# Long Time Behavior of Stochastic Thin Film Equation

Oleksiy Kapustyan<EMAIL_ADDRESS>Department of Mathematics, Taras
Shevchenko National University of Kyiv, Ukraine  Olha Martynyuk
<EMAIL_ADDRESS>Department of Mathematics, Yuriy Fedkovych Chernivtsi
National University, Chernivtsi, Ukraine  Oleksandr Misiats<EMAIL_ADDRESS>Department of Mathematics, Virginia Commonwealth University, Richmond, VA,
23284, USA  Oleksandr Stanzhytskyi<EMAIL_ADDRESS>Department of
Mathematics, Taras Shevchenko National University of Kyiv, Ukraine

###### Abstract

In this paper we consider a stochastic thin-film equation with a one
dimensional Gaussian Stratonovych noise. We establish the existence of non-
negative global weak martingale solution, and study its long time asymptotic
properties. In particular, we show the solution almost surely converges to the
average value of the initial condition. Furthermore, using the regularized
equations and adapted entropy functionals, we establish the exponential
asymptotic decay of the solution in the uniform norm.

##  1 Introduction.

We consider the stochastic thin-film equation with quadratic mobility

$du=(-\partial_{x}(u^{2}u_{xxx}))dt+\partial_{x}(u\circ d\beta)$ (1.1)

for $(t,x)\in Q_{T}:=[0,T)\times\mathbb{T}_{L}$. Here $T>0,L>0$, and
$\mathbb{T}_{L}$ is the torus on the interval $[0,L]$, with periodic boundary
conditions

$\partial_{x}^{i}u(\cdot,0)=\partial_{x}^{i}u(\cdot,L),\ i=0,1,2,3,$

and non-negative initial condition $u(0,x)=u_{0}(x)$. The term
$\partial_{x}(u\circ d\beta)$ is a stochastic perturbation in Stratonovich
form, and the process $\beta$ is a Wiener process on a complete filtered
probability space
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\},\mathcal{P}),t\in[0,T]$, with a
complete and right-continuous filtration $(\mathcal{F}_{t})_{t\in[0,T]}$.
Similarly to [12] and [17], the equation (1.1) may be re-written in the Ito’s
form

$du=\left(\partial_{x}(-u^{2}u_{xxx})+\frac{1}{2}\partial^{2}_{x}u\right)dt+\partial_{x}u\,d\beta(t).$

The equations of type (1.1) model the motion of liquid droplets of thickness
$u$, spreading over the solid surface. It can be derived from the lubrication
theory under the so-called “thin-film assumption”, that is, that the
dimensions in the horizontal directions are significantly larger than in the
vertical (normal) one. Under this assumption, the surface tension plays the
key role in the dynamics of the droplet. The regions where $u>0$ are called
the wetted regions. The equation is parabolic inside these regions, and
degenerate on the boundary of these regions. This way one may think of (1.1)
as of a fourth-order nonlinear free boundary problem inside a wetted region,
which evolves in time with finite propagation speed [3].

The mathematical analysis of equations of type (1.1) is particularly tricky
due to the lack of comparison (maximum) principle, which is one of the primary
analytical tools in classic parabolic theory. One of the pioneering works in
this direction was the paper [4] by Bernis and Friedman. In it, the authors
established the existence of a non-negative generalized weak solution using
the energy-entropy method. This solution was obtained as a limit of the
solutions of the regularized problems. It is worth noting that in this work,
the authors introduce the notion of a weak solution via an integral identity
inside wetted regions. This notion is somewhat “weaker” than the subsequent
definitions, which involve the integral identities over the entire region
$\mathbb{T}$. In [4] the authors also describe the behavior of the support of
the solution.

The work [1] addressed existence of more regular (strong or entropy)
solutions. In this work the authors also studied the asymptotic in $t$
behavior of the solution. In particular, it was shown that under certain
assumptions on the initial conditions,

$u(t,x)\to\frac{1}{L}\int_{0}^{L}u_{0}(x)\,dx,\ t\to\infty.$ (1.2)

A similar result was obtained in [5] using a somewhat different approach using
entropy estimates. Furthermore, this work established that the rate of the
aforementioned convergence is exponential. The work [28] makes use of the
energy estimates for the regularized problem to obtain a stronger result, that
is, the exponential decay of the energy

$J[u]:=\frac{1}{2}\int_{0}^{L}u_{x}^{2}(t,x)\,dx.$

The work [16] established the existence of a global trajectory attractor for
the generalized thin film equation with a nonlinear dissipative term, which
describes the relation between nonlinear absorption and spatial injection.

In this paper we consider the thin film equation with stochastic perturbation.
It is worth mentioning that one needs to be very careful when dealing with the
effect of noise on nonlocal and/or ill-posed problems. In some cases, e.g.
[20], the presence of a even small stochastic perturbation leads to a finite
time blowup while an unperturbed equation has global solution. In others [24],
the effect of the random perturbation is exactly opposite - it may lead to the
existence of a global solution while the corresponding deterministic equation
has a finite time blowup. The long time behavior of stochastically perturbed
evolution equations is typically described via the existence and properties of
invariant measures, see, e.g. [21], [23], [22], [26], [14], [18], [6].

The stochastic version of thin-film equation was first introduced in [9], it
described modeling the enhanced spreading of droplets. In the subsequent work
[13] the authors additionally take the interface potential between fluid and
substrate into account. This prevents the solution $u$ from becoming negative,
and allows to describe the coarsening and de-wetting phenomena.

The first rigorous construction of a non-negative martingale solution of the
stochastic thin film equation with Ito noise and additional interface
potential was obtained in [10]. This result was derived by constructing a
spatially discrete approximation of the solution. In [7], the author
considered a more general case of the main operator in the form
$-\partial_{x}(u^{n}u_{xxx})$ (referred as a more general mobility), and
established the conditions for the existence of a global strong solution for
this problem.

The work [12] established the existence of a nonnegative matringale solution
for (1.1) with quadratic mobility. The main tool in [12] was Trotter-Kato type
decomposition, which allowed to separate the dynamics into deterministic and
stochastic parts, which are eventually coupled. The subsequent work [8]
established the similar result for

$du=-\partial_{x}(u^{n}\partial_{x}^{3}u)\,dt+\partial_{x}(u^{\frac{n}{2}}\circ
dW)$

for $n\in[\frac{8}{3},4)$. To this end, the authors started with regularizing
the problem by replacing the mobility $u^{n}$ with
$u^{n}\to(u^{2}+\varepsilon^{2})^{n/4},\ \varepsilon>0$, which makes the
problem non-degenerate. Then, by means of Galiorkin approximations, the
authors establish the existence of a weak solution for the regularized
equation, and pass to the limit $\varepsilon\to 0$.

In [17] we established the existence of a non-negative martingale solution of
the nonlinear stochastic thin-film equation with nonlinear deterministic and
stochastic drift coefficients

$du=(-\partial_{x}(u^{2}u_{xxx})+l(u))dt+\partial_{x}(u\circ
dW)+f(u)dW_{1}(t),$

where $W$ and $W_{1}$ are independent $Q$-Wiener processes. The above equation
has both nonlinear drift $l(u)$, as well as the nonlinear stochastic Ito
perturbation $f(u)dW_{1}(t)$. Due to the presence of these two terms, the
equation is no longer in divergence form, which makes its analysis
significantly different from the one in [12] and [8].

To the best of our knowledge, the question of asymptotic long-time behavior of
solutions of stochastic thin film equation has not been addressed in
literature. Let us outline the strategy of obtaining the main result on the
long time behavior, the convergence (1.2).

Step 1. (Regularization) For any $\varepsilon>0$ consider

$du_{\varepsilon}=(-\partial_{x}(f_{\varepsilon}(u_{\varepsilon})\partial_{x}^{3}u_{\varepsilon})+\frac{1}{2}\partial_{x}^{2}u_{\varepsilon})dt+\partial_{x}u_{\varepsilon}d\beta,$
(1.3)

where

$f_{\varepsilon}(s)=\frac{s^{4}}{\varepsilon+s^{2}}$

and the initial condition

${u_{0}}_{\varepsilon}=u_{0}+\delta(\varepsilon),\ \delta(\varepsilon)\to
0\text{ as }\varepsilon\to 0.$

######  Definition 1.1.

A weak martingale solution of (1.3) with $\tilde{\mathcal{F}}_{0}$ \-
measurable initial data $u_{0,\varepsilon}\in
L^{2}(\Omega,H^{1}([0,T],\mathbb{R}_{0}^{+}))$ is a quadruple

$\\{(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathcal{F}}_{t},\tilde{\mathcal{P}}),\tilde{\beta},\tilde{u}_{0\varepsilon},\tilde{u}_{\varepsilon}\\},$

such that
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathcal{F}}_{t},\tilde{\mathcal{P}})$
is a filtered probability space, $\tilde{u}_{0\varepsilon}$ is
$\tilde{\mathcal{F}}_{0}$ measurable and has the same distribution as
$u_{0\varepsilon}$, $\tilde{\beta}$ is a real-valued
$\tilde{\mathcal{F}}_{t}$-Wiener process, and $\tilde{u}_{\varepsilon}$ is an
$\tilde{\mathcal{F}}_{t}$-adapted continuous $\mathbb{H}^{1}(\mathbb{T}_{L})$
\- valued process, such that:
1)
$\tilde{\mathbb{E}}\sup_{t\in[0,T]}\|\tilde{u}_{\varepsilon}(t)\|_{H^{1}(\mathbb{T}_{L})}^{2}<\infty$;
2) For almost all $(\omega,t)\in\tilde{\Omega}\times[0,T]$, the weak third
order derivative $\partial_{x}^{3}\tilde{u}_{\varepsilon}$ exists and
satisfies

$\tilde{\mathbb{E}}\|f_{\varepsilon}(\tilde{u}_{\varepsilon})\partial_{x}^{3}\tilde{u}_{\varepsilon}\|^{2}_{L^{2}(Q_{T})}<\infty;$

3) For any $\varphi\in C^{\infty}(\mathbb{T}_{L})$, $d\tilde{P}$ \- almost
surely, we have

$(\tilde{u}_{\varepsilon}(t,\cdot),\varphi)_{2}-(u_{0\varepsilon},\varphi)_{2}=\int_{0}^{t}\int_{0}^{L}f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{u}_{\varepsilon}(s,\cdot))\partial_{x}\varphi
dxds-\frac{1}{2}\int_{0}^{t}(\partial_{x}(\tilde{u}_{\varepsilon}(s,\cdot)),\partial_{x}\varphi)_{2}ds\\\
-\int_{0}^{t}\left(\int_{0}^{L}\tilde{u}_{\varepsilon}\partial_{x}\varphi
dx\right)d\tilde{\beta}(s).$ (1.4)

######  Definition 1.2.

A weak martingale solution of (1.1) is $\tilde{u}(t)$ which satisfies 1) and
2) of Definition 1.1, while instead of (1.4) we have

$(\tilde{u}(t,\cdot),\varphi)_{2}-(u_{0},\varphi)_{2}=\int_{0}^{t}\int_{\\{\tilde{u}(s)>0\\}}f_{\varepsilon}(\tilde{u})(\partial_{x}^{3}\tilde{u}(s,\cdot))\partial_{x}\varphi
dxds-\frac{1}{2}\int_{0}^{t}(\partial_{x}(\tilde{u}(s,\cdot)),\partial_{x}\varphi)_{2}ds\\\
-\int_{0}^{t}\left(\int_{0}^{L}\tilde{u}\partial_{x}\varphi
dx\right)d\tilde{\beta}(s)$

for all $t\in[0,T]$.

Step 2. Next we show that the solution $\tilde{u}_{\varepsilon}(t)$ can be
extended globally to $[0,\infty)$ (possibly on a different probability space).

Step 3. We apply Ito’s formula for the energy functional

$J_{\varepsilon}[\tilde{u}_{\varepsilon}]:=\frac{1}{2}\int_{0}^{L}(\partial_{x}\tilde{u}_{\varepsilon}(t,x))^{2}\,dx$

to obtain an a priori bound

$J_{\varepsilon}[\tilde{u}_{\varepsilon}(t,\cdot)]\leq
J_{\varepsilon}[\tilde{u}_{0\varepsilon}]e^{K_{\varepsilon}^{2}t}+K_{\varepsilon}C(\varepsilon)\int_{0}^{t}\int_{0}^{L}(\partial_{x}^{2}\tilde{u}_{\varepsilon})^{2}\,dxds$
(1.5)

for some positive constants $K_{\varepsilon}$ and $C(\varepsilon)$ almost
surely.

Step 4. For any $T>0$ we pass to limit as $\varepsilon\to 0$ on $[0,T]$ in
(1.3), hence establishing the existence of a global solution for (1.1) in the
sense of the Definition 1.2. Furthermore, we show
$\partial_{x}\tilde{u}_{\varepsilon}\rightharpoonup\partial_{x}\tilde{u},\
\varepsilon\to 0$, and
$\sup_{t\in[0,T]}\int_{0}^{L}(\partial_{x}^{2}\tilde{u}_{\varepsilon})^{2}\,dx$
is bounded a.s. uniformly in $\varepsilon$.

Step 5. We conclude with passage to the limit in (1.5) as $\varepsilon\to 0$
and obtain the desired result.

The paper is structured as follows: in Section 2 we introduce the notation,
list preliminary results, formulate the main results. In Sections 3 through 5
we derive the local-in-time solution of the regularized problem, and study its
properties. In Section 6 we show that the local solution of the regularized
problem, obtained in the previous sections, can be extended globally for all
$t\geq 0$. Finally, in Section 7 we pass to the limit in the regularized
problem, and therefore complete the proof of the main result.

##  2 Preliminaries and main results.

Throughout the paper, we will be using the following notation. For $u,v\in
L^{2}(\mathbb{T}_{L})$, let

$(u,v)_{2}:=\int_{0}^{L}u(x)v(x)dx:=\int_{L}u(x)v(x)dx,\text{ and
}\|u\|_{2}:=\sqrt{(u,u)_{2}},$

where $\int_{L}(\cdot)\,dx$ will be denoting $\int_{0}^{L}(\cdot)\,dx$. Next,
for $Q\in\mathbb{R}^{d}$ with $\partial Q\in C^{\infty}$, for
$s\in[0,\infty)$, $p\in[1,\infty)$, let $W^{s,p}(Q)$ be the regular Sobolev
space for $s\in\mathbb{N}$, and Sobolev-Slobodeckij space for non-integer $s$.
For $p=2$ we will denote $W^{s,p}(Q)=H^{s}(Q)=H^{s}$. If $X$ is a Banach
space, the space $C^{k+\alpha}(Q;X)$ is the space of $k$ times differentiable
functions $Q\to X$, whose $k$-th derivatives are Holder-continous with
exponent $\alpha\in(0,1)$ on compact subsets of $Q$. We also denote
$C^{k-}(Q;X)$ to be the space of $k-1$-times differentiable functions, whose
$k-1$-st derivatives are Lipschitz continuous. The space $BC^{0}(Q,X)$ is the
set of bounded continuous functions. We also denote
$H^{1}_{w}(\mathbb{T}_{L})$ to be the space of $H^{1}(\mathbb{T}_{L})$
functions endowed with the weak topology, induced by $\|\cdot\|_{1,2}$.
Finally, for $u:Q_{T}\to\mathbb{R}$ we denote

$P_{T}:=\\{(t,x)\in\bar{Q}_{T},u(t,x)>0\\}.$

Our main result is an immediate consequence of the theorem below:

######  Theorem 2.1.

Asuume $u_{0}\in H^{1}$ is such that $u_{0}\not\equiv 0$, $u_{0}\geq 0$ and

$\int_{L}|\ln u_{0}|\,dx<\infty.$ (2.1)

Then the equation (1.1) has a weak martingale solution for $t\geq 0$ such that

$J[u(t)]\leq J[u_{0}]e^{-Ct},\ t\geq 0\text{ a.s.},$ (2.2)

where $C>0$ depends only on $J[u_{0}]$.

Indeed, due to the conservation of mass property

$\overline{u_{0}}:=\frac{1}{L}\int_{L}u_{0}\,dx=\frac{1}{L}\int_{L}u(t,x)\,dx=:\overline{u(t)},\
t\geq 0,$

the Sobolev embedding and Poincare inequality imply

$\|u(t,x)-\overline{u_{0}}\|_{\infty}^{2}=\|u(t,x)-\overline{u(t)}\|_{\infty}^{2}\leq
C\|u(t,\cdot)-\overline{u(t)}\|_{H^{1}}\leq C_{1}\int_{L}u_{x}^{2}(t,x)\,dx\to
0$

with probability 1. This way, for large enough $t$ the solution of (1.1)
becomes nonnegative and converges to the average value of $u_{0}$.

######  Remark 2.2.

Theorem 2.1 can be extended to the case of random initial data $u_{0}$, which
satisfies $u_{0}\in
L^{q}(\Omega,\mathcal{F}_{0},\mathbb{P},H^{1}(\mathbb{T}_{L}))$ for $q\geq 2$
large enough (see Proposition 3.5 and formula (3.13) for the precise condition
on $q$) and $u_{0}\geq 0$ $\mathbb{P}$ \- almost surely. Furthermore, if

$\mathbb{E}\int_{L}|\ln u_{0}|\,dx<\infty,$

the estimate (2.2) holds as well.

##  3 The analysis of regularized problem.

For any $\varepsilon>0$ consider the regularized equation

$du_{\varepsilon}=\left(-\partial_{x}\left(f_{\varepsilon}(u_{\varepsilon})\partial_{x}^{3}u_{\varepsilon}\right)+\frac{1}{2}\partial_{x}^{2}u_{\varepsilon}\right)dt+\partial_{x}u_{\varepsilon}d\beta,$
(3.1)

along with the regularized initial condition

$u_{0\varepsilon}=u_{0}+\delta(\varepsilon)=u_{0}+\varepsilon^{\theta}.$ (3.2)

Here the random initial condition $u_{0\varepsilon}$ satisfies
$u_{0}(\omega)\in H^{1}$, $u_{0}\geq 0$ a.s., and
$\mathbb{E}\|u_{0}\|^{q}_{1,2}<\infty$ for sufficiently large $q\geq 2$
(specified in Proposition 3.5), as well as $\theta\in(0,\frac{2}{5})$. The
function $f_{\varepsilon}(s)=\frac{s^{4}}{\varepsilon+s^{2}}$ clearly
satisfies $f_{\varepsilon}(s)\leq s^{2}$, as well as $f_{\varepsilon}(s)$ is
non-decreasing. Furthermore, as follows from [5], $f_{\varepsilon}(s)\to
s^{2}$ and $f_{\varepsilon}^{{}^{\prime}}(s)\to 2s$ uniformly on $[0,M]$ as
$\varepsilon\to 0$, for any $M>0$.

######  Theorem 3.1.

For any $\varepsilon>0$ and $T>0$ the problem (3.1) has a weak strictly
positive martingale solution on $[0,T]$ in the sense of Definition 1.1.
Furthermore, this solution has the mass preservation property

$\int_{L}u_{\varepsilon}(t,x)\,dx=\int_{L}u_{0\varepsilon}(x)\,dx.$

###### Proof.

The strategy of the proof is to apply Trotter-Kato type decomposition of the
dynamics into deterministic and stochastic ones. This method was used by a
number of authors, e.g. [12, 19], in particular, to establish the existence of
for SPDEs with localy Lipschitz coefficients. For the convenience of the
reader, we split the proof into the steps below.
Step 1: Deterministic dynamics. Consider the following deterministic equation:

$\begin{cases}\partial_{t}v_{\varepsilon}=-\partial_{x}(f_{\varepsilon}(v_{\varepsilon})\partial_{x}^{3}v_{\varepsilon});\\\
v_{\varepsilon}(0):=v_{\varepsilon 0}.\end{cases}$ (3.3)

It follows from [4, 1] that (3.3) has a unique positive smooth solution
$v_{\varepsilon}$ satisfying

$\|\partial_{x}v_{\varepsilon}\|_{2}^{2}+2\int_{0}^{t}\int_{L}f_{\varepsilon}(v_{\varepsilon})(\partial_{x}^{3}v_{\varepsilon})^{2}\,dxds=\|\partial_{x}v_{0\varepsilon}\|_{2}^{2},\
t\in[0,T].$

Hence, for any $p\geq 2$,

$\|\partial_{x}v_{\varepsilon}\|_{2}^{p}+2\int_{0}^{t}\|\partial_{x}v_{\varepsilon}(s,\cdot)\|_{2}^{p-2}\int_{L}f_{\varepsilon}(v_{\varepsilon})(\partial_{x}^{3}v_{\varepsilon})^{2}\,dxds\\\
\leq\sup_{s\in[0,t]}\|\partial_{x}v_{\varepsilon}\|_{2}^{p-2}\left(\|\partial_{x}v_{\varepsilon}\|_{2}^{2}+2\int_{0}^{t}\int_{L}f_{\varepsilon}(v_{\varepsilon})(\partial_{x}^{3}v_{\varepsilon})^{2}\,dxds\right)\leq\|\partial_{x}v_{0\varepsilon}\|_{2}^{p-2}\|\partial_{x}v_{0\varepsilon}\|_{2}^{2}=\|\partial_{x}v_{0\varepsilon}\|_{2}^{p}.$
(3.4)

Step 2: Stochastic dynamics. Let
$(\Omega,{\mathcal{F}},({\mathcal{F}}_{t})_{t\in[0,T]},\mathbb{P})$ be a
filtered probability space, where $({\mathcal{F}}_{t})_{t\in[0,T]}$ is a
complete and right-continuous filtration, and $\beta(t)$ is standard real-
valued ${\mathcal{F}}_{t}$ -Wiener process. Consider

$\begin{cases}dw=\frac{1}{2}\partial_{x}^{2}w\,dt+\partial_{x}w\,d\beta(t)\
\text{ on }[0,T];\\\ w(0)=w_{0}\in
L^{p}(\Omega,\mathcal{F}_{0},\mathbb{P};H^{1}(\mathbb{T}_{L})).\end{cases}$
(3.5)

######  Definition 3.2.

An ${\mathcal{F}}_{t}$ \- adapted random process $w(t)$ is called a weak
solution of (3.5) on $t\in[0,T]$ if
$\mathbb{E}\sup_{t\in[0,T]}\|w(t)\|^{2}_{H^{1}}<\infty$ and for any
$\varphi\in C^{\infty}(\mathbb{T}_{L})$ we have

$(w(t),\varphi)_{2}-(w_{0},\varphi)_{2}=-\frac{1}{2}\int_{0}^{t}(\partial_{x}w,\partial_{x}\varphi)_{2}ds-\int_{0}^{t}(w,\partial_{x}\varphi)_{2}d{\beta}(s)$

$\mathbb{P}$ \- almost surely for all $t\in[0,T]$.

######  Lemma 3.3.

[12] For any $p\in[2,\infty)$ and any $w_{0}\in
L^{p}(\Omega,\mathcal{F}_{0},\mathbb{P};H^{1}(\mathbb{T}_{L}))$ there exists a
unique weak solution of (3.5) on $[0,T]$ with initial value $w_{0}$ satisfying

$\mathbb{E}\left(\sup_{t\in[0,T]}\|w(t)\|_{1,2}^{p}+\int_{0}^{T}\|w\|_{2,2}^{2}\,dt\right)<\infty,$
$\mathbb{E}\left(\sup_{t\in[0,T]}\|w(t)\|_{1,2}^{p}\right)\leq
C_{1}\mathbb{E}\|w_{0}\|_{1,2}^{p},$ (3.6)

and

$\|\partial_{x}w(t)\|_{2}^{2}=\|\partial_{x}w_{0}\|_{2}^{2},$ (3.7)

where $C_{1}>0$ is independent of $\omega$ and $\omega_{0}$. Furthermore,
1) the mass is conserved, i.e. $\int_{L}w(t,x)\,dx=\int_{L}w\,dx$ for
$t\in[0,T]$ $\mathbb{P}$-almost surely.
2) If $w_{0}\geq 0$ $\mathbb{P}$ -almost surely, then $w(t,x,\omega)\geq 0$
$\mathbb{P}$ -almost surely.

###### Proof.

Apart from (3.7), the rest of the statements follow from Proposition A.2 [12].
In order to obtain (3.7), using Ito’s formula we have

$\|\partial_{x}w(t)\|_{2}^{2}=\int_{L}(\partial_{x}w)^{2}\,dx=\int_{L}\partial_{x}w(0)\,dx-\int_{0}^{t}\int_{L}(\partial_{x}^{2}w)^{2}\,dxds+\int_{0}^{t}\int_{L}(\partial_{x}^{2}w)^{2}\,dxds\\\
+2\int_{0}^{t}\int_{L}\partial_{x}w\cdot\partial_{x}^{2}w\,dxd\beta(s)=\int_{L}\partial_{x}w(0)\,dx.$

∎

We now proceed with the realization of the Trotter-Kato splitting scheme. To
this end, for fixed $N\geq 1$, we equi-partition the time interval $[0,T]$
into intervals of length $\delta=\frac{T}{N+1}$. Then, for
$t\in[(j-1)\delta,j\delta]$ (i.e. locally) and for an arbitrary $\varphi\in
C^{\infty}(\mathbb{\mathbb{T}_{L}})$ we define:

Deterministic Dynamics (D) We look for the function $v_{N}$ which for
$t\in[(j-1)\delta,j\delta]$ satisfies

$(v_{N}^{\varepsilon}(t,\cdot),\varphi)_{2}-(v_{N}^{\varepsilon}((j-1)\delta,\cdot),\varphi)_{2}=\int_{(j-1)\delta}^{t}\int_{L}f_{\varepsilon}(v_{N}^{\varepsilon})\partial_{x}^{3}v_{N}^{\varepsilon}(s,\cdot)(\partial_{x}\varphi)dxds,$
(3.8)

where $j=1,...,N+1$ and $\varphi\in C^{\infty}(\mathbb{T}_{L})$.

Stochastic Dynamics (S) We look for the function $w_{N}$ satisfying

$(w_{N}(t,\cdot),\varphi)_{2}-(w_{N}((j-1)\delta,\cdot),\varphi)_{2}=-\frac{1}{2}\int_{(j-1)\delta}^{t}(\partial_{x}w_{N}(s,\cdot),\partial_{x}\varphi)_{2}ds\\\
-\int_{(j-1)\delta}^{t}(w_{N}(s,\cdot),\partial_{x}\varphi)_{2}d\beta(s).$
(3.9)

Deterministic-Stochastic Connection (DS) We set

$v_{N}(0):=u_{0\varepsilon},\ v_{N}(j\delta,\cdot):=\lim_{t\to
j\delta-}w_{N}(t,\cdot)\text{ and }w_{N}((j-1)\delta,\cdot):=\lim_{t\to
j\delta-}v_{N}(t,\cdot)\text{ a.s. }$ (3.10)

As we showed above, (3.8) and (3.9) are well posed. Furthermore, it is
straightforward to verify that both of these equations conserve mass, that is
$\int_{L}v_{N}^{\varepsilon}\,dx$ and $\int_{L}w_{N}\,dx$ are independent of
time. In a similar way to [12], we define the concatenated approximate
solution
$u_{N}^{\varepsilon}:[0,T)\times\mathbb{T}_{L}\times\Omega\to[0,\infty)$ as

$u_{N}^{\varepsilon}(t,\cdot):=\begin{cases}v_{N}^{\varepsilon}(2t-(\gamma-1)\delta,\cdot),\
t\in[(j-1)\delta,(j-\frac{1}{2})\delta),\\\ w_{N}(2t-j\delta,\cdot),\
t\in[(j-\frac{1}{2})\delta,j\delta),\ \ j=1,...,N+1.\end{cases}$

This way $u_{N}^{\varepsilon}(t,\cdot)$ is well defined, and satisfies the
mass preservation property
$\int_{L}u_{N}^{\varepsilon}\,dx=\int_{L}u_{0,\varepsilon}\,dx$ in $[0,T]$
$\mathbb{P}$\- almost surely, and $u_{N}^{\varepsilon}\geq 0$ for every
$t\in[0,T]$. Furthermore, the following proposition holds:

######  Proposition 3.4.

For any $p\in[2,\infty)$ there exists a constant $C>0$ such that for all
$N\geq 1$ and $\varepsilon>0$ we have

$v_{N}^{\varepsilon},\ w_{N}^{\varepsilon}\in
L^{p}(\Omega,\mathcal{F},\mathbb{P};L^{\infty}([0,T);H^{1}(\mathbb{T}_{L}))$

satisfying

$\text{ esssup }_{t\in[0,T)}\|u_{N}^{\varepsilon}(t)\|_{1,2}^{p}+\text{ esssup
}_{t\in[0,T)}\|v_{N}^{\varepsilon}(t)\|_{1,2}^{p}+\text{ esssup
}_{t\in[0,T)}\|w_{N}^{\varepsilon}(t)\|_{1,2}^{p}\\\
+\int_{0}^{t}\|v_{N}^{\varepsilon}(t)\|_{1,2}^{p-2}\int_{L}f_{\varepsilon}(v_{N}^{\varepsilon})(\partial_{x}^{3}v_{N}^{\varepsilon})^{2}\,dxdt\leq
C\|u_{0\varepsilon}\|_{1,2}^{p}.$ (3.11)

###### Proof.

By construction, $u_{N}^{\varepsilon}$ is continuous in $t$, and

$\int_{L}u_{0,\varepsilon}\,dx=\int_{L}u_{N}^{\varepsilon}\,dx=\int_{L}v_{N}^{\varepsilon}\,dx=\int_{L}w_{N}^{\varepsilon}\,dx$

for all $t\in[0,T)$, $\mathbb{P}$-almost surely. Next, we estimate the
derivatives $\partial_{x}v_{N}^{\varepsilon}(j\delta)$ and
$\partial_{x}w_{N}^{\varepsilon}(j\delta)$. Using (3.10) and (3.7) we have

$\|\partial_{x}v_{N}^{\varepsilon}(\delta)\|_{2}^{p}=\lim_{t\to\delta-}\|\partial_{x}w_{N}^{\varepsilon}(t)\|_{2}^{p}=\|\partial_{x}w_{N}^{\varepsilon}(0)\|_{2}^{p}=\lim_{t\to\delta-}\|\partial_{x}v_{N}^{\varepsilon}(t)\|_{2}^{p}\leq\|\partial_{x}v_{0,\varepsilon}\|_{2}^{p}.$

Using (3.4),

$\|\partial_{x}w_{N}^{\varepsilon}(\delta)\|_{2}^{p}=\lim_{t\to
2\delta-}\|\partial_{x}v_{N}^{\varepsilon}(t)\|_{2}^{p}\leq\|\partial_{x}v_{N}^{\varepsilon}(\delta)\|_{2}^{p}\leq\|\partial_{x}v_{0,\varepsilon}\|_{2}^{p}.$
(3.12)

Similarly,

$\|\partial_{x}v_{N}^{\varepsilon}(2\delta)\|_{2}^{p}=\lim_{t\to
2\delta-}\|\partial_{x}w_{N}^{\varepsilon}(t)\|_{2}^{p}=\|\partial_{x}w_{N}^{\varepsilon}(\delta)\|_{2}^{p}\leq\|\partial_{x}v_{0,\varepsilon}\|_{2}^{p}$

by (3.12), and

$\|\partial_{x}w_{N}^{\varepsilon}(2\delta)\|_{2}^{p}=\lim_{t\to
2\delta-}\|\partial_{x}v_{N}^{\varepsilon}(t)\|_{2}^{p}\leq\|\partial_{x}v_{N}^{\varepsilon}(\delta)\|_{2}^{p}\leq\|\partial_{x}v_{0,\varepsilon}\|_{2}^{p}.$

Therefore, for any $j=1,...,N$ we get

$\|\partial_{x}v_{N}^{\varepsilon}(j\delta)\|_{2}^{p}\leq\|\partial_{x}v_{0,\varepsilon}\|_{2}^{p}\text{
and
}\|\partial_{x}w_{N}^{\varepsilon}(j\delta)\|_{2}^{p}\leq\|\partial_{x}v_{0,\varepsilon}\|_{2}^{p}.$

We my now use (3.4) and (3.7), combined with Poincare inequality and the
conservation of mass property, we get the estimate (3.11) for
$v_{N}^{\varepsilon}$ and $w_{N}^{\varepsilon}$, and hence for
$u_{N}^{\varepsilon}$. It remains to estimate the integral term in (3.11). To
simplify the presentation, denote

$\int_{0}^{T}\|v_{N}^{\varepsilon}(t)\|_{1,2}^{p-2}\int_{L}f_{\varepsilon}(v_{N}^{\varepsilon})(\partial_{x}^{3}v_{N}^{\varepsilon})^{2}\,dxdt:=\int_{0}^{T}(\cdot)\,dt.$

It follows from (3.4) that

$2\int_{0}^{\delta}(\cdot)\,dt\leq\|\partial_{x}v_{0\varepsilon}\|_{2}^{p}-\|\partial_{x}v_{N}^{\varepsilon}(\delta-0)\|_{2}^{p},$

and

$2\int_{\delta}^{2\delta}(\cdot)\,dt\leq\|\partial_{x}v_{N}^{\varepsilon}(\delta+0)\|_{2}^{p}-\|\partial_{x}v_{N}^{\varepsilon}(2\delta-0)\|_{2}^{p}=\|\partial_{x}w_{N}^{\varepsilon}(\delta-0)\|_{2}^{p}-\|\partial_{x}v_{N}^{\varepsilon}(2\delta-0)\|_{2}^{p}\\\
=\|\partial_{x}v_{N}^{\varepsilon}(\delta-0)\|_{2}^{p}-\|\partial_{x}v_{N}^{\varepsilon}(2\delta-0)\|_{2}^{p}.$

Thus

$2\int_{\delta}^{2\delta}(\cdot)\,dt\leq\|\partial_{x}v_{0\varepsilon}\|_{2}^{p}-\|\partial_{x}v_{N}^{\varepsilon}(2\delta-0)\|_{2}^{p}.$

Next, arguing similarly,

$2\int_{2\delta}^{3\delta}(\cdot)\,dt\leq\|\partial_{x}v_{N}^{\varepsilon}(2\delta+0)\|_{2}^{p}-\|\partial_{x}v_{N}^{\varepsilon}(3\delta-0)\|_{2}^{p}=\|\partial_{x}w_{N}^{\varepsilon}(2\delta-0)\|_{2}^{p}-\|\partial_{x}v_{N}^{\varepsilon}(3\delta-0)\|_{2}^{p}\\\
=\|\partial_{x}w_{N}^{\varepsilon}(\delta+0)\|_{2}^{p}-\|\partial_{x}v_{N}^{\varepsilon}(3\delta-0)\|_{2}^{p}=\|\partial_{x}v_{N}^{\varepsilon}(2\delta-0)\|_{2}^{p}-\|\partial_{x}v_{N}^{\varepsilon}(3\delta-0)\|_{2}^{p},$

hence

$2\int_{\delta}^{3\delta}(\cdot)\,dt\leq\|\partial_{x}v_{0\varepsilon}\|_{2}^{p}-\|\partial_{x}v_{N}^{\varepsilon}(3\delta-0)\|_{2}^{p}.$

Continuing this way on the rest of the intervals, and using (3.4), (3.6) and
Poincare inequality, we get (3.11). ∎

For any Banach space $X$ and $D\subset\mathbb{R}^{d}$, denote
$B_{q}^{s,p}(D;X)$ be the Besov space, defined, e.g., in [2] or [27]. The
space $B_{q}^{s,p}(\mathbb{T}_{L};\mathbb{R})$ will be denoted with
$B(\mathbb{T}_{L})$. We will now establish the analog of Proposition 4.2 [12].

######  Proposition 3.5.

For any $p\geq 2,\sigma>0$,
$\kappa\in(2\sigma,\frac{2}{p})\cap(2\sigma,\frac{1}{2}]$ and
$q\in\left(\frac{2}{\kappa-2\delta},\infty\right)$, there exists $C>0$ such
that for all $N\geq 1$ we have

$u_{N}^{\varepsilon}\in
L^{p}(\Omega,\mathcal{F},\mathbb{P};B_{q}^{k/2-\sigma,q}([0,T);B_{q}^{1/2-2\kappa,q}(\mathbb{T}_{L})),$

and

$\mathbb{E}\|u_{N}^{\varepsilon}\|^{p}_{B_{q}^{k/2-\sigma,q}([0,T);B_{q}^{1/2-2\kappa,q}(\mathbb{T}_{L}))}\leq
C\mathbb{E}(\|u_{0,\varepsilon}\|_{1,2}^{p}+\|u_{0,\varepsilon}\|_{1,2}^{(\kappa+1)p}).$
(3.13)

###### Proof.

The proposition will be established for $v_{N}^{\varepsilon}$ and
$w_{N}^{\varepsilon}$ separately. Let us start with the estimate for
$v_{N}^{\varepsilon}$.

######  Lemma 3.6.

For any $p\geq 2$, $\sigma>0$ and $q\geq p$ there exists a constant $C>0$ such
that for all $N\geq 1$, $1\leq j\leq N+1$ and $\kappa\in(0,\frac{2}{p})$ we
have

$v_{N}^{\varepsilon}\in
B_{q}^{k/2-\sigma,q}([(j-1)\delta,j\delta);B_{q}^{1/2-2\kappa,q}(\mathbb{T}_{L})),$

with

$\mathbb{E}\left(\sum_{j=1}^{N+1}\|v_{N}^{\varepsilon}\|^{p}_{B_{q}^{k/2-\sigma,q}([0,T);B_{q}^{1/2-2\kappa,q}(\mathbb{T}_{L}))}\right)^{\frac{p}{q}}\leq
C\mathbb{E}(\|u_{0,\varepsilon}\|_{1,2}^{p}+\|u_{0,\varepsilon}\|_{1,2}^{(\kappa+1)p}).$

###### Proof.

For any $(j-1)\delta\leq t_{1}\leq t_{2}<j\delta$ and $\varphi\in
C^{\infty}(\mathbb{T}_{L})$ we have

$(v_{N}^{\varepsilon}(t_{2})-v_{N}^{\varepsilon}(t_{1}),\varphi)_{2}=\int_{t_{1}}^{t_{2}}\int_{L}f_{\varepsilon}(v_{N}^{\varepsilon})\partial_{x}^{3}v_{N}^{\varepsilon}(s,\cdot)(\partial_{x}\varphi)dxds.$

This $\mathbb{P}$ -almost surely yields

$\displaystyle\|v_{N}^{\varepsilon}(t_{2})-v_{N}^{\varepsilon}(t_{1})\|^{2}_{H^{-1}(\mathbb{T}_{L})}\leq\left(\int_{t_{1}}^{t_{2}}\left(\int_{L}f_{\varepsilon}^{2}(v_{N}^{\varepsilon})\partial_{x}^{3}(v_{N}^{\varepsilon}(s,\cdot))^{2}dx\right)^{1/2}ds\right)^{2}$
$\displaystyle\leq\left(\int_{t_{1}}^{t_{2}}\sup_{x\in[0,L]}|f_{\varepsilon}^{2}(v_{N}^{\varepsilon})|^{1/2}\left(\int_{L}f_{\varepsilon}(v_{N}^{\varepsilon})\partial_{x}^{3}(v_{N}^{\varepsilon}(s,\cdot))^{2}dx\right)^{1/2}ds\right)^{2}$
$\displaystyle\leq
C(t_{2}-t_{1})\int_{t_{1}}^{t_{2}}\|v_{N}^{\varepsilon}\|_{1,2}\int_{L}f_{\varepsilon}^{2}(v_{N}^{\varepsilon})\partial_{x}^{3}(v_{N}^{\varepsilon}(s,\cdot))^{2}dx\
ds,$

where we used the Sobolev embedding, Cauchy-Schwartz inequality, and the
inequality $f_{\varepsilon}(s)\leq s^{2}$. Thus, using (3.11) we have

$\|v_{N}^{\varepsilon}\|_{C^{1/2}([(j-1)\delta,j\delta);H^{-1}(\mathbb{T}_{L})}\leq
C\|u_{0,\varepsilon}\|_{1,2}^{2}$

$\mathbb{P}$ -almost surely, where $C$ is deterministic constant from Poincare
inequality. The rest of the proof of Lemma 3.6 and Proposition 3.5 is
analogous to the proof of Proposition 4.2 [12]. ∎

##  4 Convergence of the splitting scheme

We start with the proposition, which is an analog of Proposition 5.2 [12].

######  Proposition 4.1.

Denote

$\mathcal{X}_{u}:=BC^{0}([0,T]\times\mathbb{T}_{L}),$
$\mathcal{X}_{J}:=L^{2}([0,T]\times\mathbb{T}_{L})\text{ (with weak topology),
}$ $\mathcal{X}_{\beta}:=BC^{0}([0,T];\mathbb{R}).$

Then there exist random variables $\tilde{u}^{\varepsilon}$,
$\tilde{u}_{N}^{\varepsilon}:[0,1]\to\mathcal{X}_{u}$,
$\tilde{J}_{N}^{\varepsilon},\tilde{J}^{\varepsilon}:[0,1]\to\mathcal{X}_{J}$
and $\tilde{\beta}^{\prime}_{N},\tilde{\beta}:[0,1]\to\mathcal{X}_{\beta}$
with
$(\tilde{u}_{N}^{\varepsilon},\tilde{J}_{N}^{\varepsilon},\tilde{\beta}^{\prime}_{N})\sim(u_{N}^{\varepsilon},J_{N}^{\varepsilon},\beta)$,
where
$J_{N}:=\chi_{v_{N}^{\varepsilon}>0}f_{\varepsilon}(v_{N}^{\varepsilon})(\partial_{x}^{3}v_{N}^{\varepsilon})$.
Furthermore, there are subsequences (still indexed with $N$), such that
$\tilde{u}_{N}^{\varepsilon}(\omega)\to\tilde{u}^{\varepsilon}(\omega)$ in
$\mathcal{X}_{u}$,
$\tilde{J}_{N}^{\varepsilon}(\omega)\rightharpoonup\tilde{J}^{\varepsilon}(\omega)$
in $\mathcal{X}_{J}$ and
$\tilde{\beta}^{\prime}_{N}(\omega)\rightharpoonup\tilde{\beta}(\omega)$ in
$\mathcal{X}_{\beta}$ for every $\omega\in[0,1]$ as $N\to\infty$.

###### Proof.

Apart from the convergence of $\tilde{J}_{N}^{\varepsilon}$, the proofs of the
rest of the statements are analogous to Proposition 5.2 [12]. In order to show
the tightness of $J_{N}^{\varepsilon}$, the Sobolev embedding and the
inequality $f_{\varepsilon}(s)\leq s^{2}$ imply

$\mathbb{P}\\{\|J_{N}^{\varepsilon}\|_{L^{2}([0,T)\times\mathbb{T}_{L})}>R\\}\leq\frac{1}{R}\int_{0}^{T}\mathbb{E}\int_{L}f_{\varepsilon}^{2}(v_{N}^{\varepsilon})(\partial_{x}^{3}v_{N}^{\varepsilon})^{2}\,dxdt\\\
\leq\frac{C}{R^{2}}\mathbb{E}\int_{0}^{T}\|v_{N}^{\varepsilon}\|_{1,2}^{2}\int_{L}f_{\varepsilon}(v_{N}^{\varepsilon})(\partial_{x}^{3}v_{N}^{\varepsilon})^{2}\,dxdt\leq\frac{C}{R^{2}}\mathbb{E}\|\partial_{x}v_{0,\varepsilon}\|_{2}^{4}\to
0,\text{ as }R\to\infty,\text{ uniformly in }N,$

as well as $\\{\|J_{N}^{\varepsilon}\|_{L^{2}([0,T)\times\mathbb{T}_{L})}\leq
R\\}$ is weakly compact in $L^{2}([0,T)\times\mathbb{T}_{L})$. ∎

We will next need the analog of Proposition 5.6 from [12].

######  Proposition 4.2.

Let
$\tilde{u}_{N}^{\varepsilon},\tilde{u}^{\varepsilon},\tilde{J}_{\varepsilon}^{N}$
and $\tilde{J}_{\varepsilon}$ be as in Proposition 4.1. Then the
distributional derivative $\partial_{x}^{3}\tilde{u}_{\varepsilon}$ satisfies
$\partial_{x}^{3}\tilde{u}_{\varepsilon}\in
L^{2}(\\{\tilde{u}_{\varepsilon}>r\\})$ for any $r>0$. Furthermore,
$\tilde{J}_{N}^{\varepsilon}=\chi_{\tilde{v}_{N}^{\varepsilon}>0}f_{\varepsilon}(\tilde{v}_{N}^{\varepsilon})(\partial_{x}^{3}\tilde{v}_{N}^{\varepsilon})$
and
$\tilde{J}_{\varepsilon}=\chi_{\tilde{u}_{\varepsilon}>0}f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{u}_{\varepsilon})$
$\mathbb{P}$-almost surely.

###### Proof.

The proof of this proposition is similar to Proposition 5.6 [12] using the
inequality $f_{\varepsilon}(s)\leq s^{2}$ and monotonicity of
$f_{\varepsilon}(s)$ for every $\varepsilon>0$. ∎

######  Corollary 4.3.

For $\tilde{u}_{n}^{\varepsilon}$, $\tilde{v}_{n}^{\varepsilon}$,
$\tilde{w}_{n}^{\varepsilon}$ and $\tilde{u}_{\varepsilon}$ as in Proposition
4.1, we have

$\displaystyle\|\tilde{u}_{n}^{\varepsilon}-\tilde{u}_{\varepsilon}\|_{BC^{0}([0,T)\times\mathbb{T}_{L})}\to
0,$ (4.1)
$\displaystyle\|\tilde{v}_{n}^{\varepsilon}-\tilde{u}_{\varepsilon}\|_{L^{\infty}([0,T)\times\mathbb{T}_{L})}\to
0,$
$\displaystyle\|\tilde{u}_{n}^{\varepsilon}-\tilde{u}_{\varepsilon}\|_{L^{\infty}([0,T)\times\mathbb{T}_{L})}\to
0,$

as $N\to\infty$ $\mathbb{P}$-almost surely.

We may proceed with the scheme (D)-(S)-(DS) to conclude that for $t\in[0,T)$
and $\delta=\frac{T}{N+1}$

$\displaystyle(v_{N}^{\varepsilon}(t),\varphi)_{2}-(u_{0,\varepsilon},\varphi)_{2}=(v_{N}(t),\varphi)_{2}+\sum_{j=1}^{\left[\frac{t}{\delta}\right]}\left(-(v_{N}^{\varepsilon}(j\delta,\cdot),\varphi)_{2}+\lim_{s\to
j\delta}(w_{N}^{\varepsilon}(s,\cdot),\varphi)_{2}\right)$
$\displaystyle+\sum_{j=1}^{\left[\frac{t}{\delta}\right]}(\lim_{s\to
j\delta}(v_{N}^{\varepsilon}(s,\cdot),\varphi)_{2}-(w_{N}^{\varepsilon}((j-1)\delta,\cdot),\varphi)_{2}-(v_{N}^{\varepsilon}(0,\cdot),\varphi)_{2}$
$\displaystyle=(v_{N}^{\varepsilon}(t),\varphi)_{2}-\left(v_{N}^{\varepsilon}\left(\left[\frac{t}{\delta}\right]\delta,\cdot\right),\varphi\right)_{2}+\sum_{j=1}^{\left[\frac{t}{\delta}\right]}(\lim_{s\to
j\delta}(v_{N}^{\varepsilon}(s,\cdot),\varphi)_{2}-(v_{N}(j-1)\delta,\cdot),\varphi)_{2}$
$\displaystyle+\sum_{j=1}^{\left[\frac{t}{\delta}\right]}(\lim_{s\to
j\delta}(w_{N}^{\varepsilon}(s,\cdot),\varphi)_{2}-(w_{N}(j-1)\delta,\cdot),\varphi)_{2}=\int_{0}^{t}\int_{\\{v_{N}(s,\cdot)>0\\}}f_{\varepsilon}(v_{N}^{\varepsilon})(\partial_{x}^{3}v_{N}^{\varepsilon})(\partial_{x}\varphi)\,dxds$
$\displaystyle-\frac{1}{2}\int_{0}^{\left[\frac{t}{\delta}\right]\delta}(\partial_{x}w_{N}^{\varepsilon}(s,\cdot),\partial_{x}\varphi)_{2}\,ds-\int_{0}^{\left[\frac{t}{\delta}\right]\delta}(w_{N}^{\varepsilon}(s,\cdot),\partial_{x}\varphi)_{2}\,d\beta(s)$

for all $\varphi\in C^{\infty}(\mathbb{T}_{L})$ $\mathbb{P}$ \- almost surely.
Changing the stochastic basis to

$([0,1],\tilde{\mathcal{F}},(\tilde{\mathcal{F}})_{t\in[0,T]},\mathbb{P}),$

we obtain the existence of the in-law equivalent convergent subsequence,
denoted with $\tilde{u}_{N},\tilde{v}_{N}$ and $\tilde{w}_{N}$, such that

$\displaystyle(\tilde{v}_{N}(t,\cdot),\varphi)_{2}-(u_{0,\varepsilon},\varphi)_{2}=\int_{0}^{t}\int_{\\{\tilde{v}_{N}^{\varepsilon}(s,\cdot)>0\\}}f_{\varepsilon}(\tilde{v}_{N}^{\varepsilon})(\partial_{x}^{3}\tilde{v}_{N}^{\varepsilon})(\partial_{x}\varphi)\,dxds$
(4.2)
$\displaystyle-\frac{1}{2}\int_{0}^{\left[\frac{t}{\delta}\right]\delta}\partial_{x}(\tilde{w}_{N}(s,\cdot)),\partial_{x}\varphi)_{2}\,ds-\int_{0}^{\left[\frac{t}{\delta}\right]\delta}(\tilde{u}_{N}^{\varepsilon}(s,\cdot)),\partial_{x}\varphi)_{2}\,d\tilde{\beta}(s).$

Passing to the limit as $N\to\infty$, we get

$\displaystyle(\tilde{u}_{\varepsilon}(t,\cdot),\varphi)_{2}-(u_{0,\varepsilon},\varphi)_{2}=\int_{0}^{t}\int_{\\{\tilde{u}_{\varepsilon}(s,\cdot)>0\\}}f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{v}_{\varepsilon})(\partial_{x}\varphi)\,dxds$
(4.3)
$\displaystyle-\frac{1}{2}\int_{0}^{t}\partial_{x}(\tilde{u}_{\varepsilon}(s,\cdot)),\partial_{x}\varphi)_{2}\,ds-\int_{0}^{t}(\tilde{u}_{\varepsilon}(s,\cdot)),\partial_{x}\varphi)_{2}\,d\tilde{\beta}(s).$

The proof of convergence in (4.2) is done analogously to Lemma 5.7 [12] using
the estimates (3.11) and Propositions 3.5 through 4.2. ∎

##  5 The positivity of the solution.

It follows from (3.11) that there is $C(\omega)>0$, independent on $N$, such
that

$\text{ esssup
}_{t\in[0,T)}\|\tilde{u}_{N}^{\varepsilon}(t)\|_{1,2}^{p}+\text{ esssup
}_{t\in[0,T)}\|\tilde{v}_{N}^{\varepsilon}(t)\|_{1,2}^{p}+\text{ esssup
}_{t\in[0,T)}\|\tilde{w}_{N}^{\varepsilon}(t)\|_{1,2}^{p}\leq C(\omega).$

From Sobolev embedding we have

$0\leq\tilde{u}_{N}^{\varepsilon}(t,x\omega)+\tilde{v}_{N}^{\varepsilon}(t,x\omega)+\tilde{w}_{N}^{\varepsilon}(t,x\omega)\leq
M_{\varepsilon}(\omega)$ (5.1)

for any $(t,x)\in Q_{T}$ $\mathbb{P}$-almost surely. Define

$G_{\varepsilon}(s):=\frac{\varepsilon}{6s^{2}}-\ln(s),$

so that

$G_{\varepsilon}^{\prime\prime}(s)=\frac{\varepsilon+s^{2}}{s^{4}}.$

Multiplying (3.3) by $G_{\varepsilon}^{{}^{\prime}}$ and integrating by parts,
we arrive at the following identity for $t\in[0,\delta)$:

$\int_{L}G_{\varepsilon}(\tilde{v}_{N}^{\varepsilon})\,dx+\int_{0}^{t}\int_{L}(\partial_{x}^{2}\tilde{v}_{N}^{\varepsilon})^{2}\,dx\,ds=\int_{L}G_{\varepsilon}(u_{0,\varepsilon})\,dx.$

Next, for any $\Phi\in C^{2}(\mathbb{R})$, we apply Ito’s formula to
$\int_{0}^{L}\Phi(\tilde{w}_{N}^{\varepsilon})dx$:

$\displaystyle\int_{L}\Phi(\tilde{w}_{N}^{\varepsilon}(t))dx=\int_{L}\Phi(\tilde{w}_{N}^{\varepsilon}(0))dx+\frac{1}{2}\int_{0}^{t}\int_{L}\Phi^{{}^{\prime}}(\tilde{w}_{N}^{\varepsilon}(s))\partial_{x}^{2}(\tilde{w}_{N}^{\varepsilon}(s)dxds$
$\displaystyle+\frac{1}{2}\int_{0}^{t}\int_{L}\Phi^{{}^{\prime\prime}}(\tilde{w}_{N}^{\varepsilon}(s))(\partial_{x}\tilde{w}_{N}^{\varepsilon}(s))^{2}dxds+\int_{0}^{t}\int_{L}\Phi^{{}^{\prime}}(\tilde{w}_{N}^{\varepsilon}(s))\partial_{x}(\tilde{w}_{N}^{\varepsilon}(s)dxd\tilde{\beta}(s)$
$\displaystyle=\int_{L}\Phi(\tilde{w}_{N}^{\varepsilon}(0))dx-\frac{1}{2}\int_{0}^{t}\int_{L}\Phi^{{}^{\prime\prime}}(\tilde{w}_{N}^{\varepsilon}(s))(\partial_{x}\tilde{w}_{N}^{\varepsilon}(s))^{2}dxds$
$\displaystyle+\frac{1}{2}\int_{0}^{t}\int_{L}\Phi^{{}^{\prime\prime}}(\tilde{w}_{N}^{\varepsilon}(s))(\partial_{x}\tilde{w}_{N}^{\varepsilon}(s))^{2}dxds+\int_{0}^{t}\int_{L}\frac{d}{dx}\Phi(\tilde{w}_{N}^{\varepsilon}(s))\,dxd\beta(s)$
$\displaystyle=\int_{L}\Phi(\tilde{w}_{N}^{\varepsilon}(0))dx,$

due to the choice of the boundary conditions. Therefore, for $t\in[0,\delta)$,
we make use of (3.10) to conclude

$\int_{L}G_{\varepsilon}(\tilde{w}_{N}^{\varepsilon}(t))\,dx=\int_{L}G_{\varepsilon}(\tilde{w}_{N}^{\varepsilon}(0))\,dx=\int_{L}G_{\varepsilon}(\tilde{v}_{N}^{\varepsilon}(\delta-0))\,dx\leq\int_{L}G_{\varepsilon}(\tilde{u}_{0,\varepsilon})\,dx.$

Similarly, on $[\delta,2\delta)$ we obtain

$\int_{L}G_{\varepsilon}(\tilde{v}_{N}^{\varepsilon}(t))\,dx=\int_{L}G_{\varepsilon}(\tilde{w}_{N}^{\varepsilon}(\delta+0))\,dx=\int_{L}G_{\varepsilon}(\tilde{v}_{N}^{\varepsilon}(2\delta-0))\,dx\leq\int_{L}G_{\varepsilon}(\tilde{u}_{0,\varepsilon})\,dx$

and

$\int_{L}G_{\varepsilon}(\tilde{w}_{N}^{\varepsilon}(t))\,dx=\int_{L}G_{\varepsilon}(\tilde{w}_{N}^{\varepsilon}(\delta+0))\,dx=\int_{L}G_{\varepsilon}(\tilde{v}_{N}^{\varepsilon}(2\delta-0))\,dx\leq\int_{L}G_{\varepsilon}(\tilde{u}_{0,\varepsilon})\,dx.$

Continuing this process, for all $t\in[0,T)$ and $x\in[0,L]$ we get the
following inequality

$\int_{L}G_{\varepsilon}(\tilde{v}_{N}^{\varepsilon}(t))\,dx\leq\int_{L}G_{\varepsilon}(\tilde{u}_{0,\varepsilon})\,dx:=C(\varepsilon,\omega)$
(5.2)

uniformly in $N$. Furthermore, (5.1) implies

$\ln\frac{1}{\tilde{v}_{N}^{\varepsilon}}\geq\ln\frac{1}{M_{\varepsilon}(\omega)}.$

Since $\tilde{u}_{0,\varepsilon}=\tilde{u}_{0}+\delta(\varepsilon)$, we have

$\int_{L}\frac{dx}{(\tilde{u}_{0}+\delta)^{2}}+\int_{L}\ln\frac{1}{(\tilde{u}_{0}+\delta)}\,dx<\infty.$
(5.3)

The bounds (5.2) and (5.3)yield

$\int_{L}\frac{\varepsilon}{6(\tilde{v}_{N}^{\varepsilon}(t,x))^{2}}\,dx+\int_{L}\ln\frac{1}{\tilde{v}_{N}^{\varepsilon}(t,x)}\,dx\leq\int_{L}G_{\varepsilon}(\tilde{u}_{0,\varepsilon})\,dx<\infty.$

For all $\omega$ and $x_{0}\in[0,L]$ denote

$\delta(t,\omega,\varepsilon):=\min_{x\in
L}\tilde{v}_{N}^{\varepsilon}(t,x,\omega):=\tilde{v}_{N}^{\varepsilon}(t,x_{0}(t,\omega),\omega)$

for some $x_{0}\in[0,L].$ It follows from Sobolev’s embedding of
$H^{1}(\mathbb{T}_{L})$ into the Holder space $C^{1/2}(\mathbb{T}_{L})$ that

$\tilde{v}_{N}^{\varepsilon}(t,x,\omega)\leq\tilde{v}_{N}^{\varepsilon}(t,x_{0},\omega)+C(\omega)|x-x_{0}(t)|^{1/2}=\delta(t,\omega,\varepsilon)+C(\omega)|x-x_{0}(t)|^{1/2}.$

Thus the exists a constant $K=K(\omega,u_{0},\varepsilon)>0$ such that for all
$N\geq 1$ we have

$K\geq\varepsilon\int_{L}\frac{\varepsilon}{(\tilde{v}_{N}^{\varepsilon}(t,x))^{2}}\,dx\geq\varepsilon\int_{L}\frac{dx}{(\delta(t,\omega,\varepsilon)+C|x-x_{0}|^{1/2})^{2}}\\\
=\varepsilon\int_{L}\frac{1}{\left(1+C\left(\frac{x-x_{0}}{\delta^{2}}\right)^{1/2}\right)^{2}}d\left(\frac{x-x_{0}}{\delta^{2}}\right)\geq
C_{1}\varepsilon\int_{0}^{\frac{L}{\delta^{2}}}\frac{dy}{1+C^{2}y}=C_{2}\varepsilon\ln\left(1+\frac{C^{2}L}{\delta^{2}}\right)\geq
C_{3}\varepsilon\ln\frac{1}{\delta},$

which yields that $\delta(t,\omega,\varepsilon)\geq
e^{-\frac{K}{C_{3}\varepsilon}}$ uniformly in $N$. Since by 5.4
$\|\tilde{v}_{N}^{\varepsilon}-\tilde{u}_{\varepsilon}\|_{L^{\infty}}(Q_{T})\to
0$ as $N\to\infty$, then $\tilde{u}_{\varepsilon}(t,x,\omega)>0$
$\tilde{P}$-almost surely. Consequently, (4.3) becomes

$(\tilde{u}_{\varepsilon},\varphi)_{2}-(\tilde{u}_{0,\varepsilon},\varphi)_{2}=\int_{0}^{t}\int_{L}f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{u}_{\varepsilon})\partial_{x}\varphi
dx\,ds-\frac{1}{2}\int_{0}^{t}\int_{L}\partial_{x}\tilde{u}_{\varepsilon}(s)\partial_{x}\varphi\,dx\,ds\\\
-\int_{0}^{t}\int_{L}\tilde{u}_{\varepsilon}(s)\partial_{x}\varphi\,dx\,d\tilde{\beta}(s),$
(5.4)

which completes the proof of Theorem 3.1. ∎

##  6 Global solutions of the regularized problem and their asymptotic
behavior.

In this section we show that the local solution, obtained in Theorem 3.1, can
be extended for all $t\geq 0$. Throughout this section, in order to simplify
the notation, we will omit the tildes and denote $\Omega$ to be $[0,1]$.

######  Theorem 6.1.

Under the conditions of Theorem 3.1, the solution of (3.1) in the sense of
Definition 1.1 with the initial condition (3.2) exists for $t\in[0,\infty)$.

###### Proof.

In view of Theorem 3.1, (3.1) we have a filtered probability space
$(\Omega^{1}=[0,1],\mathcal{F}^{1},\mathcal{F}_{t}^{1},\mathbb{P}^{1})$, on
which we have a martingale solution $u_{\varepsilon}^{1}(t)$ on $[0,T)$, with
the Wiener process $\beta_{1}(t)$ and filtration
$\mathcal{F}_{t}^{1}=\sigma\\{(u_{0,\varepsilon},\beta_{1}(s)),s\leq
t,t\in[0,T)\\}$. This solution is strictly positive, and (5.2), the
convergence
$\|v_{N}^{\varepsilon}-u_{\varepsilon}^{1}\|_{L^{\infty}(Q_{T})}\to 0$,
$N\to\infty$, and Fatou Lemma yield

$\int_{L}G_{\varepsilon}(u_{\varepsilon}^{1}(T))\,dx\leq
C(\varepsilon,\omega).$

Furthermore, it follows from (3.11) that
$\|u_{\varepsilon}^{1}(T)\|_{1,2}\leq\|u_{\varepsilon,0}\|_{1,2}$ for all
$p\geq 2$. We next define a new probability space
$(\Omega^{\prime},\mathcal{F}^{\prime},\mathbb{P}^{\prime})$ and introduce a
new Wiener process $\beta^{\prime}(t,\omega^{\prime})$ on it. Finally,
introduce the product probability space

$(\Omega^{2},\mathcal{F}^{2},\mathbb{P}^{2}):=(\Omega^{1}\times\Omega^{\prime},\mathcal{F}^{1}\times\mathcal{F}^{\prime},\mathbb{P}^{1}\times\mathbb{P}^{\prime}),$

and introduce the Wiener process

$\beta_{2}(t,\omega_{1},\omega^{\prime}):=\begin{cases}\beta_{1}(t,\omega_{1}),t\in[0,T);\\\
\beta^{\prime}(t-T,\omega^{\prime})+\beta_{1}(T,\omega_{1}),t\in[T,2T].\end{cases}$

For every fixed $\omega_{1}$ the equation (3.1) has a martingale solution
$\eta_{\varepsilon}(t,\omega^{\prime})$ on $[T,2T)$ with the initial condition
$\eta_{\varepsilon}(T,\omega^{\prime})=u_{\varepsilon}^{1}(T,\omega_{1})$. The
same way as in (5.2) we conclude

$\int_{L}G_{\varepsilon}(\eta_{\varepsilon}(t,\omega^{\prime}))\,dx\leq\int_{L}G_{\varepsilon}(\eta_{\varepsilon}(T,\omega^{\prime}))\,dx=\int_{L}G_{\varepsilon}(u_{\varepsilon}^{1}(T,\omega_{1}))\,dx.$

Hence

$\int_{L}G_{\varepsilon}(\eta_{\varepsilon}(2T,\omega^{\prime}))\,dx\leq\int_{L}G_{\varepsilon}(u_{\varepsilon}^{1}(T,\omega_{1}))\,dx.$

We may now define the solution

$u_{\varepsilon}^{2}(t,\omega_{1},\omega^{\prime}):=\begin{cases}u^{1}_{\varepsilon}(t,\omega_{1}),t\in[0,T);\\\
\eta_{\varepsilon}(t,\omega^{\prime}),t\in[T,2T]\end{cases}$

on the space
$(\Omega^{2},\mathcal{F}^{2},\mathcal{F}_{t}^{2},\mathbb{P}^{2})$, where the
filtration $\mathcal{F}_{t}^{2}$ is defined as

$\mathcal{F}_{t}^{2}=\sigma\\{u_{0,\varepsilon},\beta_{2}(s),u_{\varepsilon}^{2}(s),s\leq
t\\},t\in[0,2T).$

Repeating this process $n$ times, we obtain a continuous in $t$ solution on
$[0,nT]$ for any $n\geq 1$. Their finite dimensional distributions would match
by construction. It remains to use Kolmogorov’s theorem, which guarantees the
existence of a probability space and a weak solution $u_{\varepsilon}(t),t\geq
0$, defined on it. This completes the proof of Theorem 6.1. ∎

The solution, obtained in Theorem 6.1, is strictly positive. Moreover, since
$G_{\varepsilon}^{\prime\prime}(s)=\frac{1}{f_{\varepsilon}(s)}$, by Ito’s
formula we have

$\int_{L}G_{\varepsilon}(u_{\varepsilon}(t))\,dx=\int_{L}G_{\varepsilon}(u_{0,\varepsilon})\,dx+\int_{0}^{t}\int_{L}G_{\varepsilon}^{\prime\prime}(u_{\varepsilon})f_{\varepsilon}(u_{\varepsilon})(\partial^{3}_{x}u_{\varepsilon})\partial_{x}u_{\varepsilon}\,dx\,ds\\\
-\frac{1}{2}\int_{0}^{t}\int_{L}G_{\varepsilon}^{\prime\prime}(u_{\varepsilon})(\partial_{x}u_{\varepsilon})^{2}\,dx\,ds+\frac{1}{2}\int_{0}^{t}\int_{L}G_{\varepsilon}^{\prime\prime}(u_{\varepsilon})(\partial_{x}u_{\varepsilon})^{2}\,dx\,ds+\
\int_{0}^{t}\int_{L}G_{\varepsilon}^{\prime}(u_{\varepsilon})\partial_{x}u_{\varepsilon}\,dx\,d\beta(s).$

The latter identity yields

$\int_{L}G_{\varepsilon}(u_{\varepsilon}(t))\,dx+\int_{0}^{t}\int_{L}(\partial^{2}_{x}u_{\varepsilon})^{2}\,dx\,ds=\int_{L}G_{\varepsilon}(u_{0,\varepsilon})\,dx,$
(6.1)

thus $u_{\varepsilon}\in H^{2}(\mathbb{T}_{L})$ for all $t\geq 0$ a.s.

######  Lemma 6.2.

The function $\|\partial_{x}u_{\varepsilon}\|_{2}^{2}$ is non-decreasing for
$t\geq 0$ almost surely.

###### Proof.

Using Ito’s formula for $\|\partial_{x}u_{\varepsilon}\|_{2}^{2}$ we have

$\|\partial_{x}u_{\varepsilon}(t)\|_{2}^{2}=\|\partial_{x}u_{0,\varepsilon}\|_{2}^{2}+2\int_{0}^{t}\int_{L}\partial_{x}(f_{\varepsilon}(u_{\varepsilon})(\partial^{3}_{x}u_{\varepsilon}))\partial_{x}^{2}u_{\varepsilon}\,dx\,ds\\\
-\int_{0}^{t}\int_{L}(\partial_{x}^{2}u_{\varepsilon})^{2}\,dx\,ds+\int_{0}^{t}\int_{L}(\partial_{x}^{2}u_{\varepsilon})^{2}\,dx\,ds.$

This way

$\|\partial_{x}u_{\varepsilon}(t)\|_{2}^{2}+2\int_{0}^{t}\int_{L}f_{\varepsilon}(u_{\varepsilon})(\partial^{3}_{x}u_{\varepsilon}))^{2}\,dx\,ds=\|\partial_{x}u_{0,\varepsilon}\|_{2}^{2}=\|\partial_{x}u_{0}\|_{2}^{2}$
(6.2)

which completes the proof. ∎

Furthermore, using the Sobolev embedding, Poincare inequality and the
conservation of mass property we have

$u_{\varepsilon}(t,x)-\overline{u_{0,\varepsilon}}\leq
C\int_{L}(\partial_{x}u_{\varepsilon})^{2}\,dx\leq
C\|\partial_{x}u_{0}\|^{2},$

which, in turn, yields the following corollaries.

######  Corollary 6.3.

There is a nonrandom constant $K>0$, independent of $\varepsilon$, such that
for all $x\in[0,L]$ and $t\geq 0$ we have

$0<u_{\varepsilon}(t,x)\leq K\text{ almost surely.}$ (6.3)

######  Corollary 6.4.

For any $p\geq 2$, by the same reasoning as in Lemma 6.2, there is $C>0$ such
that for all $t\geq 0$ we have

$\sup_{s\in[0,t]}\|u_{\varepsilon}(s)\|_{1,2}^{p}+2\int_{0}^{t}\|u_{\varepsilon}(s)\|_{1,2}^{p-2}\int_{L}f_{\varepsilon}(u_{\varepsilon})(\partial^{3}_{x}u_{\varepsilon}))^{2}\,dx\,ds\leq
C\|\partial_{x}u_{0,\varepsilon}\|_{1,2}^{p}.$

The bound (6.3) implies that the $\ln\frac{1}{u_{\varepsilon}(t)}$ is bounded
from below for all $t\geq 0$ and $x\in[0,L]$ $\mathbb{P}$ \- almost surely.
Hence $\int_{L}G_{\varepsilon}(u_{\varepsilon}(t))\,dx$ is bounded from above
uniformly in $\varepsilon\in(0,1)$. As for the upper bound for this quantity,
using (6.1) as well as (2.1), we have

$\int_{L}G_{\varepsilon}(u_{\varepsilon}(t))\,dx\leq\int_{L}G_{\varepsilon}(u_{0,\varepsilon})\,dx=\int_{L}\frac{\varepsilon\,dx}{6(u_{0}+\delta(\varepsilon))^{2}}+\int_{L}\ln\left(\frac{1}{u_{0}+\delta(\varepsilon)}\right)\,dx\\\
\leq L\varepsilon^{1-2\theta}+\int_{L}\ln\left(\frac{1}{u_{0}}\right)\,dx\leq
L+\int_{L}\ln\left(\frac{1}{u_{0}}\right)\,dx<\infty.$

This leads us to the following corollary:

######  Corollary 6.5.

There is a nonrandom constant $C>0$, independent of $\varepsilon$, such that

$\int_{0}^{\infty}\int_{L}(\partial_{x}^{2}u_{\varepsilon})^{2}\,dx\,ds<C.$
(6.4)

We now proceed with establishing the estimate (1.5). To this end, we will
modify the approach from [28] to stochastic case. Firstly, note that due to
the choice of periodic boundary conditions, we have the analogs of Lemmas 3
and 4 from [28]:

######  Lemma 6.6.

There is $C>0$ such that for any $\beta\in\mathbb{R}$ and any strictly
positive $v\in H^{2}(\mathbb{T}_{L})$ we have

$\int_{L}v^{\beta}(\partial_{x}^{2}v)^{2}\,dx\geq
C(1-\beta)^{2}\int_{L}v^{\beta-2}(\partial_{x}v)^{4}\,dx.$

######  Lemma 6.7.

There is $C>0$ such that for any $v\in H^{2}(\mathbb{T}_{L})$ with positive
mass we have

$\int_{L}v^{2}(\partial_{x}^{2}v)^{2}\,dx\geq
C\left(\int_{L}v(x)\,dx\right)^{2}\int_{L}(\partial_{x}v)^{2}\,dx.$

Note also that in view of (6.3) and Corollary 6.5, the solution
$u_{\varepsilon}(t,x,\omega)$ satisfies the conditions of Lemma 6.6 and Lemma
6.7 $\mathbb{P}$-almost surely. The next proposition pretty much follows the
lines of Theorem 3 [28]. Nevertherless, for the convenience of the reader, we
shall highlight the key ingredients of the proof below.

######  Proposition 6.8.

Assume $u_{0}\in H^{1}(\mathbb{T}_{L})$ and the condition (2.1) holds. Then

$J_{\varepsilon}[u_{\varepsilon}(t)]:=\frac{1}{2}\int_{L}(\partial_{x}u_{\varepsilon}(t))^{2}\,dx$

satisfies

$J_{\varepsilon}[u_{\varepsilon}(t)]\leq
e^{-K_{\varepsilon}^{2}t}\int_{L}(\partial_{x}u_{0}(t))^{2}\,dx+C\varepsilon^{1/2}K_{\varepsilon}\int_{0}^{t}\int_{L}(\partial_{x}u_{\varepsilon}(s))^{2}\,dxds\text{
a.s., }$ (6.5)

where $C>0$ is a non-random constant, independent of $\varepsilon$, and
$K_{\varepsilon}>0$ is a deterministic constant satisfying

$\lim_{\varepsilon\to 0}K_{\varepsilon}=K<\infty.$ (6.6)

###### Proof.

For $t\geq 0$ denote

$I_{\varepsilon}(t):=\int_{L}f_{\varepsilon}(u_{\varepsilon})(\partial^{3}_{x}u_{\varepsilon}))^{2}\,dx.$

By Schwartz inequality,

$(2I_{\varepsilon}J_{\varepsilon})^{1/2}\geq-\int_{L}f_{\varepsilon}^{1/2}(u_{\varepsilon})(\partial_{x}u_{\varepsilon})(\partial_{x}^{3}u_{\varepsilon})\,dx.$

The same way as in [28], Theorem 3, we get

$(2I_{\varepsilon}J_{\varepsilon})^{1/2}\geq-\int_{L}f_{\varepsilon}^{1/2}(u_{\varepsilon})(\partial_{x}^{2}u_{\varepsilon})^{2}\,dx-\frac{2}{3}\varepsilon^{2}\int_{L}(\varepsilon+u_{\varepsilon}^{2})^{-\frac{5}{2}}(\partial_{x}u_{\varepsilon})^{4}\,dx.$

Thus, from (6.3) we have

$f_{\varepsilon}^{1/2}(u_{\varepsilon})\geq\frac{u_{\varepsilon}^{2}}{(\varepsilon+K^{2})^{\frac{1}{2}}}$

for almost all $t\geq 0$ and $x\in[0,L]$. Then from Lemma 6.6 and Lemma 6.7
there exist $C>0$ and $C_{1}>0$ such that

$C_{1}K_{\varepsilon}^{2}J_{\varepsilon}\leq
I_{\varepsilon}+C\varepsilon^{1/2}K_{\varepsilon}\int_{L}(\partial_{x}^{2}u_{\varepsilon})^{2}\,dx,$
(6.7)

where

$K_{\varepsilon}=\frac{\pi^{2}(1+2\varepsilon^{\theta})^{2}}{16(\pi+1)^{2}(\varepsilon+K^{2})^{1/2}}.$

From (6.2) we thus get

$\frac{d}{dt}J_{\varepsilon}[u_{\varepsilon}(t)]=-\int_{L}f_{\varepsilon}(u_{\varepsilon})(\partial_{x}^{3}u_{\varepsilon})^{2}\,dx.$

The latter equality, combined with (6.7), implies

$\frac{d}{dt}J_{\varepsilon}[u_{\varepsilon}(t)]\leq-
C_{1}K_{\varepsilon}^{2}J_{\varepsilon}[u_{\varepsilon}]+C\varepsilon^{1/2}K_{\varepsilon}\int_{L}(\partial_{x}^{2}u_{\varepsilon})^{2}\,dx,$

and thus (6.5) follows from the Gronwall inequality. ∎

##  7 Proof of the main result.

###  7.1 Compactness and convergence of a subsequence.

In Theorem 6.1 we constructed a weak solution $u_{\varepsilon}$ of the
equation (3.1) on $[0,\infty)$ for any $\varepsilon>0$. Each of these
solutions exists on its own probability space
$(\Omega_{\varepsilon},\mathcal{F}_{\varepsilon},{\mathcal{F}}_{t,\varepsilon},\mathbb{P}_{\varepsilon}).$
In order to pass to the limit as $\varepsilon\to 0$, we will take a sequence
$\varepsilon=\frac{1}{n}$ and consider $(\beta_{\varepsilon},u_{\varepsilon})$
on the common probability space

$(\Omega,\mathcal{F},\mathcal{F}_{t},\mathbb{P}):=\Pi_{\varepsilon}(\Omega_{\varepsilon},\mathcal{F}_{\varepsilon},\mathcal{F}_{t,\varepsilon},\mathbb{P}_{\varepsilon}).$

Fix $T>0$. We shall pass to the limit as $\varepsilon\to 0$ in (3.1) on
$[0,T]$ using the techniques from [8]. Note, however, that in [8] the authors
used the regularization
$f_{\varepsilon}(s)=(s^{2}+\varepsilon^{2})^{n/4},n\in[8/3,4)$ while in this
work we use $f_{\varepsilon}(s)=\frac{s^{4}}{s^{2}+\varepsilon}$.

######  Lemma 7.1.

Suppose $\varepsilon\in(0,1]$, $p>1$ and $u_{0}\in
L^{4p}(\Omega,\mathbb{F}_{0},\mathbb{P};H^{1}(\mathbb{T}_{L}))$ satisfies
$u_{0}\geq 0$ $\mathbb{P}$-almost surely, and

$\mathbb{E}\left(\int_{L}u_{0}(x)\,dx\right)^{4p}<\infty.$

Then, for any $p^{\prime}\in[1,2p)$, the weak solution $u_{\varepsilon}$,
whose existence was established in Theorem 6.1, satisfies

$u_{\varepsilon}\in
L^{p^{\prime}}(\Omega,\mathcal{F},\mathbb{P};C^{1/4}([0,T];L^{2}(\mathbb{T}_{L})),$

with

$\|u_{\varepsilon}\|_{L^{p^{\prime}}(\Omega,\mathcal{F},\mathbb{P};C^{1/4}([0,T];L^{2}(\mathbb{T}_{L}))}\leq
C\left[\left(\mathbb{E}\int_{L}u_{0}(x,\omega)\,dx)^{2p^{\prime}}\right)+\left(\mathbb{E}\|\partial_{x}u_{0}\|_{2}^{2p}\right)^{p^{\prime}/2p}+\mathbb{E}\|\partial_{x}u_{0}\|_{2}^{2p^{\prime}}\right],$
(7.1)

where the constant $C$ depends only on $p,p^{\prime},L$ and $T$.

###### Proof.

It follows from (5.4) that for any $\varphi\in H^{1}(\mathbb{T}_{L})$, and any
$t_{1}\leq t_{2}\in[0,T]$, the identity

$(u_{\varepsilon}(t_{2})-u_{\varepsilon}(t_{1}),\varphi)_{2}+\int_{t_{1}}^{t_{2}}\int_{L}f_{\varepsilon}(u_{\varepsilon})(\partial_{x}^{3}u_{\varepsilon})\partial_{x}\varphi
dx\,ds+\frac{1}{2}\int_{t_{1}}^{t_{2}}\int_{L}\partial_{x}u_{\varepsilon}(s)\partial_{x}\varphi\,dx\,ds\\\
=\int_{t_{1}}^{t_{2}}\int_{L}\partial_{x}u_{\varepsilon}(s)\varphi\,dx\,d\beta(s)$

holds $\mathbb{P}$-almost surely. Since $H^{1}(\mathbb{T}_{L})$ is separable,
using the approximation argument we may conclude that the $\mathbb{P}$-zero
set can be chosen independently of $\varphi$, in other words,
$\mathbb{P}$-almost surely we have the following bound

$(u_{\varepsilon}(t_{2})-u_{\varepsilon}(t_{1}),\varphi)_{2}+\int_{t_{1}}^{t_{2}}\int_{L}f_{\varepsilon}(u_{\varepsilon})(\partial_{x}^{3}u_{\varepsilon})\partial_{x}\varphi
dx\,ds+\frac{1}{2}\int_{t_{1}}^{t_{2}}\int_{L}\partial_{x}u_{\varepsilon}(s)\partial_{x}\varphi\,dx\,ds\\\
\leq\sup_{\|\psi\|_{2}\leq
1}|(I_{\varepsilon}(t_{2})-I_{\varepsilon}(t_{1}),\psi)_{2}|$ (7.2)

for all $\varphi\in H^{1}(\mathbb{T}_{L})$ with $\|\varphi\|_{2}=1$, where

$I_{\varepsilon}(t)=\int_{0}^{t}\partial_{x}u_{\varepsilon}(s)\,d\beta(s).$

Setting

$\varphi:=\frac{u_{\varepsilon}(t_{2})-u_{\varepsilon}(t_{1})}{\|u_{\varepsilon}(t_{2})-u_{\varepsilon}(t_{1})\|_{2}}$

in (7.2), we have

$\displaystyle\frac{\|u_{\varepsilon}(t_{2})-u_{\varepsilon}(t_{1})\|_{2}^{2}}{\|u_{\varepsilon}(t_{2})-u_{\varepsilon}(t_{1})\|_{2}}\leq\frac{1}{\|u_{\varepsilon}(t_{2})-u_{\varepsilon}(t_{1})\|_{2}}\left|\int_{t_{1}}^{t_{2}}\int_{L}f_{\varepsilon}(u_{\varepsilon})(\partial_{x}^{3}u_{\varepsilon})\partial_{x}(u_{\varepsilon}(t_{2})-u_{\varepsilon}(t_{1}))dx\,ds\right|$
$\displaystyle+\frac{1}{2\|u_{\varepsilon}(t_{2})-u_{\varepsilon}(t_{1})\|_{2}}\left|\int_{t_{1}}^{t_{2}}\int_{L}\partial_{x}u_{\varepsilon}(s)\partial_{x}(u_{\varepsilon}(t_{2})-u_{\varepsilon}(t_{1}))\,dx\,ds\right|+\|I_{\varepsilon}(t_{2})-I_{\varepsilon}(t_{1})\|_{2}.$

Multiplying the latter inequality with
$\|u_{\varepsilon}(t_{2})-u_{\varepsilon}(t_{1})\|_{2}$ and applying Young’s
inequality, we get that there is a $C>0$, independent of $\varepsilon$, such
that

$\displaystyle\|u_{\varepsilon}(t_{2})-u_{\varepsilon}(t_{1})\|_{2}^{2}\leq
C\left|\int_{t_{1}}^{t_{2}}\int_{L}f_{\varepsilon}(u_{\varepsilon})(\partial_{x}^{3}u_{\varepsilon})\partial_{x}(u_{\varepsilon}(t_{2})-u_{\varepsilon}(t_{1}))dx\,ds\right|$
(7.3)
$\displaystyle+\left|\int_{t_{1}}^{t_{2}}\int_{L}\partial_{x}u_{\varepsilon}(s)\partial_{x}(u_{\varepsilon}(t_{2})-u_{\varepsilon}(t_{1}))\,dx\,ds\right|+\|I_{\varepsilon}(t_{2})-I_{\varepsilon}(t_{1})\|_{2}^{2}$
$\displaystyle:=C(J_{1}^{2}(t_{1},t_{2})+J_{2}^{2}(t_{1},t_{2}))+\|I_{\varepsilon}(t_{2})-I_{\varepsilon}(t_{1})\|_{2}^{2}.$

Let us now estimate each term separately. Using Lemma 2.1 [11], for $p\geq 2$
and $\alpha<\frac{1}{2}$ we have for any $T>0$

$\mathbb{E}\left\|\int_{0}^{T}\partial_{x}(u_{\varepsilon}(s))d\beta(s)\right\|^{p}_{W^{\alpha,p}([0,T],L^{2}(\mathbb{T}_{L}))}\leq
C(p,\alpha)\mathbb{E}\left(\int_{0}^{T}\|\partial_{x}u_{\varepsilon}\|_{2}^{p}\,ds\right)\leq
C(p,\alpha)T\mathbb{E}\|\partial_{x}u_{0}\|_{2}^{p},$ (7.4)

where we used (6.2). From (7.4) and the continuous embedding

$W^{\alpha,p}([0,T],L^{2}(\mathbb{T}_{L}))\subset
C^{\alpha-\frac{1}{p}}([0,T];L^{2}(\mathbb{T}_{L}))$

for all $\alpha\in(0,\frac{1}{2})$, we get

$\mathbb{E}\|I_{\varepsilon}\|_{L^{p^{\prime}}(\Omega,C^{\frac{1}{2}-\sigma}([0,T];L^{2}(\mathbb{T}_{L}))}\leq
C\left(\mathbb{E}\|\partial_{x}u_{0}\|_{2}^{2p}\right)^{\frac{1}{2p}}$

for all $0<\sigma<\frac{1}{2}$, $1\leq p^{\prime}<2p$ and $p\geq 2$ satisfying
the assumptions of Lemma 2.1 [11]. Therefore, for such $\sigma$ and
$p^{\prime}$,

$\mathbb{E}\sup_{t_{1},t_{2}\in[0,T]}\left(\frac{|I_{\varepsilon}(t_{2})-I_{\varepsilon}(t_{1})|}{|t_{2}-t_{1}|^{1/2-\sigma}}\right)^{p^{\prime}}\leq
C\left(\mathbb{E}\|\partial_{x}u_{0}\|_{2}^{2p}\right)^{\frac{p^{\prime}}{2p}}.$
(7.5)

Let us estimate $J_{1}$ in (7.3). We have

$J_{1}(t_{1},t_{2})=\left(\int_{t_{1}}^{t_{2}}\int_{L}f_{\varepsilon}(u_{\varepsilon})(\partial_{x}^{3}u_{\varepsilon})\partial_{x}(u_{\varepsilon}(t_{2})-u_{\varepsilon}(t_{1}))dx\,ds\right)^{\frac{1}{2}}\\\
\leq\left(\int_{t_{1}}^{t_{2}}\int_{L}f_{\varepsilon}(u_{\varepsilon})(\partial_{x}(u_{\varepsilon}(t_{2})-u_{\varepsilon}(t_{1})))^{2}\,dx\,ds\right)^{\frac{1}{4}}\left(\int_{t_{1}}^{t_{2}}\int_{L}f_{\varepsilon}(u_{\varepsilon})(\partial_{x}^{3}u_{\varepsilon})^{2}\,dx\,ds\right)^{\frac{1}{4}}.$

Using the inequality $f_{\varepsilon}(s)\leq s^{2}$, (6.2), (6.3) and using
the conservation of mass property, we have

$\displaystyle J_{1}(t_{1},t_{2})\leq
C|t_{2}-t_{1}|^{\frac{1}{4}}\left(1+\|\partial_{x}u_{0}\|_{2}+\frac{1}{L}\int_{L}u_{0,\varepsilon}\,dx\right)\left(\int_{0}^{T}\int_{L}f_{\varepsilon}(u_{\varepsilon})(\partial_{x}^{3}u_{\varepsilon})^{2}\,dx\,ds\right)^{\frac{1}{4}}$
$\displaystyle\leq
C_{1}|t_{2}-t_{1}|^{\frac{1}{4}}\left(1+\|\partial_{x}u_{0}\|_{2}+\frac{1}{L}\int_{L}u_{0,\varepsilon}\,dx\right)\|\partial_{x}u_{0}\|_{2}^{\frac{1}{2}}$
$\displaystyle\leq
C_{2}|t_{2}-t_{1}|^{\frac{1}{4}}\left(1+\|\partial_{x}u_{0}\|_{2}^{2}+(\overline{u_{0,\varepsilon}})^{2}\right).$

Hence,

$\mathbb{E}\sup_{t_{1},t_{2}\in[0,T]}\left(\frac{J_{1}(t_{1},t_{2})}{|t_{2}-t_{1}|^{\frac{1}{4}}}\right)^{p^{\prime}}\leq
C_{3}\mathbb{E}\left(1+\|\partial_{x}u_{0}\|_{2}^{2}+(\overline{u_{0,\varepsilon}})^{2}\right).$
(7.6)

In order to estimate $J_{2}$, using (6.2), we have

$2J_{2}(t_{1},t_{2})=\left|\int_{t_{1}}^{t_{2}}\int_{L}\partial_{x}u_{\varepsilon}(s)\partial_{x}(u_{\varepsilon}(t_{2})-u_{\varepsilon}(t_{1}))\,dx\,ds\right|^{\frac{1}{2}}\\\
\leq
CT^{\frac{1}{4}}|t_{2}-t_{1}|^{\frac{1}{4}}\left(\sup_{t\in[0,T]}\|\partial_{x}u_{\varepsilon}(t)\|_{2}^{2}\right)^{\frac{1}{2}}\leq
C_{1}|t_{2}-t_{1}|^{\frac{1}{4}}\|\partial_{x}u_{0}\|_{2}.$

Therefore,

$\mathbb{E}\sup_{t_{1},t_{2}\in[0,T]}\left(\frac{J_{1}(t_{1},t_{2})}{|t_{2}-t_{1}|^{\frac{1}{4}}}\right)^{p^{\prime}}\leq
C\mathbb{E}\|\partial_{x}u_{0}\|_{2}^{p^{\prime}}.$ (7.7)

Combining (7.5), (7.6) and (7.7), and choosing $\delta=\frac{1}{4}$, we get

$\|u_{\varepsilon}\|^{p^{\prime}}_{L^{p^{\prime}}(\Omega,C^{\frac{1}{4}}([0,T];L^{2}(\mathbb{T}_{L})))}\leq\tilde{C}\left[\left(\mathbb{E}\|\partial_{x}u_{0}\|_{2}^{2p}\right)^{\frac{p^{\prime}}{2p}}+1+\mathbb{E}\|\partial_{x}u_{0}\|_{2}^{2p^{\prime}}+\mathbb{E}(\overline{u_{0,\varepsilon}})^{2p^{\prime}}+\mathbb{E}\|\partial_{x}u_{0}\|_{2}^{p^{\prime}}\right]\\\
\leq\tilde{C}_{1}\left[1+\left(\mathbb{E}\|\partial_{x}u_{0}\|_{2}^{2p}\right)^{\frac{p^{\prime}}{2p}}+\mathbb{E}\|\partial_{x}u_{0}\|_{2}^{2p^{\prime}}+\mathbb{E}(\overline{u_{0}})^{2p^{\prime}}\right],$

where the constant $\tilde{C}_{1}>0$ is independent of $\varepsilon$.
Similarly to [8], we apply the interpolation argument, (7.1) and the
continuous embedding $H^{1}(\mathbb{T}_{L})$ into $C^{1/2}(\mathbb{T}_{L})$ we
have

######  Corollary 7.2.

Under the conditions of Lemma 7.1, the solution $u_{\varepsilon}$ in Holder
continuous $\mathbb{P}$-almost surely. In particular, there is a constant
$C>0$, independent of $\varepsilon>0$, such that

$\mathbb{E}\left(\|u_{\varepsilon}\|^{p}_{C^{\frac{1}{8},\frac{1}{2}}(Q_{T})}\right)\leq
C$ (7.8)

for any $p^{\prime}\in[1,2p).$

Denote

$C^{\frac{1}{8}-,\frac{1}{2}-}(Q_{T}):=\cap_{\gamma\in\left(0,\frac{1}{2}\right)}C^{\frac{\gamma}{4},\gamma}(Q_{T}).$

We endow $C^{\frac{1}{8}-,\frac{1}{2}-}(Q_{T})$ with the topology generated by
the metric

$\rho(u,v):=\sum_{n=1}^{\infty}2^{-n}(\|u-v\|_{C^{\frac{\gamma_{n}}{4},\gamma_{n}}(Q_{T})}\wedge
1),$

where $\gamma_{n}=\frac{1}{2}-\frac{1}{2^{n+1}}$. The space
$C^{\frac{1}{8}-,\frac{1}{2}-}(Q_{T})$ is a Polish space, moreover, the
embedding $C^{\frac{1}{8},\frac{1}{2}}(Q_{T})\subset
C^{\frac{1}{8}-,\frac{1}{2}-}(Q_{T})$ is compact, as shown in [8]. The
following proposition is the analog of Proposition 5.4 [8].

######  Proposition 7.3.

Let $T\in[0,\infty),\ \varepsilon\in(0,1],\ p>1$ and $u_{0}\in
L^{4p}(\Omega,\mathcal{F}_{0},\mathbb{P};H^{1}(\mathbb{T}_{L}))$ be such that
$u_{0}\geq 0$ $\mathbb{P}$-almost surely. Theorem 6.1 guarantees the existence
of a weak solution $u_{\varepsilon}$ with such initial condition. Define
$J_{\varepsilon}:=f_{\varepsilon}(u_{\varepsilon})\partial_{x}^{3}u_{\varepsilon}$.
Then there is a subsequence in $\varepsilon$, and the random variables

$\tilde{u},\tilde{u}_{\varepsilon}:[0,1]\to
C^{\frac{1}{8}-,\frac{1}{2}-}(Q_{T}),$
$\tilde{J},\tilde{J}_{\varepsilon}:[0,1]\to L^{2}(Q_{T}),$
$\tilde{\beta},\tilde{\beta}_{\varepsilon}:[0,1]\to C([0,T],\mathbb{R}),$

with

$(\tilde{u}_{\varepsilon},\tilde{J}_{\varepsilon},\tilde{\beta}_{\varepsilon})\sim(u_{\varepsilon},J_{\varepsilon},\beta_{\varepsilon}),$

such that

$\tilde{u}_{\varepsilon}\to\tilde{u}\text{ as }\varepsilon\to 0\text{ in
}C^{\frac{1}{8}-,\frac{1}{2}-}(Q_{T}),$ (7.9)
$\tilde{J}_{\varepsilon}\rightharpoonup\tilde{J}\text{ as }\varepsilon\to
0\text{ in }L^{2}(Q_{T})$ (7.10)

and

$\tilde{\beta}_{\varepsilon}\to\tilde{\beta}\text{ as }\varepsilon\to 0\text{
in }C([0,T],\mathbb{R})$

for every $\omega\in[0,1]$. Furthermore,

$\tilde{u}\in
L^{p}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}};C^{\frac{\gamma}{4},\gamma}(Q_{T}))\text{
for all }\gamma\in\left(0,\frac{1}{2}\right)\text{ and }p^{\prime}\in[1,2p).$
(7.11)

###### Proof.

By Skorokhod’s Theorem [25] and its generalization [29] it suffices to show
the tightness of the measures corresponding to the process
$u_{\varepsilon},J_{\varepsilon}$ and $\beta_{\varepsilon}$. The tightness of
the law $\mu_{\beta_{\varepsilon}}$ follows from the fact that it is a Radon
measure in the Polish space $C([0,T];\mathbb{R})$. The tightness of the law
for $\mu_{u_{\varepsilon}}$ on $C^{\frac{1}{8}-,\frac{1}{2}-}(Q_{T})$ follows
from (7.8) and the compactness of the embedding
$C^{\frac{1}{8},\frac{1}{2}}(Q_{T})\subset
C^{\frac{1}{8}-,\frac{1}{2}-}(Q_{T}).$ The tightness of the law for
$J_{\varepsilon}$ follows from (6.2). Finally, (7.11) follows from (7.9),
(7.8) and Fatou’s Lemma, which completes the proof of Proposition 7.3.
Furthermore, from (7.9) and the positivity of
$\tilde{u}_{\varepsilon}(t,x,\omega)$ we have $\tilde{u}(t,x,\omega)\geq 0$
$\mathbb{P}$-almost surely. ∎

######  Proposition 7.4.

Let $T\in(0,\infty)$, $\varepsilon\in(0,1]$ and $u_{0}\in
L^{2}(\Omega,\tilde{\mathcal{F}}_{0},\mathbb{P},H^{1}(\mathbb{T}_{L}))$ is
such that $u_{0}\geq 0$ $\mathbb{P}$-almost surely. Then there is a
subsequence, still denoted $\tilde{u}_{\varepsilon}$, such that

$\partial_{x}\tilde{u}_{\varepsilon}\rightharpoonup\partial_{x}\tilde{u}\text{
in
}L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}},L^{2}([0,T)\times\mathbb{T}_{L}))\text{
as }\varepsilon\to 0.$

Furthermore, the distributional derivative $\partial_{x}^{3}\tilde{u}$ is in
$L^{2}(\\{\tilde{u}>r\\})$, and we can identify
$\tilde{J}_{\varepsilon}=f_{\varepsilon}(\tilde{u}_{\varepsilon})\partial_{x}^{3}\tilde{u}_{\varepsilon}$
and $\tilde{J}=\chi_{\\{\tilde{u}>0\\}}\tilde{u}^{2}\partial_{x}^{3}\tilde{u}$
$\mathbb{P}$-almost surely.

###### Proof.

It follows from (6.2) we have $\partial_{x}\tilde{u}_{\varepsilon}$ is weakly
compact in
$L^{2}(\tilde{\Omega},\tilde{\mathcal{F}}_{0},\mathbb{P},L^{2}([0,T)\times\mathbb{T}_{L}))$.
Thus, for any $\varphi\in
L^{\infty}(\tilde{\Omega},\tilde{\mathcal{F}}_{0},\mathbb{P};C^{1}(Q_{T}))$ we
have

$\tilde{E}\int_{Q_{T}}\partial_{x}\tilde{u}_{\varepsilon}\varphi\,dxdt\to\tilde{E}\int_{Q_{T}}\eta\cdot\varphi\,dxdt\text{
as }\varepsilon\to 0.$ (7.12)

On the other hand, it follows from (6.3), (7.9), dominated convergence
theorem, as well as from the fact that $u_{\varepsilon}$ and
$\tilde{u}_{\varepsilon}$ have the same distribution, that

$\tilde{E}\int_{Q_{T}}\tilde{u}_{\varepsilon}\partial_{x}\varphi\,dxdt\to\tilde{E}\int_{Q_{T}}\tilde{u}\partial_{x}\varphi\,dxdt\text{
as }\varepsilon\to 0.$

Combining the latter convergence with (7.12), we get
$\eta=\partial_{x}\tilde{u}$ $\mathbb{P}$-almost surely. Similarly, since the
joint distribution laws of $(u_{\varepsilon},J_{\varepsilon})$ and
$(\tilde{u}_{\varepsilon},\tilde{J}_{\varepsilon})$ are the same, for any
$\varphi\in C^{\infty}(Q_{T})$,

$0=\mathbb{E}\left|\int_{Q_{T}}J_{\varepsilon}\,\varphi
dxdt-\int_{Q_{T}}f_{\varepsilon}(u_{\varepsilon})(\partial_{x}^{3}u_{\varepsilon})\,dxdt\right|=\tilde{\mathbb{E}}\left|\int_{Q_{T}}\tilde{J}_{\varepsilon}\,\varphi
dxdt-\int_{Q_{T}}f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{u}_{\varepsilon})\,dxdt\right|,$

which implies that
$\tilde{J}_{\varepsilon}=f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{u}_{\varepsilon})$.
Furthermore, since $f_{\varepsilon}(s)$ is non-decreasing in $s$, for any
$r>0$, using (6.2) we have

$\tilde{\mathbb{E}}\int_{Q_{T}}(\partial_{x}^{3}u_{\varepsilon})^{2}\chi_{\\{\|\tilde{u}_{\varepsilon}-\tilde{u}\|_{L^{\infty}(Q_{T})}<\frac{r}{2}\\}\cap\\{\tilde{u}>r\\}}\,dxdt\leq\frac{1}{f_{\varepsilon}(\frac{r}{2})}\tilde{\mathbb{E}}\int_{0}^{T}\int_{\tilde{u}_{\varepsilon}(t)>\frac{r}{2}}f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{u}_{\varepsilon})^{2}\,dx\,dt\leq
C(r,u_{0}),$

where $C(r,u_{0})<\infty$ is independent of $\varepsilon$. Thus, up to a
subsequence, still denoted with $\tilde{u}_{\varepsilon}$, we have

$\partial_{x}^{3}u_{\varepsilon}\chi_{\\{\|\tilde{u}_{\varepsilon}-\tilde{u}\|_{L^{\infty}(Q_{T})}<\frac{r}{2}\\}\cap\\{\tilde{u}>r\\}}\,dxdt\rightharpoonup\eta(r)\chi_{\\{\tilde{u}>r\\}}$
(7.13)

as $\varepsilon\to 0$ in
$L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}};L^{2}(Q_{T}))$.
In order to identify $\eta(r)$, for any $\varphi\in
L^{2}(\tilde{\Omega},C^{\infty}(Q_{T}))$ with $\text{supp}_{(t,x)\in
Q_{T}}\varphi\subset\\{\tilde{u}>r\\}$ for a.e. $\tilde{\omega}$,

$\tilde{\mathbb{E}}\int_{Q_{T}}(\partial_{x}^{3}u_{\varepsilon})\chi_{\\{\|\tilde{u}_{\varepsilon}-\tilde{u}\|_{L^{\infty}(Q_{T})}<\frac{r}{2}\\}\cap\\{\tilde{u}>r\\}}\varphi\,dxdt=-\tilde{E}\int_{Q_{T}}u_{\varepsilon}\chi_{\\{\|\tilde{u}_{\varepsilon}-\tilde{u}\|_{L^{\infty}(Q_{T})}<\frac{r}{2}\\}\cap\\{\tilde{u}>r\\}}\partial_{x}^{3}\varphi\,dxdt\\\
\to-\tilde{\mathbb{E}}\int_{Q_{T}}\tilde{u}\chi_{\\{\tilde{u}>r\\}}\partial_{x}^{3}\varphi\,dxdt$
(7.14)

as $\varepsilon\to 0$, for any $r>0$. The limit in (7.14) follows from the
Dominated Convergence theorem, (6.3), the fact that $u_{\varepsilon}$ and
$\tilde{u}_{\varepsilon}$ have the same distributions, as well as the fact
that the constant $K$ in (6.3) is non-random. Hence, we have

$\eta(r)=\partial_{x}^{3}\tilde{u}\text{ on }\\{\tilde{u}>r\\}\text{ for every
}r>0.$ (7.15)

Now, since $\tilde{u}\geq 0$, for any $\varphi\in
L^{\infty}(\tilde{\Omega},\tilde{F},\tilde{P};L^{\infty}(Q_{T})),r>0$ and
$\varepsilon\in(0,1]$ we have

$\displaystyle
I_{\varepsilon}=\tilde{\mathbb{E}}\int_{Q_{T}}f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{u}_{\varepsilon})\varphi\,dxdt=\tilde{\mathbb{E}}\int_{Q_{T}}f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{u}_{\varepsilon})\chi_{\\{\|\tilde{u}_{\varepsilon}-\tilde{u}\|_{L^{\infty}(Q_{T})}<\frac{r}{2}\\}\cap\\{\tilde{u}>r\\}}\,\varphi\,dxdt$
$\displaystyle+\tilde{\mathbb{E}}\int_{Q_{T}}f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{u}_{\varepsilon})\chi_{\\{\|\tilde{u}_{\varepsilon}-\tilde{u}\|_{L^{\infty}(Q_{T})}\geq\frac{r}{2}\\}\cup\\{\tilde{u}\leq
r\\}}\,\varphi\,dxdt:=J_{1}(r,\varepsilon)+J_{2}(r,\varepsilon).$ (7.16)

Since $f_{\varepsilon}(s)$ converges to $s^{2}$, as well as
$f_{\varepsilon}^{\prime}(s)$ converges to $2s$ uniformly on any interval
$[0,M]$, it follows from (6.3), (7.13) and (7.15), that

$J_{1}(r,\varepsilon)\to\tilde{\mathbb{E}}\int_{Q_{T}}\tilde{u}^{2}(\partial_{x}^{3}\tilde{u})\chi_{\hat{u}>r}\varphi\,dxdt\text{
as }\varepsilon\to 0.$ (7.17)

As for the second term $J_{2}(r,\varepsilon)$, we have

$|J_{2}(r,\varepsilon)|\leq\|\varphi\|_{L^{\infty}(\tilde{\Omega},\tilde{F},\tilde{P};L^{\infty}(Q_{T}))}\left(\tilde{\mathbb{E}}\int_{Q_{T}}f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{u}_{\varepsilon})^{2}\,dx\,dt\right)^{\frac{1}{2}}\\\
\times\left(\tilde{\mathbb{E}}\int_{Q_{T}}f_{\varepsilon}(\tilde{u}_{\varepsilon})\chi_{\\{\|\tilde{u}_{\varepsilon}-\tilde{u}\|_{L^{\infty}(Q_{T})}\geq\frac{r}{2}\\}\cup\\{\tilde{u}\leq
r\\}}dx\,dt\right)^{\frac{1}{2}}.$ (7.18)

Since $f_{\varepsilon}(\tilde{u}_{\varepsilon})$ converges to $\tilde{u}^{2}$
as $\varepsilon\to 0$, it follows from (6.3), that

$\tilde{\mathbb{E}}\int_{Q_{T}}f_{\varepsilon}(\tilde{u}_{\varepsilon})\chi_{\\{\|\tilde{u}_{\varepsilon}-\tilde{u}\|_{L^{\infty}(Q_{T})}\geq\frac{r}{2}\\}\cup\\{\tilde{u}\leq
r\\}}dx\,dt\to\tilde{\mathbb{E}}\int_{Q_{T}}\tilde{u}^{2}\chi_{\\{\tilde{u}\leq
r\\}}\,dx\,dt=O(r^{2})$ (7.19)

as $\varepsilon\to 0$. Thus, (7.18) and (7.19), combined with (6.2), yield

$J_{2}(r,\varepsilon)\leq
Cr\|\varphi\|_{L^{\infty}(\tilde{\Omega},\tilde{F},\tilde{P};L^{\infty}(Q_{T}))},$

for some $C>0$. From (7.10), for any $\varphi\in
L^{\infty}(\tilde{\Omega},\tilde{F},\tilde{P};L^{\infty}(Q_{T}))$ we have

$I_{\varepsilon}\to\int_{Q_{T}}\eta\varphi\,dx\,dt:=I\text{ as }\varepsilon\to
0,\mathbb{P}-\text{ a.s. },$ (7.20)

where $\eta=\eta(t,x,\omega)\in L^{2}(Q_{T})$. Using (6.2) and (6.3) once
again,

$\tilde{\mathbb{E}}\left(\int_{Q_{T}}f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{u}_{\varepsilon})\varphi\,dx\,dt\right)^{2}\\\
\leq\|\varphi\|^{2}_{L^{\infty}(\tilde{\Omega},\tilde{F},\tilde{P};L^{\infty}(Q_{T}))}\sqrt{\tilde{\mathbb{E}}\left(\int_{Q_{T}}f_{\varepsilon}(\tilde{u}_{\varepsilon})\,dx\,dt\right)^{2}}\sqrt{\tilde{\mathbb{E}}\left(\int_{Q_{T}}f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{u}_{\varepsilon})^{2}\,dx\,dt\right)^{2}}\\\
\leq
C\|\varphi\|^{2}_{L^{\infty}(\tilde{\Omega},\tilde{F},\tilde{P};L^{\infty}(Q_{T}))}\sqrt{\tilde{\mathbb{E}}\|\partial_{x}u_{0}\|_{2}^{2}}<\infty.$
(7.21)

By Vitali’s Convergence Theorem, (7.20) and (7.21) yield

$I_{\varepsilon}=\tilde{\mathbb{E}}\int_{Q_{T}}f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{u}_{\varepsilon})\varphi\,dx\,dt\to\tilde{\mathbb{E}}\int_{Q_{T}}\eta\varphi\,dx\
dt=I.$

Due to (7.1) and (7.17),

$\lim_{\varepsilon\to 0}I_{\varepsilon}=\limsup_{\varepsilon\to
0}I_{\varepsilon}=\lim_{\varepsilon\to
0}I_{1}(r,\varepsilon)+\limsup_{\varepsilon\to
0}I_{2}(r,\varepsilon)=\tilde{\mathbb{E}}\int_{Q_{T}}\tilde{u}^{2}(\partial_{x}^{3}\tilde{u})\chi_{\hat{u}>r}\varphi\,dxdt+\limsup_{\varepsilon\to
0}I_{2}(r,\varepsilon).$

This way

$\left|I-\tilde{\mathbb{E}}\int_{Q_{T}}\tilde{u}^{2}(\partial_{x}^{3}\tilde{u})\chi_{\hat{u}>r}\varphi\,dxdt\right|\leq\limsup_{\varepsilon\to
0}I_{2}(r,\varepsilon)=O(r).$ (7.22)

It remains to pass to the limit in (7.22) as $r\to 0$. Using (7.13), (7.15)
and the uniform convergence of $f_{\varepsilon}(\tilde{u}_{\varepsilon})$ to
$\tilde{u}^{2}$ $\mathbb{P}$ -almost surely, we have

$\left|\tilde{\mathbb{E}}\int_{Q_{T}}\tilde{u}^{2}(\partial_{x}^{3}\tilde{u})\chi_{\hat{u}>r}\varphi\,dxdt\right|=\lim_{\varepsilon\to
0}\left|\tilde{\mathbb{E}}\int_{Q_{T}}f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{u}_{\varepsilon})\chi_{\\{\|\tilde{u}_{\varepsilon}-\tilde{u}\|_{L^{\infty}(Q_{T})}<\frac{r}{2}\\}\cap\\{\tilde{u}>r\\}}\,\varphi\,dxdt\right|$

for any $\varphi\in
L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}};L^{2}(Q_{T}))$.
For any $\varepsilon>0$, (6.2) and (6.3) imply

$\displaystyle\left|\tilde{\mathbb{E}}\int_{Q_{T}}f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{u}_{\varepsilon})\chi_{\\{\|\tilde{u}_{\varepsilon}-\tilde{u}\|_{L^{\infty}(Q_{T})}<\frac{r}{2}\\}\cap\\{\tilde{u}>r\\}}\,\varphi\,dxdt\right|\leq\tilde{\mathbb{E}}\int_{Q_{T}}f_{\varepsilon}(\tilde{u}_{\varepsilon})|\partial_{x}^{3}\tilde{u}_{\varepsilon}|\,|\varphi|\,dxdt$
$\displaystyle\leq\left(\tilde{\mathbb{E}}\int_{Q_{T}}f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{u}_{\varepsilon})^{2}\,dxdt\right)^{\frac{1}{2}}\left(\tilde{\mathbb{E}}\int_{Q_{T}}f_{\varepsilon}(\tilde{u}_{\varepsilon})\varphi^{2}\,dxdt\right)^{\frac{1}{2}}\leq
C\left(\tilde{\mathbb{E}}\int_{Q_{T}}\varphi^{2}\,dxdt\right)^{\frac{1}{2}},$

where $C$ is independent of $r$ and $\varepsilon$. This way, the formula

$\mathbb{E}\int_{Q_{T}}\tilde{u}^{2}(\partial_{x}^{3}\tilde{u})\chi_{\\{\tilde{u}>r\\}}\varphi\,dxdt$

defines a linear continuous functional on
$L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}};L^{2}(Q_{T}))$,
and thus, by Rietz Representation Theorem,

$\mathbb{E}\int_{Q_{T}}\tilde{u}^{4}(\partial_{x}^{3}\tilde{u})^{2}\chi_{\\{\tilde{u}>r\\}}\,dxdt\leq
C.$ (7.23)

Combined with the convergence
$\tilde{u}^{2}(\partial_{x}^{3}\tilde{u})\chi_{\\{\tilde{u}>r\\}}\to\tilde{u}^{2}(\partial_{x}^{3}\tilde{u})\chi_{\\{\tilde{u}>0\\}}$
as $r\to 0$, and Vitali’s Convergence Theorem, (7.23) yields

$\tilde{J}=\tilde{u}^{2}(\partial_{x}^{3}\tilde{u})\chi_{\\{\tilde{u}>0\\}}$

$\mathbb{P}$\- almost surely, which concludes the proof of the Proposition. ∎

Combined with (6.2), Proposition 7.4 yields the following Corollary:

######  Corollary 7.5.

We have

$\tilde{u}\in L^{\infty}(0,T;H^{1}(\mathbb{T}_{L}))\cap
C^{\frac{\gamma}{4},\gamma}(Q_{T})$

for $\gamma<\frac{1}{2}$, $\tilde{\mathbb{P}}$ almost surely.

###  7.2 Passing to the limit in the regularized problem

In this subsection we are going to pass to the limit in (1.4) as
$\varepsilon\to 0$. For any $\varphi\in C^{\infty}(\mathbb{T}_{L})$ we have

$(\tilde{u}_{\varepsilon}(t,\cdot),\varphi)_{2}-(u_{0\varepsilon},\varphi)_{2}=\int_{0}^{t}\int_{L}f_{\varepsilon}(\tilde{u}_{\varepsilon}(\partial_{x}^{3}\tilde{u}_{\varepsilon}(s,\cdot))\partial_{x}\varphi
dxds-\frac{1}{2}\int_{0}^{t}(\partial_{x}(\tilde{u}_{\varepsilon}(s,\cdot)),\partial_{x}\varphi)_{2}ds\\\
-\int_{0}^{t}\left(\int_{L}\tilde{u}_{\varepsilon}\partial_{x}\varphi
dx\right)d\tilde{\beta}(s),$

for all $t\in[0,T]$. In the same way as in [8], we have, for all $t\in[0,T]$,

$(\tilde{u}_{\varepsilon}(t),\varphi)_{2}\to(\tilde{u}(t),\varphi)_{2},$
(7.24)
$\int_{0}^{t}\int_{L}f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{u}_{\varepsilon}(s))\partial_{x}\varphi
dxds\to\int_{0}^{t}\int_{\\{\tilde{u}(s)>0\\}}\tilde{u}^{2}(\partial_{x}^{3}\tilde{u}(s))\partial_{x}\varphi
dxds$ (7.25)

and

$\int_{0}^{t}(\partial_{x}\tilde{u}_{\varepsilon},\partial_{x}\varphi)_{2}\,ds\to\int_{0}^{t}(\partial_{x}\tilde{u},\partial_{x}\varphi)_{2}\,ds$
(7.26)

as $\varepsilon\to 0$, $\mathbb{P}$-almost surely. From (7.9) we get
$\|\tilde{u}_{\varepsilon}-\tilde{u}\|_{C(Q_{T})}\to 0$ as $\varepsilon\to 0$
$\mathbb{P}$-almost surely, and (7.24) follows. The convergence (7.25) follows
from the weak convergence

$\tilde{J}_{\varepsilon}=f_{\varepsilon}(\tilde{u}_{\varepsilon})\partial_{x}^{3}\tilde{u}_{\varepsilon}\rightharpoonup\tilde{J}=\chi_{\\{\tilde{u}>0\\}}\tilde{u}^{2}(\partial_{x}^{3}\tilde{u})$

in $L^{2}(Q_{T})$ (see (7.10) and Proposition 7.2). In order to establish
(7.26), note that

$\int_{0}^{t}(\partial_{x}\tilde{u}_{\varepsilon},\partial_{x}\varphi)_{2}\,ds=-\int_{0}^{t}\int_{L}\tilde{u}_{\varepsilon}(\partial_{x}^{2}\varphi)\,dx\
ds\to-\int_{0}^{t}\int_{L}\tilde{u}(\partial_{x}^{2}\varphi)\,dx\
ds=\int_{0}^{t}\int_{L}\partial_{x}\tilde{u}\partial_{x}\varphi\,dx\ ds,$

since $\tilde{u}\in H^{1}(\mathbb{T}_{L})$. Thus

$M_{\varepsilon,\varphi}(t)=-\int_{0}^{t}(\tilde{u}_{\varepsilon}(s),\partial_{x}\varphi)_{2}\,d\tilde{\beta}_{\varepsilon}(s)=(\tilde{u}_{\varepsilon},\varphi)_{2}-(\tilde{u}_{\varepsilon}(0),\varphi)_{2}-\int_{0}^{t}(f_{\varepsilon}(\tilde{u}_{\varepsilon})(\partial_{x}^{3}\tilde{u}_{\varepsilon}),\partial_{x}\varphi)_{2}\,ds\\\
+\frac{1}{2}\int_{0}^{t}(\partial_{x}\tilde{u}_{\varepsilon},\partial_{x}\varphi)_{2}\,ds\to(\tilde{u},\varphi)_{2}-(\tilde{u}(0),\varphi)_{2}-\int_{0}^{t}\int_{\\{\tilde{u}(s)>0\\}}\tilde{u}^{2}(\partial_{x}^{3}\tilde{u}),\partial_{x}\varphi\,dxds\\\
+\frac{1}{2}\int_{0}^{t}(\partial_{x}\tilde{u}_{\varepsilon},\partial_{x}\varphi)_{2}\,ds,\
\ \varepsilon\to 0.$

In this case, it is sufficient to show that for all $t\in[0,T]$

$M_{\varphi}(t)=-\int_{0}^{t}(\tilde{u}(s),\partial_{x}\varphi)_{2}\,d\tilde{\beta}(s).$

The proof of this fact follows the lines of [8], p.229, using [15],
Proposition A.1. Hence, we established the existence of a martingale solution
of (1.1) on any interval $[0,T],T>0$. In the same way as in the proof of
Theorem 6.1, this solution may be extended globally. In order to complete the
proof of the main result, we need to pass to the limit as $\varepsilon\to 0$
in the inequality (6.5). From Corollary 7.5 we get that $u$ is continuous and
bounded $\mathbb{P}$-almost surely on $Q_{T}$. Since $u\in
L^{\infty}(0,T;H^{1}(\mathbb{T}_{L}))$, for fixed $t_{0}\in[0,T]$ we can find
$t_{n}(\omega)\to t_{0}$ such that $\tilde{u}(t_{n},\cdot)\in
H^{1}(\mathbb{T}_{L})$ and
$\|\tilde{u}(t_{n},\cdot)\|_{H^{1}(\mathbb{T}_{L})}$ are uniformly bounded in
$n$ $\mathbb{P}$-almost surely. For any $\varphi\in
C^{\infty}(\mathbb{T}_{L})$, using the continuity of $\tilde{u}$ and Lebesgue
Dominated Convergence Theorem, we have

$\int_{L}\tilde{u}(t_{n},x)\partial_{x}\varphi\,dx\to\int_{L}\tilde{u}(t_{0},x)\partial_{x}\varphi\,dx,\
n\to\infty.$ (7.27)

On the other hand, since $\\{\tilde{u}_{x}(t_{n},\cdot)\\}_{n}$ is bounded in
$L^{2}(\mathbb{T}_{L})$ $\mathbb{P}$-almost surely, for every
$\omega\in\tilde{\Omega}$ we have the existence of a subsequence, along which

$\int_{L}\partial_{x}\tilde{u}(t_{n},x)\varphi(x)\,dx\to\int_{L}\eta\varphi(x)\,dx,\
n\to\infty$ (7.28)

$\mathbb{P}$-almost surely for some $\eta\in L^{2}(Q_{T}).$ Taking

$\int_{L}\tilde{u}(t_{n},x)\partial_{x}\varphi\,dx=-\int_{L}\partial_{x}\tilde{u}(t_{n},x)\varphi\,dx$

into account, (7.27) and (7.28) yield

$\int_{L}\tilde{u}(t_{0},x)\partial_{x}\varphi
dx=-\int_{L}\eta\varphi(x)\,dx.$

Thus $\partial_{x}\tilde{u}(t_{0},x)=\eta\in L^{2}(Q_{T})$ $\mathbb{P}$-almost
surely. The measurability of $\partial_{x}\tilde{u}(t_{0},x,\omega)$ follows
from the Proposition 7.4. Hence $\tilde{u}(t,\cdot)\in H^{1}(\mathbb{T}_{L})$
for all $t\in[0,T]$. The convergence of $\tilde{u}_{\varepsilon}$ to
$\tilde{u}$ also implies

$\int_{L}\tilde{u}_{\varepsilon}(t,x)(\partial_{x}\varphi)\,dx\to\int_{L}\tilde{u}(t,x)(\partial_{x}\varphi)\,dx\text{
for all }t\in[0,T]$

for any $\varphi\in C^{\infty}(\mathbb{T}_{L})$ $\tilde{\mathbb{P}}$\- almost
surely. The latter is equivalent to

$\partial_{x}\tilde{u}_{\varepsilon}(t,\cdot)\rightharpoonup\partial_{x}\tilde{u}(t,\cdot)\text{
in }L^{2}(\mathbb{T}_{L})\text{ as }\varepsilon\to 0$

$\tilde{\mathbb{P}}$\- almost surely, hence

$\int_{L}(\partial_{x}\tilde{u}(t,x))^{2}\,dx\leq\liminf_{\varepsilon\to
0}\int_{L}(\partial_{x}\tilde{u}_{\varepsilon}(t,x))^{2}\,dx.$

Thus, using (6.4) and passing to the limit in (6.5) as $\varepsilon\to 0$, and
taking into account (6.6), the inequality (2.2) follows, which completes the
proof. ∎

## Acknowledgments

The research of Oleksandr Misiats was supported by Simons Collaboration Grant
for Mathematicians No. 854856. The work of Oleksandr Stanzhytskyi and Oleksiy
Kapustyan was partially supported by the Ukrainian Government Scientific
Research Grant No. 210BF38-01.

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