n,1}$ and
$\mathcal{I}_{n,2}$ from $(\ref{In_Eqn})$. Indeed, observe that since $Z$ is
bounded we have by Hölder’s Inequality with Lemma 4.1 that

$|\mathcal{I}_{n,1}|\leq
C\iint_{Q_{T}}\left[(\frac{B_{n}}{A_{n}})^{1/2}|A-A_{n}|\right]\left[(\frac{B_{n}}{A_{n}})^{1/2}|\nabla\cdot(m\nabla\varphi_{n})|\right]\leq
C||(B_{n}/A_{n})^{1/2}(A-A_{n})||_{L^{2}(Q_{T})}$
$\leq\frac{Cn^{1/2}}{n}=C/n^{1/2}.$

Also we have the following bound from Lemma 4.1

$|\mathcal{I}_{n,2}|\leq
C\iint_{Q_{T}}|B_{n}-B|\left(\frac{A_{n}}{B_{n}}\right)^{1/2}\left(\frac{B_{n}}{A_{n}}\right)^{1/2}|\nabla\cdot(m\nabla\varphi_{n})|$
$\leq C||(A_{n}/B_{n})^{1/2}(B_{n}-B)||_{L^{2}(Q_{T})}\leq C/n^{1/2}.$

So by letting $n\rightarrow\infty$ in (4.5), we see that for any $\psi\in
C^{\infty}_{c}(Q_{T})$ that

$\iint_{{\mathbb{R}}^{d}}(\varrho_{1}-\varrho_{2})\psi=0.$

That is we have shown that $\varrho_{1}=\varrho_{2}$ almost everywhere. Now
using the density equation in $(\ref{Hele-Shaw})$ with
$\varrho_{1}=\varrho_{2}$ a.e., we conclude that for all $\varphi\in
C^{\infty}_{c}(Q_{T})$

$\iint_{Q_{T}}\nabla\cdot(m\nabla\varphi)(p_{1}-p_{2})=0.$

By an approximation argument we may take $\varphi=p_{1}-p_{2}$ to conclude by
integration by parts that

$\iint_{Q_{T}}m|\nabla(p_{1}-p_{2})|^{2}=0,$

so from $m\geq\delta>0$, we see this implies from $p_{i}$ being compactly
supported that $p_{1}=p_{2}$ a.e., so we conclude that:

######  Theorem 4.2 (Uniqueness).

There is at most one weak solution of $(\ref{Hele-Shaw})$ $(\varrho,p)$ in the
regularity class $L^{\infty}((0,T);L^{1}({\mathbb{R}}^{d})\cap
L^{\infty}({\mathbb{R}}^{d}))\times L^{2}([0,T];H^{1}({\mathbb{R}}^{d})\cap
L^{\infty}(Q_{T}))$ with compact support and
$\varrho(x,0)=\varrho_{\infty}^{(0)}(x)$.

Note that if Assumptions 1, 2, and 3 are met then our limiting density and
pressure $(\varrho_{\infty},p_{\infty})$ satisfy the hypothesis of
$(\ref{uniqueness})$. Hence, they are the unique solutions of the density
equation in $(\ref{Hele-Shaw})$ in this regularity class with
$\varrho_{\infty}(x,0)=\varrho_{\infty}^{(0)}(x)$. This immediately implies
with Urysohn’s subsequence principle the following corollary:

######  Corollary 4.3.

Assume that Assumptions 1, 2, and 3 are met, then the entire sequence
$(\varrho_{k},p_{k})$ converges to $(\varrho_{\infty},p_{\infty})$ in
$L^{1}(Q_{T})\times L^{1}(Q_{T})$.

This concludes the proof of Theorem 2.1. In addition, an analogous proof of
the above dual problem shows the following comparison principle for bounded
domains: (see [12] Proposition 5.1 for the case of $m\equiv 1$ with
$\vec{b}=0$)

######  Theorem 4.4 (Density Comparison Principle For Limiting Equation).

Let $\Omega\subset{\mathbb{R}}^{d}$ be a open and connected subset of
${\mathbb{R}}^{d}$ with smooth boundary and $0\leq t_{1}<t_{2}<\infty$. Assume
that $(\varrho_{1},p_{1})$ and $(\varrho_{2},p_{2})$ are weak solutions of
$(\ref{Hele-Shaw})$ in $\Omega\times(t_{1},t_{2})$ such that
$(\varrho_{i},p_{i})\in$ $L^{\infty}((t_{1},t_{2});L^{1}(\Omega)\cap
L^{\infty}(\Omega))\times L^{2}((t_{1},t_{2});H^{1}(\Omega)\cap
L^{\infty}(\Omega))$ with compact support such that
$\varrho_{1}(t=t_{1})\geq\varrho_{2}(t=t_{1})$ and $p_{1}\geq p_{2}$ on
$\partial\Omega\times[t_{1},t_{2}]$. Then we have that

$\varrho_{1}(x,t)\geq\varrho_{2}(x,t)\text{ a.e. in
}\Omega\times[t_{1},t_{2}].$

##  5\. The Complimentary Relationship on the Pressure

Now we focus on deriving the complimentary relationship. That is we will show
that $p_{\infty}$ is a weak solution to the pressure equation in $(\ref{Hele-
Shaw})$. To derive the complimentary relationship we modify the arguments in
[8] to obtain a uniform $L^{4}(Q_{\tau,T})$ bounds on $|\nabla p_{k}|$. To
achieve such bounds, we first prove the following technical lemma:

######  Lemma 5.1.

Let $f\in C^{\infty}_{c}({\mathbb{R}}^{d})$ then there is a constant $C>0$
independent of $f$ such that

$\int_{{\mathbb{R}}^{d}}|\nabla f|^{4}\leq
C\int_{{\mathbb{R}}^{d}}|f|^{2}|\nabla
f|^{2}+C\int_{{\mathbb{R}}^{d}}\frac{|f|^{2}}{m}|\nabla\cdot(m\nabla
f)|^{2}+C\sum_{i,j=1}^{d}\int_{{\mathbb{R}}^{d}}m|f|^{2}|\partial^{2}_{x_{i}x_{j}}f|^{2}.$

###### Proof.

Observe that from integration by parts that

$\int_{{\mathbb{R}}^{d}}|\nabla f|^{4}=\int_{{\mathbb{R}}^{d}}\left(|\nabla
f|^{2}\nabla f\right)\cdot\nabla f=-\underbrace{\int_{{\mathbb{R}}^{d}}f\Delta
f|\nabla f|^{2}}_{\mathcal{I}_{1}}-\underbrace{\int_{{\mathbb{R}}^{d}}f(\nabla
f)\cdot\nabla(|\nabla f|^{2})}_{\mathcal{I}_{2}}.$

Expaning out $\mathcal{I}_{2}$ shows us that

$\mathcal{I}_{2}=\sum_{i,j=1}^{d}2\int_{{\mathbb{R}}^{d}}f(\partial_{x_{j}}f)(\partial_{x_{i}}f)(\partial^{2}_{x_{i}x_{j}}f).$

Hence, by applying Young’s Inequality, we have that

$|\mathcal{I}_{2}|\leq\frac{1}{4}\int_{{\mathbb{R}}^{d}}|\nabla
f|^{4}+4\sum_{i,j=1}^{d}\int_{{\mathbb{R}}^{d}}f^{2}|\partial^{2}_{x_{i}x_{j}}f|^{2}.$

Now using that $\frac{1}{m}\nabla\cdot(m\nabla f)=\nabla\lambda\cdot\nabla
f+\Delta f$ implies

$\mathcal{I}_{1}=\int_{{\mathbb{R}}^{d}}\frac{f}{m}\nabla\cdot(m\nabla
f)|\nabla f|^{2}-\int_{{\mathbb{R}}^{d}}f(\nabla\lambda\cdot\nabla f)|\nabla
f|^{2}.$

Now by applying Young’s Inequality and Cauchy-Schwarz we have that

$|\mathcal{I}_{1}|\leq\frac{1}{2}\int_{{\mathbb{R}}^{d}}\frac{|f|^{2}}{m^{2}}|\nabla\cdot(m\nabla
f)|^{2}+\frac{2}{3}\int_{{\mathbb{R}}^{d}}|\nabla
f|^{4}+C\int_{{\mathbb{R}}^{d}}|f|^{2}|\nabla f|^{2}.$

By using $m\geq\delta>0$ and combining terms we obtain the desired bound. ∎

This gives us the following $L^{4}(Q_{T})$ control of the pressure gradient by
an approximation argument due to our $L^{\infty}(Q_{T})$ control of $p_{k}$
and $L^{2}(Q_{T})$ control of $\nabla p_{k}$ (see Lemma 3.1 and Lemma 3.3):

######  Corollary 5.2.

If Assumptions 1, 2, and 3 are met, then there is a constant $C=C(T)$ such
that

(5.1) $\iint_{Q_{T}}|\nabla p_{k}|^{4}\leq
C+C\iint_{Q_{T}}\frac{p_{k}}{m}|\nabla\cdot(m\nabla
p_{k})|^{2}+C\sum_{i,j=1}^{d}\iint_{Q_{T}}mp_{k}|\partial^{2}_{x_{i}x_{j}}p_{k}|^{2}.$

Now we control the terms arising in Corollary 5.1 using (2.2).

######  Lemma 5.3.

If Assumptions 1, 2, 3, and 4 are met, then there is a constant $C$ and
$\tilde{C}$ such that for $k>19/3$ we have

$\frac{1}{12}\sum_{i,j=1}^{d}\iint_{Q_{T}}mp_{k}|\partial^{2}_{x_{i}x_{j}}p_{k}|^{2}+(k-\frac{19}{3})\iint_{Q_{T}}\frac{p_{k}}{m}|\nabla\cdot(m\nabla
p_{k})+mF|^{2}\leq C\int_{{\mathbb{R}}^{d}}|\nabla
p_{k}(x,0)|^{2}dx+C\leq\tilde{C}.$

######  Remark 5.1.

This estimate with our smooth approximations of $p_{k}$ constructed in Section
9 implies that $p_{k}\in H^{2}_{loc}(\\{p_{k}>0\\}).$ Then we interpret
$p_{k}|\partial^{2}_{x_{i}x_{j}}p_{k}|^{2}$ classically on $\\{p_{k}>0\\}$ and
as 0 on $\\{p_{k}=0\\}$.

###### Proof.

We will drop the subscript $k$ for the rest of this proof. Multiplying the
pressure equation (2.2) by $\omega:=\nabla\cdot(m\nabla p)+mF$ and integrating
over $Q_{T}$ yields

(5.2)
$\underbrace{\iint_{Q_{T}}\omega\partial_{t}p}_{\mathcal{I}_{1}}=\underbrace{\iint_{Q_{T}}\omega|\nabla
p|^{2}}_{\mathcal{I}_{2}}+\underbrace{\iint_{Q_{T}}\omega(\nabla
p\cdot\vec{b})}_{\mathcal{I}_{3}}+\iint_{Q_{T}}(k-1)\frac{p}{m}|\omega|^{2}.$

We integrate each term separately. Observe

$\mathcal{I}_{1}=\iint_{Q_{T}}\left[\nabla\cdot(m\nabla
p)\right]\partial_{t}p+\iint_{Q_{T}}mF(\partial_{t}p)=\iint_{Q_{T}}-m\nabla
p\cdot\partial_{t}(\nabla p)+mF(\partial_{t}p)$
$=\iint_{Q_{T}}-\frac{1}{2}\partial_{t}(m|\nabla
p|^{2})+\frac{1}{2}m_{t}|\nabla p|^{2}+mF(\partial_{t}p).$

So now using $mF(\partial_{t}p)=\partial_{t}(pmF)-p\partial_{t}(mF)$ and
integrating gives

$=\int_{{\mathbb{R}}^{d}}\frac{1}{2}\left[m(x,0)|\nabla
p(x,0)|^{2}-m(x,T)|\nabla
p(x,T)|^{2}+m(x,T)F(x,T)p(x,T)-m(x,0)F(x,0)p(x,0)\right]+...$
$+\iint_{Q_{T}}\frac{1}{2}m_{t}|\nabla p|^{2}-p\partial_{t}(mF).$

So our $L^{\infty}(Q_{T})$ bounds on $p_{k}$ (Lemma 3.1) and $L^{2}(Q_{T})$
bounds on $\nabla p_{k}$ (Lemma 3.3) implies

(5.3) $\boxed{\mathcal{I}_{1}\leq C\int_{{\mathbb{R}}^{d}}|\nabla
p(x,0)|^{2}+C.}$

Now we deal with $\mathcal{I}_{2}$. Observe that

(5.4) $\mathcal{I}_{2}=\underbrace{\iint_{Q_{T}}\nabla\cdot(m\nabla p)|\nabla
p|^{2}}_{\mathcal{I}_{2,1}}+\iint_{Q_{T}}mF|\nabla p|^{2},$

so integrating by parts twice gives

$\mathcal{I}_{2,1}=\iint_{Q_{T}}p\nabla\cdot(m\nabla(|\nabla
p|^{2}))=\underbrace{\iint_{Q_{T}}p(\nabla m\cdot\nabla(|\nabla
p|^{2}))}_{\mathcal{J}_{1}}+\underbrace{\iint_{Q_{T}}mp\Delta(|\nabla
p|^{2})}_{\mathcal{J}_{2}}.$

Now using that $\Delta(|\nabla
p|^{2})=2\sum_{i,j=1}^{d}|\partial^{2}_{x_{i}x_{j}}p|^{2}+2\nabla
p\cdot\nabla(\Delta p)$ gives

$\mathcal{J}_{2}=2\iint_{Q_{T}}mp\sum_{i,j=1}^{d}|\partial^{2}_{x_{i}x_{j}}p|^{2}+\underbrace{2\iint_{Q_{T}}mp(\nabla
p\cdot\nabla(\Delta p))}_{\mathcal{J}_{2,1}},$

and integration by parts also implies

(5.5) $\mathcal{J}_{2,1}=-2\iint_{Q_{T}}\nabla\cdot(p(m\nabla p))\Delta
p=-2\iint_{Q_{T}}m|\nabla p|^{2}\Delta p+p\Delta p\left[\nabla\cdot(m\nabla
p)\right].$

Now we want to simplify the expression for $\mathcal{J}_{2,1}$ using our
previous expressions. First notice from the definition of $\mathcal{I}_{2,1}$
in (5.4) that

(5.6) $\mathcal{I}_{2,1}-\iint_{Q_{T}}[\nabla m\cdot\nabla p]|\nabla
p|^{2}=\iint_{Q_{T}}m\Delta p|\nabla p|^{2}.$

So we obtain from (5.6) and (5.5) that

(5.7) $\mathcal{J}_{2,1}=-2\mathcal{I}_{2,1}+2\iint_{Q_{T}}[\nabla
m\cdot\nabla p]|\nabla p|^{2}-p\Delta p[\nabla\cdot(m\nabla p)].$

Observe that by unpacking our definitions that

$\mathcal{I}_{2,1}=\mathcal{J}_{1}+2\iint_{Q_{T}}mp\sum_{i,j=1}^{d}|\partial^{2}_{x_{i}x_{j}}p|^{2}+\mathcal{J}_{2,1},$

so by using (5.7), we see that

(5.8)
$\mathcal{I}_{2,1}=\frac{1}{3}\mathcal{J}_{1}+\frac{2}{3}\iint_{Q_{T}}mp\sum_{i,j=1}^{d}|\partial^{2}_{x_{i}x_{j}}p|^{2}+\iint_{Q_{T}}\frac{2}{3}[\nabla
m\cdot\nabla p]|\nabla p|^{2}-\frac{2}{3}p\Delta p[\nabla\cdot(m\nabla p)].$

Now our goal is to compute lower bounds of $\mathcal{I}_{2,1}$. To deal with
the $\iint_{Q_{T}}[\nabla m\cdot\nabla p]|\nabla p|^{2}$ term, we first obtain
bounds on $\iint_{Q_{T}}p\Delta p(\nabla p\cdot\nabla m)$. Integrating by
parts gives that

(5.9) $\iint_{Q_{T}}p\Delta p(\nabla p\cdot\nabla m)=-\iint_{Q_{T}}|\nabla
p|^{2}[\nabla p\cdot\nabla m]-\underbrace{\iint_{Q_{T}}p[\nabla
p\cdot\nabla(\nabla p\cdot\nabla m)]}_{\mathcal{J}_{2,2}}.$

Note that $p[\nabla p\cdot\nabla(\nabla p\cdot\nabla m)]=p[\nabla
p^{T}D^{2}p\nabla m+\nabla p^{T}D^{2}m\nabla p]$ so Young’s Inequality with
$\varepsilon$ gives

$|\mathcal{J}_{2,2}|\leq\sum_{i,j=1}^{d}\frac{\delta}{2}\iint_{Q_{T}}p|\partial^{2}_{x_{i}x_{j}}p|^{2}+C\iint_{Q_{T}}p|\nabla
p|^{2}+\iint_{Q_{T}}p|(\nabla p)^{T}(D^{2}m)(\nabla p)|$
$\leq\sum_{i,j=1}^{d}\frac{1}{2}\iint_{Q_{T}}mp|\partial^{2}_{x_{i}x_{j}}p|^{2}+\tilde{C}\iint_{Q_{T}}p|\nabla
p|^{2}$ (5.10)
$\leq\sum_{i,j=1}^{d}\frac{1}{2}\iint_{Q_{T}}mp|\partial^{2}_{x_{i}x_{j}}p|^{2}+C.$

In this final inequality we used our $L^{2}(Q_{T})$ bounds on $\nabla p$
(Lemma 3.3) and $L^{\infty}(Q_{T})$ bounds on $p$ (Lemma 3.1).

And to deal with the $[\nabla m\cdot\nabla p]|\nabla p|^{2}$ term in
$\mathcal{J}_{2,1}$, we observe from (5.9) and (5.10) that

(5.11) $\iint_{Q_{T}}|\nabla p|^{2}[\nabla p\cdot\nabla
m]\geq-C-\iint_{Q_{T}}p\Delta p(\nabla p\cdot\nabla
m)-\frac{1}{2}\sum_{i,j=1}^{d}\iint_{Q_{T}}mp|\partial^{2}_{x_{i}x_{j}}p|^{2}.$

Now using (5.11) in (5.8) gives us

$\mathcal{I}_{2,1}\geq-C+\frac{1}{3}\mathcal{J}_{1}+\frac{1}{3}\left[\iint_{Q_{T}}mp\sum_{i,j=1}^{d}|\partial^{2}_{x_{i}x_{j}}p|^{2}\right]-\frac{2}{3}\left[\iint_{Q_{T}}p\Delta
p[\nabla\cdot(m\nabla p)]+p\Delta p(\nabla p\cdot\nabla m)\right].$

Now we use that

$p\Delta p[\nabla\cdot(m\nabla p)]+p\Delta p(\nabla p\cdot\nabla m)=mp|\Delta
p|^{2}+2p\Delta p(\nabla p\cdot\nabla m)$ $\leq mp|\Delta p|^{2}+2p\Delta
p(\nabla p\cdot\nabla m)+\frac{p}{m}|\nabla p\cdot\nabla
m|^{2}=\frac{p}{m}|\nabla\cdot(m\nabla p)|^{2}.$

Hence, we have that

(5.12)
$\mathcal{I}_{2,1}\geq-C+\frac{1}{3}\mathcal{J}_{1}+\frac{1}{3}\iint_{Q_{T}}mp\sum_{i,j=1}^{d}|\partial^{2}_{x_{i}x_{j}}p|^{2}-\left[\frac{2}{3}\iint_{Q_{T}}\frac{p}{m}|\nabla\cdot(m\nabla
p)|^{2}\right].$

We use integration by parts to see

$\mathcal{J}_{1}=-\iint_{Q_{T}}\left([\nabla p\cdot\nabla m]+p\Delta
m\right)|\nabla p|^{2}.$

Hence, by using Young’s Inequality with $\varepsilon$ and our $L^{2}(Q_{T})$
bounds on $\nabla p$ (Lemma 3.3) and $L^{\infty}(Q_{T})$ bounds on the
pressure (Lemma 3.1), we see that for $\varepsilon>0$

$|\mathcal{J}_{1}|\leq\varepsilon\iint_{Q_{T}}|\nabla
p|^{4}+C(\varepsilon)\iint_{Q_{T}}||\nabla m||^{2}_{\infty}|\nabla
p|^{2}+C\iint_{Q_{T}}|\nabla p|^{2}$ $\leq\varepsilon\iint_{Q_{T}}|\nabla
p|^{4}+C(\varepsilon).$

Now using Corollary 5.1 with an approximation argument and a fixed
$\varepsilon>0$ chosen such that for the constant $C$ in from Corollary 5.1
satisfies $\varepsilon C\leq 1/2$ gives

(5.13) $|\mathcal{J}_{1}|\leq
C+\frac{1}{2}\iint_{Q_{T}}\frac{p}{m}|\nabla\cdot(m\nabla
p)|^{2}+\frac{1}{2}\sum_{i,j=1}^{d}mp|\partial^{2}_{x_{i}x_{j}}p|^{2}.$

Then by using (5.13) in (5.12) we obtain

$\mathcal{I}_{2,1}\geq\frac{1}{6}\sum_{i,j=1}^{d}\iint_{Q_{T}}mp|\partial^{2}_{x_{i}x_{j}}p|^{2}-\frac{5}{6}\iint_{Q_{T}}\frac{p}{m}|\nabla\cdot(m\nabla
p)|^{2}-C.$

Now using that

$|\nabla\cdot(m\nabla p)|^{2}\leq 4|\nabla\cdot(m\nabla p)+mF|^{2}+4|mF|^{2},$

and that $F$ is bounded due to our assumptions implies

$\mathcal{I}_{2,1}\geq\frac{1}{6}\sum_{i,j=1}^{d}\iint_{Q_{T}}mp|\partial^{2}_{x_{i}x_{j}}p|^{2}-\frac{10}{3}\iint_{Q_{T}}\frac{p}{m}|\nabla\cdot(m\nabla
p)+mF|^{2}-C.$

So now using (5.4) and our $L^{2}(Q_{T})$ bound on $\nabla p$ (Lemma 3.3), we
deduce that

(5.14)
$\boxed{\mathcal{I}_{2}\geq-C+\frac{1}{6}\sum_{i,j=1}^{d}\iint_{Q_{T}}mp|\partial^{2}_{x_{i}x_{j}}p|^{2}-\frac{10}{3}\iint_{Q_{T}}\frac{p}{m}|\nabla\cdot(m\nabla
p)+mF|^{2}.}$

So it remains to deal with $\mathcal{I}_{3}$ to obtain the desired bounds.
Observe that

(5.15) $\mathcal{I}_{3}=\underbrace{\iint_{Q_{T}}(\nabla
p\cdot\vec{b})(\nabla\cdot(m\nabla
p))}_{\mathcal{I}_{3,1}}+\underbrace{\iint_{Q_{T}}mF(\nabla
p\cdot\nabla\vec{b})}_{\mathcal{I}_{3,2}}.$

Now Cauchy-Schwarz and our $L^{1}(Q_{T})$ control of $\nabla p_{k}$ (Lemma 3.1
and Lemma 3.3) implies

(5.16) $\mathcal{I}_{3,2}\geq-C\iint_{Q_{T}}|\nabla p|\geq-\tilde{C}.$

Now we compute from integration by parts that

$\mathcal{I}_{3,1}=-\iint_{Q_{T}}p\nabla\cdot(\vec{b}\left[\nabla\cdot(m\nabla
p)\right])=-\underbrace{\iint_{Q_{T}}p(\nabla\cdot\vec{b})[\nabla\cdot(m\nabla
p)]}_{\mathcal{K}_{1}}-\underbrace{\iint_{Q_{T}}p(\vec{b}\cdot\nabla\left[\nabla\cdot(m\nabla
p)\right])}_{\mathcal{K}_{2}}.$

We apply Young’s Inequality to see that

(5.17)
$|\mathcal{K}_{1}|\leq\frac{1}{2||m||_{\infty}}\iint_{Q_{T}}p|\nabla\cdot(m\nabla
p)|^{2}+\frac{||m||_{\infty}}{2}\iint_{Q_{T}}p|\nabla\cdot\vec{b}|^{2}\leq
C+\frac{1}{2}\iint_{Q_{T}}\frac{p}{m}|\nabla\cdot(m\nabla p)|^{2}.$

Now we compute to see that

$\mathcal{K}_{2}=\underbrace{\sum_{i,j=1}^{d}\iint_{Q_{T}}pb_{i}\left[(\partial^{2}_{x_{i}x_{j}}m)(\partial_{x_{j}}p)\right]}_{\mathcal{K}_{2,1}}+\underbrace{\sum_{i,j=1}^{d}\iint_{Q_{T}}pb_{i}\left[(\partial_{x_{j}}m)(\partial^{2}_{x_{i}x_{j}}p)\right]}_{\mathcal{K}_{2,2}}+\underbrace{\sum_{i,j=1}^{d}\iint_{Q_{T}}p(mb_{i})\left[\partial^{3}_{x_{j}x_{i}x_{j}}p.\right]}_{\mathcal{K}_{2,3}}+$
$+\iint_{Q_{T}}(\vec{b}\cdot\nabla m)\Delta p.$

The last term can be bounded by integration by parts:

(5.18) $\left|\iint_{Q_{T}}(\vec{b}\cdot\nabla m)\Delta
p\right|=\left|\iint_{Q_{T}}-\nabla(\vec{b}\cdot\nabla m)\cdot\nabla
p\right|\leq C\iint_{Q_{T}}|\nabla p|\leq C.$

Now we bound the other terms. Observe that from Young’s Inequality that

(5.19)
$|\mathcal{K}_{2,1}|\leq\frac{1}{2}\sum_{i,j=1}^{d}\iint_{Q_{T}}p|\partial_{x_{i}}p|^{2}+p|b_{i}|^{2}|\partial^{2}_{x_{i}x_{j}}m|^{2}\leq
C(T),$

where we used our $L^{2}(Q_{T})$ bounds on $\nabla p$ (Lemma (3.3)) and our
$L^{\infty}(Q_{T})$ bounds on $p$ (Lemma (3.1)). Then by using Young’s
Inequality again

(5.20)
$|\mathcal{K}_{2,2}|\leq\iint_{Q_{T}}\frac{\delta}{24}\sum_{i,j=1}^{d}p|\partial^{2}_{x_{i}x_{j}}p|^{2}+\frac{12}{\delta}\sum_{i,j=1}^{d}p|b_{i}|^{2}|\partial_{x_{j}}m|^{2}\leq\frac{1}{24}\iint_{Q_{T}}\sum_{i,j=1}^{d}mp|\partial^{2}_{x_{i}x_{j}}p|^{2}+C(T).$

Now we deal with $\mathcal{K}_{2,3}$. Let us define
$\vec{c}:=m\vec{b}=(c_{1},c_{2},...,c_{d})^{T}$ and integrate by parts to see

(5.21)
$\mathcal{K}_{2,3}=-\underbrace{\sum_{i,j=1}^{d}\iint_{Q_{T}}c_{i}(\partial_{x_{j}}p)(\partial^{2}_{x_{i}x_{j}}p)}_{\mathcal{K}_{2,4}}-\underbrace{\sum_{i,j=1}^{d}\iint_{Q_{T}}p(\partial_{x_{j}}c_{i})(\partial^{2}_{x_{j}x_{i}}p)}_{\mathcal{K}_{2,5}}.$

By Young’s Inequality we have that

(5.22)
$|\mathcal{K}_{2,5}|\leq\iint_{Q_{T}}\frac{\delta}{24}\sum_{i,j=1}^{d}p|\partial^{2}_{x_{i}x_{j}}p|^{2}+\frac{12}{\delta}p|\partial_{x_{j}}c_{i}|^{2}\leq\frac{1}{24}\sum_{i,j=1}^{d}\iint_{Q_{T}}mp|\partial^{2}_{x_{i}x_{j}}p|^{2}+C(T).$

Now we observe that

$\mathcal{K}_{2,4}=\frac{1}{2}\iint_{Q_{T}}\vec{c}\cdot\nabla(|\nabla
p|^{2})=\frac{1}{2}\iint_{Q_{T}}\nabla\cdot(\vec{c}|\nabla
p|^{2})-(\nabla\cdot\vec{c})|\nabla p|^{2}$
$=-\frac{1}{2}\iint_{Q_{T}}(\nabla\cdot\vec{c})|\nabla p|^{2}.$

Hence, we have from our $L^{2}(Q_{T})$ control of $\nabla p$ (Lemma 3.3)

(5.23) $|\mathcal{K}_{2,4}|\leq C(T).$

So now combining our inequalities on $\mathcal{K}_{1}$ and $\mathcal{K}_{2}$
(see (5.17), (5.19), (5.18), (5.20), (5.23), (5.22)) gives

$|\mathcal{I}_{3,1}|\leq
C(T)+\frac{1}{12}\sum_{i,j=1}^{d}\iint_{Q_{T}}mp|\partial^{2}_{x_{i}x_{j}}p|^{2}+\frac{1}{2}\iint_{Q_{T}}\frac{p}{m}|\nabla\cdot(m\nabla
p)|^{2}.$

Now again using

$|\nabla\cdot(m\nabla p)|^{2}\leq 4|\nabla\cdot(m\nabla p)+mF|^{2}+4|mF|^{2},$

we deduce that

(5.24) $|\mathcal{I}_{3,1}|\leq
C(T)+\frac{1}{12}\sum_{i,j=1}^{d}\iint_{Q_{T}}mp|\partial^{2}_{x_{i}x_{j}}p|^{2}+2\iint_{Q_{T}}\frac{p}{m}|\nabla\cdot(m\nabla
p)+mF|^{2}.$

So by using (5.24) and (5.16) in (5.15) we arrive at

(5.25)
$\boxed{\mathcal{I}_{3}\geq-C(T)-\frac{1}{12}\sum_{i,j=1}^{d}\iint_{Q_{T}}mp|\partial^{2}_{x_{i}x_{j}}p|^{2}-2\iint_{Q_{T}}\frac{p}{m}|\nabla\cdot(m\nabla
p)+F|^{2}.}$

That is we have from (5.3), (5.14), (5.25), (5.2), and Assumption (4) that

$\frac{1}{12}\sum_{i,j=1}^{d}\iint_{Q_{T}}mp|\partial^{2}_{x_{i}x_{j}}p|^{2}+(k-\frac{19}{3})\iint_{Q_{T}}\frac{p}{m}|\nabla\cdot(m\nabla
p)+F|^{2}\leq C\int_{{\mathbb{R}}^{d}}|\nabla p(x,0)|^{2}+C\leq\tilde{C}.$

∎

Now by combining Lemma 5.3 with Lemma 5.1 we obtain uniform $L^{4}(Q_{T})$
bounds on $\nabla p$ for $k\geq 7$ due to an approximation argument.

######  Corollary 5.4 ($L^{4}(Q_{T})$ bounds on $|\nabla p|$).

If Assumptions 1, 2, 3, and 4 are met, then for $k\geq 7$ there is a constant
$C=C(T)$ such that

$\iint_{Q_{T}}|\nabla p|^{4}\leq C.$

Now we have enough compactness for strong $L^{2}(Q_{T})$ convergence of
$|\nabla p_{k}|$.

######  Lemma 5.5.

Assume that Assumptions 1, 2, 3, and 4 are met, then $\nabla
p_{k}\rightarrow\nabla p_{\infty}$ in $L^{r}(Q_{\tau,T})$ for any $\tau>0$ and
$r\in[1,4)$.

###### Proof.

This proof is essentially identical to the proof of Lemma 3.5 in [8], so we
omit the proof. ∎

Now we use (2.2) to obtain $L^{2}(Q_{T})$ convergence of $\nabla p_{k}$.

######  Theorem 5.6.

Assume that Assumptions 1, 2, 3, and 4 are met then $\nabla
p_{k}\rightarrow\nabla p_{\infty}$ in $L^{2}(Q_{T})$.

###### Proof.

By Lemma 5.5 we know that $\nabla p_{k}\rightarrow\nabla p_{\infty}$ in
$L^{2}(Q_{\tau,T})$ for any $\tau>0$. So it remains to control $||\nabla
p_{k}||_{L^{2}(Q_{\tau})}$ for small $\tau>0$. Integrating (2.2) over
$Q_{\tau}$ gives

$(k-2)\iint_{Q_{\tau}}|\nabla
p_{k}|^{2}=\int_{{\mathbb{R}}^{d}}p_{k}(x,0)-p_{k}(x,\tau)+\iint_{Q_{\tau}}\nabla
p_{k}\cdot(\vec{b}+(k-1)p_{k}\nabla\lambda)+(k-1)p_{k}F.$

By applying Young’s Inequality with $\varepsilon>0$ we arrive at

$(k-2)\iint_{Q_{\tau}}|\nabla
p_{k}|^{2}\leq\int_{{\mathbb{R}}^{d}}p_{k}(x,0)+\iint_{Q_{\tau}}|\nabla
p_{k}|^{2}+(k-1)\iint_{Q_{\tau}}\varepsilon||p_{k}||^{2}_{L^{\infty}(Q_{T})}|\nabla
p_{k}|^{2}+C|(\Omega\times[0,T])\cap Q_{\tau}|+$
$+C(\varepsilon)(k-1)\iint_{Q_{\tau}}p_{k},$

where $\Omega\times[0,T]=:\Omega_{T}$ is a compact set such that
$\text{supp}(p_{k})\subset\Omega_{T}$ for all $k>1$ and $|A|$ is the Lebesgue
measure of $A\subset{\mathbb{R}}^{d+1}$. Choose $\varepsilon>0$ so small such
that
$\varepsilon||p_{k}||_{L^{\infty}(Q_{T})}||\nabla\lambda||_{\infty}\leq\frac{1}{2}$
for all $k$, we arrive at

$\iint_{Q_{\tau}}|\nabla
p_{k}|^{2}\leq\frac{2}{k-5}\int_{{\mathbb{R}}^{d}}p_{k}(x,0)+C|\Omega_{T}\cap
Q_{\tau}|.$

So for $k\geq 6$, we have from Lemma 3.1 that

(5.26) $\iint_{Q_{\tau}}|\nabla p_{k}|^{2}\leq\frac{2C}{k-5}+C|\Omega_{T}\cap
Q_{\tau}|,$

where $|A|$ is the Lebesgue measure of $A\subset{\mathbb{R}}^{d+1}$. Note that
this constant $C$ is independent of both $\tau$ and $k$.

Now if $\varepsilon^{\prime}>0$ is given, there exists a $K^{\prime}$
sufficiently large with $\tau$ sufficiently small and $\nabla p_{\infty}\in
L^{2}(Q_{T})$ such that

$\iint_{Q_{\tau}}|\nabla p_{k}|^{2}+|\nabla
p_{\infty}|^{2}\leq\varepsilon^{\prime}/2$

for all $k\geq K^{\prime}$ thanks to (5.26). Now by making $K^{\prime}$ even
larger if necessary, Lemma 5.5 shows us that $\iint_{Q_{\tau,T}}|\nabla
p_{k}-\nabla p_{\infty}|^{2}\leq\varepsilon^{\prime}/2$ for all $k\geq
K^{\prime}$. So now from the triangle inequality, we obtain

$\iint_{Q_{T}}|\nabla p_{k}-\nabla p_{\infty}|^{2}\leq\varepsilon^{\prime}$

for $k\geq K^{\prime}$. ∎

Now we have enough compactness to show that $p_{\infty}$ is a weak solution to
the pressure equation in (1.4). Indeed, this follows from similar arguments as
in [22] Lemma 2.5 that showed when $m\equiv 1$ the complimentary relationship
is equivalent to $L^{2}(Q_{T})$ convergence of $\nabla p_{k}$. Or we could
proceed as in [7] and take the limit of the weak form of the pressure equation
(2.2) to conclude the proof of Theorem 2.2.

##  6\. Velocity Law

###  6.1. Velocity Law

Now we aim to characterize the movement of $\\{p_{\infty}(t)>0\\}$. We will
show the velocity law (1.5) in the viscosity sense. Our arguments and
definitions on the velocity law and classical barriers closely follow the ones
in [12].

######  Definition 6.1 (Classical Super-Solutions).

Let $B$ be any space ball in ${\mathbb{R}}^{d}$ and $0\leq t_{1}<t_{2}<\infty$
and $\phi\in C_{c}(\overline{B}\times[t_{1},t_{2}])$ be non-negative. We also
assume that $\phi\in C^{2}(\\{\overline{\phi>0\\}})$ and that
$\partial\\{\phi>0\\}$ is $C^{2}$ in space and $C^{1}$ in time. Then we choose
a $\rho^{E}\in C^{1,1}(\\{\phi=0\\})$ such that

$\partial_{t}\rho^{E}\geq\nabla\cdot(\rho^{E}\vec{b})+f\rho^{E}\text{ in
}\\{\phi=0\\}$

with $\rho^{E}\leq m$ on $\\{\phi=0\\}$. And we also assume that

(6.1)
$\begin{cases}-\nabla\cdot(m\nabla\phi)\geq\nabla\cdot(m\vec{b})-m_{t}+mf\text{
in }\\{\phi>0\\}\\\
(m-\rho^{E})V_{\phi}\geq\left[-m\nabla\phi-\vec{b}(m-\rho^{E})\right]\cdot\vec{\nu}\text{
on }\partial\\{\phi>0\\}\end{cases}$

where $V_{\phi}$ is the normal velocity of $\partial\\{\phi(t)>0\\}$ and
$\vec{\nu}$ is the outward normal of $\partial\\{\phi(t)>0\\}$. Then we define

$\rho:=m\chi_{\\{\phi>0\\}}+\rho^{E}\chi_{\\{\phi=0\\}}.$

Then the pair $(\rho,\phi)$ is called a classical super-solution.

######  Lemma 6.1.

If the pair $(\rho,\phi)$ is a classical super-solution, then they are also a
weak super-solution of the limiting density equation $(\ref{Hele-Shaw})$ in
$B\times(t_{1},t_{2})$ with initial data $\rho_{1}:=\rho(t=t_{1})$. In
particular, for all $\varphi\in C_{c}^{\infty}(B\times[t_{1},t_{2}))$ that is
non-negative we have

$\iint_{B\times[t_{1},t_{2}]}-\varphi_{t}\rho+\nabla\varphi\cdot(m\nabla\phi)\geq\iint_{B\times[t_{1},t_{2}]}\varphi
f\rho-\nabla\varphi\cdot(\rho\vec{b})+\int_{B}\varphi(t=t_{1})\rho_{1}$

###### Proof.

The proof is essentially identical to the proof of Proposition 7.1 in [12], so
we omit the details. ∎

And analogously, we have the following definition for classical sub-solutions.

######  Definition 6.2 (Classical Sub-Solutions).

Let $B$ be any space ball and ${\mathbb{R}}^{d}$ and $0\leq
t_{1}<t_{2}<\infty$ and $\phi\in C_{c}(\overline{B}\times[t_{1},t_{2}])$ be
non-negative. We also assume that $\phi\in C^{2}(\\{\overline{\phi>0\\}})$ and
that $\partial\\{\phi>0\\}$ is $C^{2}$ in space and $C^{1}$ in time. Then we
choose a $\rho^{E}\in C^{1,1}(\\{\phi=0\\})$ such that

$\partial_{t}\rho^{E}\leq\nabla\cdot(\rho^{E}\vec{b})+f\rho^{E}\text{ in
}\\{\phi=0\\}$

and $\rho^{E}\leq m$ on $\\{\phi=0\\}$. And we also assume that

(6.2)
$\begin{cases}-\nabla\cdot(m\nabla\phi)\leq\nabla\cdot(m\vec{b})-m_{t}+mf\text{
in }\\{\phi>0\\}\\\
(m-\rho^{E})V_{\phi}\leq\left[-m\nabla\phi-\vec{b}(m-\rho^{E})\right]\cdot\vec{\nu}\text{
on }\partial\\{\phi>0\\}\end{cases}$

where $V_{\phi}$ is the normal velocity of $\partial\\{\phi(t)>0\\}$ and
$\vec{\nu}$ is the outward normal of $\partial\\{\phi(t)>0\\}$. Now if we
define

$\rho:=m\chi_{\\{\phi>0\\}}+\rho^{E}\chi_{\\{\phi=0\\}},$

then the pair $(\varrho,\phi)$ is called a classical sub-solution.

Similarly we have the following relation between classical sub-solutions and
weak sub-solutions of the limiting density equation $(\ref{Hele-Shaw})$.

######  Lemma 6.2.

If the pair $(\rho,\phi)$ is a classical sub-solution, then they are a weak
sub-solution of the limiting equation $(\ref{Hele-Shaw})$ in
$B\times(t_{1},t_{2})$ with initial data $\rho_{1}:=\rho(t=t_{1})$. In
particular, for all $\varphi\in C_{c}^{\infty}(B\times[t_{1},t_{2}))$ that is
non-negative we have

$\iint_{B\times[t_{1},t_{2}]}-\varphi_{t}\rho+\nabla\varphi\cdot(m\nabla\phi)\leq\iint_{B\times[t_{1},t_{2}]}\varphi
f\rho-\nabla\varphi\cdot(\rho\vec{b})+\int_{B}\varphi(t=t_{1})\rho_{1}$

Recall that the limiting pair $(\varrho_{\infty},p_{\infty})$ is a weak
solution to $(\ref{Hele-Shaw})$ in the sub-domain $B\times(t_{1},t_{2})$ with
the initial density condition $\varrho_{\infty}(t=t_{1})$ and pressure
boundary conditions $p_{\infty}$ on $\partial B\times[t_{1},t_{2}]$. So we
deduce from the weak density comparison principle Lemma 4.4 for the limiting
density equation that we have

######  Proposition 6.3.

Let the pair $(\rho,\phi)$ be a classical super solution in
$B\times(t_{1},t_{2})$, then if

  1. (1)

$\varrho_{\infty}(t=t_{1})\leq\rho(t=t_{1})$ on  B

  2. (2)

$p_{\infty}\leq\phi$ on  $\partial B\times[t_{1},t_{2}]$ ,

then we have that $\varrho_{\infty}\leq\rho$ on $B\times[t_{1},t_{2}].$

So in the sense of comparison with barriers (or viscosity sense) we have that

$(m-\varrho^{E})V_{p_{\infty}}\geq\left[-m\nabla\phi-\vec{b}(m-\varrho^{E})\right]\cdot\vec{\nu}\text{
on }\partial\\{p_{\infty}>0\\}.$

Analogously, we have the following proposition:

######  Proposition 6.4.

Let the pair $(\rho,\phi)$ be a classical sub-solution on
$B\times(t_{1},t_{2})$, then if

  1. (1)

$\varrho_{\infty}(t=t_{1})\geq\rho(t=t_{1})$ on  B

  2. (2)

$p_{\infty}\geq\phi$ on  $\partial B\times[t_{1},t_{2}]$ ,

then we have that $\varrho_{\infty}\geq\rho$ on $B\times[t_{1},t_{2}].$

So in the sense of comparison with barriers (or viscosity sense) we have that

$(m-\varrho^{E})V_{p_{\infty}}\leq\left[-m\nabla\phi-\vec{b}(m-\varrho^{E})\right]\cdot\vec{\nu}\text{
on }\partial\\{p_{\infty}>0\\}$

Hence, we have derived the velocity law in the viscosity sense. That is we
have proven Theorem 2.3.

##  7\. Congestion Case

In this section we focus on the congestion case. That is we assume throughout
this section that (1.6) holds. So now if $\Omega$ is a compact subset of
${\mathbb{R}}^{d}$ such that $p_{k}$ is uniformly supported in
$\Omega\times[0,T]$ then there exists by continuity an $\varepsilon>0$ such
that

(7.1) $m_{t}+\varepsilon\leq mf+\nabla\cdot(m\vec{b})\text{ in
}\Omega\times[0,T].$

First we recall the definition of the streamlines. Since the external density
solves a transport equation, the natural coordinates are the streamlines (or
characteristics). For any $x_{0}\in{\mathbb{R}}^{d}$, we let the streamline
$X(\cdot,x_{0})$ be the unique solution to

$\begin{cases}\partial_{t}X(t,x_{0})=-\vec{b}(X(t,x_{0}))\text{ for
}t\in{\mathbb{R}}\\\ X(0,x_{0})=x_{0}\end{cases}.$

As $\vec{b}$ is Lipschitz, we know that there is a unique solution for all
time and is continuously differentiable in time. This uniqueness actually
implies that for any $t_{0}\in{\mathbb{R}}$ that
$X(t_{0},x):{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}^{d}$ is invertible and its
inverse is given by $X(-t_{0},x)$ (see [15]).

We will show that $\\{p_{\infty}(t)>0\\}:=\Sigma(t)$ is non-decreasing with
respect to the streamlines. That is if $x\in\Sigma(t)$, then
$X(s,X(-t,x))\in\Sigma(s)$ for all $s\geq t$ (the streamline containing $x$ at
time $t$ is in $\Sigma(s)$ for all $s\geq t$). This will follow from deriving
a uniform positive retention principle for $p_{k}$ along streamlines through a
refined Aronson-Bénilan estimate. We refine the barrier used in Section 3.3.

To derive such an Aronson-Bénilan estimate, we first drop the subscript $k$
and recall that $w:=\frac{1}{m}\nabla\cdot(m\nabla p)$. Then we define the
elliptic operator for sufficiently large $C>0$

$\mathcal{L}[\phi]:=-C-C|\nabla p|^{2}+2\nabla\phi\cdot\nabla
p+\nabla\phi\cdot\vec{b}+(k-1)(\phi(\phi+F)+(\nabla\phi+\nabla
F)\cdot(p\nabla\lambda+2\nabla p)+p(\Delta\phi+\Delta F)).$

Then when $C$ is large enough, by using (3.6), (3.7), (3.8) with a similar
computation as in Section 3.7 we see that $\partial_{t}w\geq\mathcal{L}[w]$.

Now $\tau>0$ and for $\alpha,\beta>0$ we define our barrier

$\psi(x,t;\tau):=-F+\frac{\alpha p-\beta}{k-1}-\frac{1}{(k-1)(t-\tau)}.$

We note that our constants $C$ and $\tilde{C}$ in estimating
$\partial_{t}\psi$ and $\mathcal{L}[\psi]$ are also independent of $\alpha$
and $\beta$. Plugging in $\psi$ into $\mathcal{L}$ along with Young’s
Inequality on the $2\nabla\psi\cdot\nabla p$ term implies there exists a large
enough constant $\tilde{C}\geq C$ such that

$\mathcal{L}[\psi]\geq-\tilde{C}-\tilde{C}|\nabla
p|^{2}+\frac{2\alpha}{k-1}|\nabla p|^{2}+\frac{\alpha}{k-1}\nabla
p\cdot\vec{b}+\alpha pw+2\alpha|\nabla p|^{2}+(k-1)[\psi^{2}+\psi F].$

Then from using (2.2)

$\partial_{t}\psi\leq\tilde{C}+\frac{\alpha}{k-1}|\nabla
p|^{2}+\frac{\alpha}{k-1}(\nabla p\cdot\vec{b})+\alpha pw+\alpha
pF+\frac{1}{(k-1)(t-\tau)^{2}}.$

This implies that

(7.2) $\partial_{t}\psi-\mathcal{L}[\psi]\leq 2\tilde{C}+|\nabla
p|^{2}(\tilde{C}-\frac{\alpha}{k-1}-2\alpha)+\alpha
pF+\frac{1}{(k-1)(t-\tau)^{2}}-(k-1)[\psi^{2}+\psi F].$

So we first we choose $\beta>2\alpha\sup_{k>1}||p_{k}||_{L^{\infty}(Q_{T})}$,
so that $\alpha p-\beta\leq 0$. This combined with $F>0$ gives

(7.3) $(\psi^{2}+\psi
F)=(\psi+\frac{F}{2})^{2}-\frac{F^{2}}{4}\geq(-\frac{F}{2}+\frac{\alpha
p-\beta}{k-1})^{2}+\frac{1}{(k-1)^{2}(t-\tau)^{2}}-\frac{F^{2}}{4}$ (7.4)
$=\left(\frac{\alpha p-\beta}{k-1}\right)^{2}-F\frac{(\alpha
p-\beta)}{k-1}+\frac{1}{(k-1)^{2}(t-\tau)^{2}}.$

So using (7.3) and (7.4) in (7.2)

$\partial_{t}\psi-\mathcal{L}[\psi]\leq 2\tilde{C}+|\nabla
p|^{2}(\tilde{C}-\frac{\alpha}{k-1}-2\alpha)+F(2\alpha p-\beta)-\frac{(\alpha
p-\beta)^{2}}{k-1}.$

Now because $2\alpha p-\beta\leq 0$ due to our choice of $\beta$ and
$F\geq\varepsilon>0$ inside $\Omega\times[0,T]$ we have that

$\partial_{t}\psi-\mathcal{L}[\psi]\leq 2\tilde{C}+|\nabla
p|^{2}(\tilde{C}-\frac{\alpha}{k-1}-2\alpha)+\varepsilon(2\alpha
p-\beta)-\frac{(\alpha p-\beta)^{2}}{k-1}\text{ in }\Omega\times[0,T].$

Now we let $\alpha=\tilde{C}/2$ and
$\beta=2\alpha\sup_{k>1}||p_{k}||_{L^{\infty}(Q_{T})}+2\tilde{C}/\varepsilon$,
then we see that

$\partial_{t}\psi-\mathcal{L}[\psi]\leq 0\text{ in }\Omega\times[0,T].$

That is $\psi$ is a sub-solution of a parabolic equation where $w$ is a super
solution of in $\Omega\times[0,T]$. And recalling that $p$ is compactly
supported inside $\Omega\times[0,T]$ gives us $w\equiv 0$ on
$\partial\Omega\times[0,T]$. Also we have $\psi\leq 0$ on
$\partial\Omega\times[0,T]$, so now as $\lim_{\tau\rightarrow
0+}\psi(x,0;\tau)=-\infty$, we deduce from the comparison principle that

$\frac{1}{m}\nabla\cdot(m\nabla
p)=w(x,t)\geq\psi(x,t;0)\geq-F-\frac{\beta}{k-1}-\frac{1}{(k-1)t}.$

Hence, we have shown

######  Lemma 7.1 (Refined Aronson-Bénilan Estimate).

Assume we are in the congestion case $(\ref{congestion})$ with Assumptions 1,
2, and 3, then we have the following refined semi-convexity estimate for all
$k>1$

(7.5) $\frac{1}{m}\nabla\cdot(m\nabla
p_{k})\geq-F-\frac{\beta}{k-1}-\frac{1}{(k-1)t}$

for some constant $\beta>0$ independent of $k$ in the sense of distributions.

This Aronson-Bénilan Estimate implies the following estimates by using (7.5)
in (2.2)

######  Corollary 7.2.

Assume we are in the congestion case $(\ref{congestion})$ with Assumptions 1,
2, and 3, then for all $k>1$ we have for some constant $\beta>0$ independent
of $k$ that

$\frac{d}{dt}p_{k}(X(t,x_{0}),t)\geq-\left(\beta+\frac{1}{t}\right)p_{k}(X(t,x_{0}),t)$

and

$\frac{d}{dt}v_{k}(X(t,x_{0}),t)\geq-\frac{1}{k-1}\left(\beta+\frac{1}{t}\right)v(X(t,x_{0}),t)$

in the sense of distributions.

The normalized density estimate implies the following monotonicity property:

######  Lemma 7.3.

The limiting normalized density $v_{\infty}:=\varrho_{\infty}/m$ is non-
decreasing along streamlines for $t>0$.

Note that as the streamlines $X$ are a lipschitz bijection of $Q_{T}$, $X$
maps sets of full measure to sets of full measure. This along with the
pressure estimate in Corollary 7.2 combined with Grönwall’s Inequality gives
us:

######  Lemma 7.4 (Retention Along Streamlines For Positive Time).

Assume that Assumptions 1, 2, and 3 are met, then we have the following
estimate for all $k>1$ and $\tau>0$

$p_{k}(X(t,x_{0}),t)\geq
p_{k}(X(\tau,x_{0}),\tau)\exp[-(\beta+\frac{1}{\tau})(t-\tau)]$

for some constant $\beta>0$ independent of $k$ for a.e.
$(x_{0},t),(x_{0},\tau)\in Q_{T}$ such that $t\geq\tau$. This implies that for
a.e. $(x_{0},t)$ and $(x_{0},\tau)\in Q_{T}$ such that $t\geq\tau$ that we
have the estimate

$p_{\infty}(X(t,x_{0}),t)\geq
p_{\infty}(X(\tau,x_{0}),\tau)\exp[-(\beta+\frac{1}{\tau})(t-\tau)].$

This implies that for a.e. $(x,t)\in Q_{T}$ with $t>0$ that $\Sigma(t)$ is
non-decreasing with respect to streamlines.

To obtain the retention property down to $t=0$, we use the classical sub-
solution $|\Pi|_{+}=\max(\Pi,0)$, as described in Section (8), to the pressure
equation for all $k>1$. This barrier was inspired by the one in [15].

######  Lemma 7.5 (Retention Property Along Streamlines For $t=0$).

Assume that $p_{k}(x,0)\rightarrow p_{\infty}(x,0)$ uniformly such that
$\Sigma(0)$ is open, we are in the congestion case $(\ref{congestion})$, and
Assumptions 1, 2, and 3. Then if $p_{\infty}(x,0)>0$ we have that
$p_{\infty}(X(t,x),t)>0$ for a.e. $(x,t)\in Q_{T}.$ That is for a.e. $(x,t)\in
Q_{T}$ that $X(t,\Sigma(0))\subset\Sigma(t)$.

###### Proof.

Let $x_{0}\in\Sigma(0)$, then there is an $r_{1}>0$ such that
$B_{r_{1}}(x_{0})\subset\Sigma(0)$. By choosing $\gamma$ and $r$ sufficiently
small in the barrier $|\Pi|_{+}$ constructed in Lemma 8.4, we can assume that
for all $k$ sufficiently large that

$|\Pi|_{+}(x,0)\leq p_{k}(x,0).$

Hence, by the comparison principle, we conclude that for all large $k$ that

$|\Pi|_{+}(x,t)\leq p_{k}(x,t)\Rightarrow|\Pi|_{+}(x,t)\leq
p_{\infty}(x,t)\text{ for a.e. }(x,t)\in Q_{T}.$

But notice $|\Pi(X(t,x_{0}),t)|_{+}=\gamma^{2}>0\Rightarrow
p_{\infty}(X(t,x_{0}),t)\geq\gamma^{2}>0$ for a.e. $(x_{0},t)\in Q_{T}$. That
is we have $X(t,\Sigma(0))\subset\Sigma(t)$ for a.e. $(x,t)\in Q_{T}$. ∎

By combining these two lemmas, we have that

######  Theorem 7.6 ($\Sigma(t)$ Increases Along Streamlines).

Assume that $p_{k}(x,0)\rightarrow p_{\infty}(x,0)$ uniformly such that
$\Sigma(0)$ is open, we are in the congestion case $(\ref{congestion})$, and
Assumptions 1, 2, and 3 are met. Then $\Sigma(t)=\\{p_{\infty}(t)>0\\}$ is
non-decreasing with respect to streamlines for a.e. $(x,t)\in Q_{T}$.

Now we can prove Theorem 2.4.

###### Proof.

As $\\{p_{\infty}(x,t)>0\\}\subset\\{\varrho_{\infty}(x,t)=m(x,t)\\}$, it
suffices to show if $(x_{0},t_{0})\in\text{Int}(\\{p_{\infty}(x,t)=0\\})$ then
$\varrho_{\infty}(x,t)=\varrho^{E}(x,t)$. So let
$(x_{0},t_{0})\in\text{Int}(\\{p_{\infty}(x,t)=0\\})$, then by Lemma 7.6, we
conclude that $X(t,X(-t_{0},x_{0}))\notin\Sigma(t)$ for $0\leq t\leq t_{0}$
(the streamline containing $(x_{0},t_{0})$ is not in $\Sigma(t)$ for previous
time). Combining this with $\varrho_{\infty}$ weakly solving

$\partial_{t}\varrho_{\infty}=\nabla\cdot(\varrho_{\infty}\vec{b})+f\varrho_{\infty}\text{
in }\text{Int}(\\{p_{\infty}(x,t)=0\\})$

we conclude that $\varrho_{\infty}(x_{0},t_{0})=\varrho^{E}(x_{0},t_{0})$. ∎

This shows Theorem 2.4. Note that Theorem 2.4 implies that patch solutions
stay patch solutions for all times for congestion.

##  8\. Appendix: Barrier

In this section we will construct some barriers for (2.2) that we used
throughout the article.

###  8.1. Super Solutions For Radial $m$ or $d=1$

We first construct a super solution to the pressure equation (2.2) for all
$k>1$ when $m$ is radial or we are in one spatial dimension.

######  Lemma 8.1.

Assume that either $m$ is radial $(m(x,t)=m(r,t)$ where $r=|x|$), or we are
working in one spatial dimension ($d=1$). Then if Assumption 1 is met , then
the radial function

$\varphi(r,t):=\frac{1}{d}\int_{0}^{r}\frac{r^{\prime}}{m(r^{\prime},t)}dr^{\prime}$

solves

(8.1) $\frac{1}{m}\nabla\cdot(m\nabla\varphi)=1\text{ in }{\mathbb{R}}^{d}$

such that there exists $C_{1},C_{2},C_{3}>0$ with
$C_{1}r^{2}\leq|\varphi(r,t)|\leq C_{2}r^{2}$ and $|\nabla\varphi(r,t)|\leq
C_{3}|r|.$

With the above $\varphi$ we can construct a viscosity super solution to the
pressure equation (2.2) for all $k>1$ when $d=1$ or $m$ is radial.

######  Definition 8.1 (Viscosity Super-Solutions).

We say that $\psi$ is a viscosity super solution to (2.2) if for all $\phi\in
C^{\infty}({\mathbb{R}}^{d})$ that touch $\psi$ from below at $(x_{0},t_{0})$
then

(8.2)
$\partial_{t}\phi\geq|\nabla\phi|^{2}+\nabla\phi\cdot\vec{b}+(k-1)\phi(\frac{1}{m}\nabla\cdot(m\nabla\phi)+F)\text{
at the point }(x_{0},t_{0}).$

######  Lemma 8.2 (Viscosity Super Solutions To The Pressure Equation for
Radial $m$ or in 1D).

Assume that $m$ is radial or we are in one spatial dimension with Assumption 1
met. Then let $\varphi(x,t)$ be as in Lemma 8.1. Now if we fix a $\gamma>1$
then there exists a large $\alpha>0$ and a smooth increasing $R(t)$ with
$R(0)=\gamma$ such that

$Z(x,t):=\alpha|R(t)-\varphi(x,t)|_{+}$

is viscosity super solution to (2.2) for all $k>1$.

###### Proof.

We note that in this lemma, it is important to keep track of the constants
that are not $\alpha$. Let us denote $U(t):=\\{x:Z(x,t)>0\\}$. We first show
that $Z$ is a classical super solution of (2.2) in $\\{Z>0\\}$. First thanks
to Lemma 8.1, we have that

$\frac{1}{m}\nabla\cdot(m\nabla Z)=-\alpha/m\text{ in }U(t).$

This implies if $\alpha$ is sufficiently large that

$\frac{1}{m}\nabla\cdot(m\nabla Z)+F\leq 0\text{ in }U(t)\quad\forall t>0.$

So by (2.2), it suffices to check that

$\partial_{t}Z\geq\alpha^{2}|\nabla\varphi|^{2}-\alpha(\nabla\varphi\cdot\vec{b})\text{
in }U(t)$

to show that $Z$ is a classical super solution in $\\{Z>0\\}$. Now by Young’s
Inequality with $M:=||\vec{b}|||_{\infty}^{2}/2$, it suffices to show that

(8.3) $\partial_{t}Z\geq\alpha^{2}\frac{3}{2}|\nabla\varphi|^{2}+M\text{ in
}U(t).$

Now we compute bounds on $\partial_{t}\varphi$ and $|\nabla\varphi|^{2}$. By
Lemma 8.1 we have that $|\nabla\varphi(r,t)|^{2}\leq C_{3}^{2}r^{2}$ with
$C_{1}r^{2}\leq|\varphi(r,t)|,$ so by choosing $K_{\varphi}:=C_{1}C_{3}^{2}$,
we have

(8.4) $|\nabla\varphi|^{2}\leq K_{\varphi}|\varphi|.$

Now to bound $\partial_{t}\varphi$, we observe from Lemma (8.1) that

$\partial_{t}\varphi(r,t)=-\frac{1}{d}\int_{0}^{r}\frac{m_{t}r^{\prime}}{m^{2}}dr^{\prime},$

so we have the bound

(8.5) $|\partial_{t}\varphi(r,t)|\leq C_{4}r^{2}\leq M_{\varphi}\varphi(r,t)$

for some $C_{4},M_{\varphi}>0$.

Now by (8.4) and (8.5) to show (8.3) it suffices to choose an $R$ such that

$\partial_{t}R\geq(\frac{3}{2}\alpha^{2}K_{\varphi}+M_{\varphi})R(t)+M.$

In particular, we can choose

$R(t)=(\gamma+M)\exp((\frac{3}{2}\alpha^{2}K_{\varphi}+M_{\varphi})t)-M,$

which satisfies all the desired conditions. Thus, $Z$ is a classical super
solution inside $\\{Z>0\\}$.

Because $0$ is a solution of the pressure equation (2.2) and $Z$ is a
classical super solution in $U(t)$, it suffices to show the viscosity touching
condition (8.2) on $\partial\\{Z>0\\}$ to conclude $Z$ is a viscosity super
solution.

Let $\phi$ be a smooth test function that touches $Z$ from below at
$(x_{0},t_{0})\in\partial\\{Z>0\\}$. We may assume that we have
$|\nabla\phi(x_{0},t_{0})|\neq 0$ as then (8.2) would be trivial because
$\partial_{t}\phi(x_{0},t_{0})\geq 0$. Indeed, observe

$Z(x_{0},t_{0}+h)-Z(x_{0},t_{0})\geq\phi(x_{0},t_{0}+h)-\phi(x_{0},t_{0}),$

which implies $\partial_{t}\phi(x_{0},t_{0})\geq 0.$

So now assume that $\phi$ touches $Z$ from below at
$(x_{0},t_{0})\in\partial\\{Z>0\\}$ such that $|\nabla\phi(x_{0},t_{0})|>0$.
First observe from (8.3) that by undoing our Young’s Inequality that

(8.6) $\partial_{t}Z\geq|\nabla Z|^{2}+(|\nabla Z|)||\vec{b}||_{\infty}\text{
in }\\{Z>0\\}.$

Now let $V_{\phi}$ and $V_{Z}$ denote the normal velocity of the free boundary
of $\partial\\{x:\phi(x,t)>0\\}$ and $\partial\\{x:Z(x,t)>0\\}$ respectively
at $(x_{0},t_{0})$ then from (8.6)

$\frac{\partial_{t}\phi(x_{0},t_{0})}{|\nabla\phi(x_{0},t_{0})|}=V_{\phi}\geq
V_{Z}\geq|\nabla Z(x_{0},t_{0})|+||\vec{b}||_{\infty}.$

Note that $\nabla Z(x_{0},t_{0})$ refers to the limit of the $\nabla Z(x,t)$
from inside $\\{Z>0\\}$ to the point $(x_{0},t_{0})$ and is non-zero because
$|\nabla\phi(x_{0},t_{0})|\leq|\nabla Z(x_{0},t_{0})|$ because $\phi$ is
touching $Z$ from below. Then this implies

$\partial_{t}\psi(x_{0},t_{0})\geq|\nabla\psi(x_{0},t_{0})|^{2}+\nabla\psi(x_{0},t_{0})\cdot\vec{b}(x_{0},t_{0}),$

which allows us to conclude that $Z$ is a viscosity super solution. ∎

######  Remark 8.2.

Assume that we can construct a $u$ such that $\nabla\cdot(m\nabla u)\geq 1$
where $C_{1}|x|^{2}\leq u(x,t)\leq C_{2}|x|^{2}$ and $|\nabla u(x,t)|\leq
C|x|$ with $|\partial_{t}u(x,t)|\leq C|x|^{2}+C$. Then the proof of Lemma
(8.2) implies that we can construct a viscosity super solution of (2.2) for
all $k>1$ that is uniformly compact support in $Q_{T}$.

###  8.2. Super Solutions When $\nabla m$ Decays Fast Enough At $\infty$

Now we use this for the case of $|\nabla m(x,t)|\leq\varepsilon/|x|$ for
$|x|\geq R/2$ to construct a positive sub-solution of the uniformly elliptic
operator $\nabla\cdot(m(x,t)\nabla u)$ in ${\mathbb{R}}^{d}$. We first
consider the unique solution of

(8.7) $\begin{cases}\nabla\cdot(m(x,t)\nabla[\phi(x,t)])=1\text{ in
}B_{R}(0)\times[0,T]\\\ \phi(x,t)=1\text{ on }\partial B_{R}(0)\end{cases}.$

Now we use a priori estimates to show that we may assume that $\phi(x,t)>0$ in
$B_{R}(0)\times[0,T]$.

By standard a priori estimates for uniformly elliptic PDEs (see [13, 11]), we
have that there is a constant $C=C(m,\nabla m,R,d)$ such that

$\sup_{(x,t)\in B_{R}(0)\times[0,T]}|\phi(x,t)|\leq C.$

So we may assume that $\phi(x,t)>0$ on $B_{R}(0)\times[0,T]$ since otherwise
we may consider $\phi(x,t)+2C$, which solves the same PDE with constant
boundary data.

Next we will obtain regularity of $\phi$ in time. First by global Schauder
estimates (see [11]), there is a $C=C(m,\nabla m,R,d)$ such that

(8.8) $\sup_{(x,t)\in B_{R}(0)\times[0,T]}|\phi(x,t)|+\sup_{(x,t)\in
B_{R}(0)\times[0,T]}\sum_{i=1}^{d}|\partial_{x_{i}}\phi(x,t)|+\sup_{(x,t)\in
B_{R}(0)\times[0,T]}\sum_{i,j=1}^{d}|\partial^{2}_{x_{i}x_{j}}\phi(x,t)|\leq
C.$

Hence, by differentiating $(\ref{ellipitic_pressure_L})$ in time, we have that

$\begin{cases}\nabla\cdot(m_{t}\nabla\phi)+\nabla\cdot(m\nabla\phi_{t})=0\text{
in }B_{R}(0)\times[0,T]\\\ \phi_{t}(x,t)=0\text{ on }\partial
B_{R}(0)\end{cases}.$

The first term, $\nabla\cdot(m_{t}\nabla\phi)$ is smooth in space and bounded
in $\overline{B_{R}(0)}\times[0,T]$, so by treating this as a known term, we
can conclude again by standard a priori arguments for uniformly elliptic PDE
that there is a $C=C(m,\nabla m,R,d)$ such that

$|\phi_{t}(x,t)|\leq C.$

Observe that for $|x|\geq R$ by Cauchy-Schwarz we have that

$\nabla\cdot\left[m\nabla(|x|^{2})\right]=2dm+2\nabla m\cdot\vec{x}\geq
2d\delta-2\varepsilon\geq\delta.$

Note that we used $m(x,t)\geq\delta>0$, $|\nabla m(x,t)|\leq\varepsilon/|x|$
for $|x|\geq R/2$, and assumed $\varepsilon=\varepsilon(\delta)$ is
sufficiently small. From $(\ref{phi-bounds})$, we can choose a $C\geq
1/\delta$ so large such that for $w(x):=1+C(|x|^{2}-R^{2})$ and if
$x\in\partial B_{R}(0)$, we have that the normal derivatives are ordered:

$\frac{\partial w}{\partial r}(x)>\frac{\partial\phi}{\partial r}(x),$

which implies

$\varphi(x,t):=\begin{cases}\phi(x,t)\text{ for }|x|\leq R\\\ w(x)\text{ for
}|x|\geq R\end{cases}$

is a viscosity solution of $\nabla\cdot(m\nabla u)\geq 1$ (since the maximum
of two viscosity sub-solutions is a viscosity sub-solution [6] and $\varphi$
cannot be touched from above on $\partial B_{R}(0)$). Also notice that
$\partial_{t}\varphi=0$ for $|x|\geq R$, and
$|\partial_{t}\varphi|=|\partial_{t}\phi|\leq C$ in $B_{R}(0)$, so
$\partial_{t}\varphi$ is bounded.

Now note that $\varphi$ satisfies all the requirements of Remark 8.2, so we
conclude

######  Lemma 8.3 (Super Solutions To The Pressure Equation when $\nabla m$
decays fast enough at $\infty$).

Assume that Assumption 1 is met and there is an $R>0$ such that $|\nabla
m(x,t)|\leq\varepsilon/|x|$ for $|x|\geq R/2$ and
$\varepsilon=\varepsilon(\delta)$ sufficiently small and non-negative. Then
for any $\gamma>1$ we have that there is an $R$ with exponential growth such
that $R(0)=\gamma$ with $\partial_{t}R\geq 0$ and

$Z(x,t):=\alpha|R(t)-\varphi(x,t)|_{+}$

is a viscosity super solution to the pressure equation (2.2) when $\alpha>0$
is sufficiently large and $k>1$. And $Z$ is bounded in $L^{\infty}(Q_{T})$ and
is compactly supported for any fixed time.

###  8.3. Subsolutions For Congestion Case

Now we construct a barrier for the pressure equation when we are in the
congestion case $(\ref{congestion})$. By a similar argument as in [15] Lemma
A.4 we have the following:

######  Lemma 8.4.

Let $x_{0}\in{\mathbb{R}}^{d}$, $L$ be the Lipschitz Constant of $\vec{b}$,
and for sufficiently small $0<\gamma\ll r$, then define

$\Pi(x,t):=\gamma^{2}-r^{2}e^{2Lt}|x-X(t,x_{0})|^{2}$

Then if we are in the congestion case $(\ref{congestion})$ and we have
Assumption 1 then $\Pi(x,t)$ is a classical sub-solution of the pressure
equation (2.2) on the set $\\{\Pi>0\\}\cap\\{0<t<T\\}$ for all $k>1$.

######  Lemma 8.5.

Let $\Pi$ be as in Lemma 8.4. Then $|\Pi|_{+}:=\max\\{\Pi,0\\}$ is a viscosity
sub-solution to the pressure equation for all $k>1$.

###### Proof.

This follows from an analogous proof in Lemma 8.2 using the normal velocity of
$\partial\\{x:\Pi(x,t)>0\\}.$ ∎

##  9\. Appendix: Weak Solutions of mPME

In this appendix, we will sketch the proofs of properties of weak solutions to
(1.1).

Note that (1.1) is a parabolic equation that degenerates on the zero level
set, so we still expect a comparison principle to hold for weak sub and super
solutions. We sketch the proof below, which is based on the proof in [19].

######  Theorem 9.1 (Comparison Principle for mPME).

Let $\overline{u}$ and $\underline{u}$ respectively be a weak super and sub
solution of (1.1) with initial datas $\overline{u}_{0}$ and
$\underline{u}_{0}$ respectively. Assume further that
$\overline{u}_{0}\geq\underline{u}_{0}$ almost everywhere, then we have
$\overline{u}\geq\underline{u}$ almost everywhere.

###### Proof.

The proof is based on a Hilbert duality argument. Let
$w:=\overline{u}-\underline{u}$, then we have for any non-negative test
function $\varphi$ that

$0\leq\iint_{Q_{T}}w\left(\partial_{t}\varphi+A\nabla\cdot(m\nabla\varphi)+f\varphi-\nabla\varphi\cdot\vec{b}\right),$

where $0\leq
A:=\frac{\overline{v}^{k}-\underline{v}^{k}}{\overline{u}-\underline{u}}$ with
$\overline{v}:=\overline{u}/m$ and $\underline{v}:=\underline{u}/m$. Then if
$\xi$ is a non-negative test function, then we want to solve the following
degenerate backwards parabolic PDE

(9.1)
$\begin{cases}\partial_{t}\varphi+A\nabla\cdot(m\nabla\varphi)+f\varphi-\nabla\varphi\cdot\vec{b}=\xi\text{
in }Q_{T}\\\ \varphi(x,T)=0\end{cases}.$

Note that solutions of (9.1) will be non-negative because 0 is a sub-solution
of (9.1). Then if we could solve (9.1), then we would have

$\iint_{Q_{T}}w\xi\geq 0$

for all non-negative test functions, which would imply $w\geq 0$ almost
everywhere.

To solve (9.1), we have to approximate $A$ by a smooth approximation
$A_{\varepsilon}$ where $\varepsilon\leq
A_{\varepsilon}\leq\frac{1}{\varepsilon}$ with $\varepsilon>0$. One can then
solve (9.1) with $A$ replaced by $A_{\varepsilon}$ by using uniform parabolic
theory. Then by following similar arguments as in [19] Theorem 3.4, one sees
that as $\varepsilon\rightarrow 0$ we obtain that $\iint_{Q_{T}}w\xi\geq 0$. ∎

Our comparison principle immediately implies that we have uniqueness of weak
solutions of (1.1). So we now focus on existence.

Our existence proof is based on the methods in [24]. We first fix a large
spatial domain $\Omega$ and construct weak solutions on a bounded domain. We
let $\Omega_{T}:=\Omega\times(0,T)$, then we want to construct a solution to

(9.2) $\begin{cases}\partial_{t}\varrho_{k}=\nabla\cdot(\varrho_{k}(\nabla
p_{k}+\vec{b}))+f\varrho_{k}\text{ in }\Omega_{T}\\\
\varrho_{k}(x,0)=\varrho_{k}^{0}(x)\text{ in }\Omega\\\
\varrho_{k}(x,t)=0\text{ on }\partial\Omega\times[0,T]\end{cases},$

where the initial data $\varrho_{k}^{0}(x)$ is compactly supported inside
$\Omega$.

######  Definition 9.1 (Weak Solutions to mPME on $\Omega_{T}$).

We say that $\varrho_{k}$ is a weak solution to (9.2) with initial data
$\varrho_{k}^{0}\in L^{\infty}(\Omega)$ that is compactly supported in
$\Omega$ if $\varrho_{k}\in L^{2}(\Omega_{T})$ and $v_{k}:=\varrho_{k}/m$
satisfies $v_{k}^{k}\in H_{0}^{1}(\Omega_{T})$ such that for all test
functions $\varphi\in C^{\infty}_{c}(\Omega\times[0,\infty))$

(9.3)
$\iint_{\Omega_{T}}-\varrho_{k}\partial_{t}\varphi=\int_{\Omega}\varphi(x,0)\varrho_{k}^{0}(x)+\iint_{\Omega_{T}}-m\nabla\varphi\cdot\nabla
v_{k}^{k}-\varrho_{k}(\nabla\varphi\cdot\vec{b})+f\varphi\varrho_{k}.$

To construct a weak solution, we solve the following approximate solution. For
$n\in\mathbb{N}$, we consider the approximate problem

(9.4)
$\begin{cases}\partial_{t}\varrho^{(n)}_{k}=\nabla\cdot(\varrho_{k}^{(n)}(\nabla
p_{k}^{(n)}+\vec{b}))+f\varrho_{k}^{(n)}\text{ in }\Omega_{T}\\\
\varrho^{(n)}_{k}(x,0)=\varrho_{k}^{0}(x)+\frac{m(x,0)}{n}\text{ in }\Omega\\\
\varrho^{(n)}_{k}(x,t)=m(x,t)/n\text{ on
}\partial\Omega\times[0,T]\end{cases}.$

We can compare $\varrho_{k}^{(n)}$ to the sub-solution $\delta
m(x,t)\exp(-\alpha t)$ for small enough $\delta>0$ and large enough
$\alpha>0$, to conclude $\varrho_{k}^{(n)}\geq\varepsilon_{n}>0$. Then this
implies $\varrho_{k}^{(n)}$ solves a uniformly parabolic PDE, so by following
the arguments in [24] Chapter 3, we know that $\varrho_{k}^{(n)}$ is a smooth
and is a classical solution to (9.2).

Now from the comparison principle, we have that
$\varrho_{k}^{(n)}\geq\varrho_{k}^{(n+1)}$, so there exists a limit
$\varrho_{k}:=\lim_{n\rightarrow\infty}\varrho_{k}^{(n)}$. By the monotone
convergence theorem, we have that $\varrho_{k}^{(n)}\rightarrow\varrho_{k}$ in
$L^{p}(\Omega_{T})$ for $p\in[1,\infty)$. So from the weak formulation of
(9.2), it suffices to show that if $v_{k}^{(n)}:=\varrho_{k}^{(n)}/m$, then
$(v_{k}^{(n)})^{k}\rightharpoonup v_{k}^{k}$ in $H^{1}(\Omega_{T})$ where
$v_{k}:=\varrho_{k}/m$. In particular, it suffices to show that
$||\nabla(v^{(n)}_{k})^{k}||_{L^{2}(\Omega_{T})}\leq C$ where $C$ is
independent of $n$. To simplify our notations, let us write
$q_{n}:=(v_{k}^{(n)})^{k}$ and $u_{n}:=\varrho_{k}^{(n)}$. Then we can
multiply (1.1) by $(q_{n}-\frac{1}{n^{k}})$ which is 0 on $\partial\Omega$ to
see by integration that $||\nabla q_{n}||_{L^{2}(\Omega_{T})}\leq C$. This
concludes the existence of weak solutions of (9.2).

Notice in particular, that this shows we can approximate our weak solutions of
(9.2) with positive smooth solutions of (9.2). Then by using our smooth
approximations we have a comparison principle for viscosity super solutions
and our weak solutions on $\Omega_{T}$. Now to go from $\Omega_{T}$ to
$Q_{T}$, we use our barriers constructed in Section 8. These barriers show
that if we choose $\Omega$ sufficiently large, then our weak solution of (9.3)
will be compactly supported inside $\Omega$. Then we can extend our weak
solution by defining to be zero outside of $\Omega$, which gives a weak
solution of (1.1).

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