# Non-Uniqueness in Plane Fluid Flows

Heiko Gimperlein Engineering Mathematics, University of Innsbruck, Innsbruck,
AustriaDepartment of Mathematical, Physical and Computer Sciences, University
of Parma, 43124 Parma, Italy Michael Grinfeld Department of Mathematics and
Statistics, University of Strathclyde, Glasgow, G1 1XH, UK Robin J. Knops The
Maxwell Institute of Mathematical Sciences and School of Mathematical and
Computing Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, Scotland, UK
Marshall Slemrod Department of Mathematics, University of Wisconsin, Madison,
WI 53706, USA

###### Abstract

Examples of dynamical systems proposed by Z. Artstein and C. M. Dafermos admit
non-unique solutions that track a one parameter family of closed circular
orbits contiguous at a single point. Switching between orbits at this single
point produces an infinite number of solutions with the same initial data.
Dafermos appeals to a maximal entropy rate criterion to recover uniqueness.

These results are here interpreted as non-unique Lagrange trajectories on a
particular spatial region. The corresponding velocity is proved consistent
with plane steady compressible fluid flows that for specified pressure and
mass density satisfy not only the Euler equations but also the Navier-Stokes
equations for specially chosen volume and (positive) shear viscosities. The
maximal entropy rate criterion recovers uniqueness.

Key words: Non-uniqueness, entropy rate criterion, Euler equations, Navier-
Stokes equations.
MSC classes: 76N10 (primary), 34A12, 35Q35 (secondary).

##  1 Introduction

This paper derives explicit non-unique continuous Lagrange trajectories
related to certain Euler and Navier-Stokes steady compressible plane fluid
flow. The orbits tracked by the trajectories belong to a one parameter family
of closed contiguous circular paths. Non-uniqueness occurs when trajectories
switch orbits at the common point of intersection. Even so, uniqueness follows
for the problems considered here by appeal to an entropy rate criterion that
isolates a single preferred trajectory.

The non-unique behaviour is not dissimilar to that occurring in the convex
integration investigations by De Lellis and Szèkelyhidi [8, 9] and Buckmaster
and Vicol [4] for the incompressible Euler and Navier-Stokes systems. It is
therefore reasonable to speculate whether a unique solution also might be
selected by application of an entropy rate criterion. Brief comment on this
aspect is given in Section 5.

The present analysis is partly inspired by an example in control theory
proposed by Artstein [1, see eqn. (6.1)] who, rather than non-uniqueness,
emphasises non-existence of suitable controls that otherwise stabilise the
system and ensure solutions asymptotically approach zero. The primary
motivation, however, comes from what Dafermos [7] refers to as an artificial
example of a nonlinear oscillator governed by an ordinary differential
equation. The associated trajectories considered by both Artstein and Dafermos
are circular, and touch at their common point of intersection taken to be the
origin. Consequently, trajectories can be switched at the origin by
instantaneous variation of an _entropy_ function that is constant along
individual trajectories. In this manner, irrespective of initial conditions,
an uncountable set of different solutions can be chosen to establish non-
uniqueness.

Dafermos [7] recovers uniqueness by selecting that particular solution which
dissipates entropy at maximum rate. Such a solution is not only unique but
physically relevant in the sense of Hadamard’s definition of well-posedness.
Similar entropy rate criteria, originally formulated to treat shock wave
propagation, include the maximal entropy rate proposed by Ziegler [23] and the
minimum entropy rate due to Prigogine [18, Chapt. V]. Both principles are
reviewed in the monograph by Moroz [16, Chapt. 1]. Glimm, Lazarev and Chen
[12] remark that the subtle difference between these two principles may cause
confusion. Ziegler’s principle is relevant to closed thermodynamic systems
while Prigogine’s principle applies to open thermodynamic systems. Evidently,
Dafermos’s investigation relates to a closed system.

Dafermos’s example, however, is far from trivial. It is shown in this paper
that solutions, interpreted as steady Lagrange trajectories (particle paths),
are related to steady plane flows for compressible Euler and Navier-Stokes
equations with variable mass density and positive shear viscosity. The
extension to fluid problems demonstrates that tracking arbitrarily prescribed
discrete entropy profiles produces an infinite number of continuous Lagrange
trajectories with the same initial conditions. Consequently, trajectories are
not unique. Nevertheless, a single unique trajectory is identified by the
maximal entropy rate admissibility criterion proposed by Dafermos [7]. An
interpretation of this result is that “wild” initial data can be produced for
steady solutions to an initial boundary problem for both the Euler and Navier-
Stokes equations. The term “wild” is understood in the sense that there are an
infinite number of Lagrange trajectories for the same initial conditions. The
entropy rate criterion identifies a unique trajectory and renders the notion
of “wildness” redundant.

The conclusion contrasts with the uniqueness results for weak solutions to the
incompressible Navier-Stokes equations obtained by Robinson and Sadowski [19,
20]. These authors show for sufficiently regular velocity that the Lagrange
trajectories are unique and smooth in time for almost every initial data. On
the other hand, the velocity for the Lagrange trajectories considered here
lacks Lipschitz continuity at one point and therefore does not satisfy the
continuity conditions imposed by Robinson and Sadowski.

An alternative admissibility procedure, introduced by Brenier [3], relies upon
the concave maximization of a certain functional to derive smooth solutions to
Euler’s equation. Further procedures include that by Lasarzik [14] who adopts
a maximal entropy rate to conclude that “dissipative solutions” as defined by
P.-L. Lions [15] exist, are unique, and depend continuously upon initial data
for both the Euler and Navier-Stokes equations of incompressible fluid
dynamics. It must be noted, however, that such dissipative solutions are not
in general weak solutions. A different notion of dissipative solution is
treated by Breit, Feireisl and Hofmanova [2] who replace the concept of a weak
solution by a semi-flow satisfying a maximal entropy rate production
criterion. By construction, the semi-flow implies modified well-posedness of
the Euler system.

Also of importance is whether the maximal entropy rate criterion can be
accurately simulated by numerical analysis. Glimm, Lazarev and Chen [12] note
that this comparatively delicate issue is of substantial physical importance.
A numerical method that preserves qualitative behaviour is required and, in
fact, it is shown in Section 4 that the symplectic Euler method successfully
ensures convergence to the unique solution selected by the criterion. The
explicit Euler method fails in this respect.

As already remarked, based upon the Artstein-Dafermos examples, the non-
uniqueness described in this paper for the Euler and Navier-Stokes equations
is due to the corresponding Lagrange trajectories switching at the origin from
one circular path to another along each of which there is different constant
_entropy_. Arbitrary prescription of these entropies and the order in which
the trajectories are described results in an uncountable number of distinct
solutions satisfying the same initial conditions. The behaviour has features
in common with that respectively established by De Lellis and Szèkelyhidi [8,
9] and by Buckmaster and Vicol [4] for weak solutions on a torus to the
incompressible Euler and Navier-Stokes equations. These authors use convex
integration techniques to prove that an arbitrary number of such weak
solutions exist possessing both the same assigned smooth global energy and the
same initial conditions. Consequently, non-uniqueness is established.
(Extension to the compressible Euler equations is due to Chiodaroli and Kreml
[6], and to Feireisl [11]).

Arbitrary specification of an appropriate profile is thus common to both
developments and invites exploration of the possible application of a maximal
entropy rate criterion to identify a unique solution from among the many weak
solutions determined by convex integration methods.

Of separate interest is the physical relevance of weak solutions obtained by
convex integration to the Euler and Navier-Stokes equations. Such solutions
are the limit of increasingly high frequency oscillatory perturbations that
eventually may contravene the basic continuum hypotheses assumed in the
derivation of the equations. In this respect, the Lagrange trajectories
considered here are explicit and indeed are continuously differentiable except
at the origin. Clearly, apart from this singularity, they accord with
continuum hypotheses.

Section 2 commences with Dafermos’s construction of non-unique orbits for a
nonlinear oscillator that involves the arbitrary prescription of an _entropy_
function constant on each path. A theorem conveniently summarises relevant
conclusions. Two admissibility criteria proposed by Dafermos are then
described that recover a physically meaningful unique trajectory. The first
criterion augments the system of equations by a friction term. Solutions to
the penalised system tend to a unique solution in the limit as friction tends
to zero. The second, or maximal entropy rate, criterion requires maximum
dissipation of the individual entropies as trajectories are successively
switched and leads to the same path obtained by the limiting friction
argument. Section 3 establishes that the velocity occurring in the nonlinear
oscillator satisfies the plane compressible Euler equations with specified
pressure and particular non-uniform mass density. Moreover, the corresponding
Lagrange trajectories are geometrically similar to those obtained from the
nonlinear oscillator and therefore possess similar non-uniqueness features.
Isolation of a unique trajectory follows from the maximal entropy rate
criterion. The analysis is extended to the corresponding plane compressible
Navier-Stokes equations with specially chosen (positive) shear and volume
viscosities. These viscosity coefficients are computed in the Appendices.
Section 4 discusses the numerical approximations of solutions to the Dafermos
system considered in Section 2. The objective is to employ a numerical method
that not only preserves the qualitative behaviour of constant entropy along
individual trajectories, but is also capable of simulating trajectories
selected by the maximal entropy rate criterion. Convergence to a unique
solution would then be implied. The symplectic Euler method satisfies these
requirements since it determines the approximation to the entropy function to
within an error of the order of the step size $h>0$. The approach is
diagramatically illustrated and arguments are given that the discrete flow
selects the solution given by the entropy rate admissibility criterion.
Section 5 comments on the resemblance between the results of Section 3 and
those in the convex integration literature that for a given smooth global
energy construct an infinite number of solutions to the Euler equations on a
torus. A final brief remark concerns the fundamental issue of whether the
eventual irregularity of such solutions contravenes basic continuum
hypotheses.

Standard notation is employed throughout the main text apart from Remark 3.1
where bold type indicates vectors. The Appendices introduce an indicial
notation accompanied by the summation and comma conventions.

##  2 Motivation and admissibility criteria

This section reviews previous contributions that inspired the present
generalisation to the Euler and Navier-Stokes equations. The discussion also
explains how an entropy rate criterion recovers a unique solution.

Let $(x(t),y(t))\in\mathrm{I\\!R\\!}^{2},\,t\in\mathrm{I\\!R\\!},$ be state
variables corresponding to the Cartesian coordinates of a point moving in the
plane as time varies. Dafermos [7] studies motions whose state variables
satisfy the system of ordinary differential equations for a nonlinear
oscillator given by

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle y,$ (2.1)
$\displaystyle\dot{y}$ $\displaystyle=$
$\displaystyle\frac{(y^{2}-x^{2})}{2x},$ (2.2)

where a superposed dot indicates differentiation with respect to time $t$.

Initial data are specified by

$(x(0),\,y(0))=(x_{0},\,y_{0}),$ (2.3)

where $(x_{0},\,y_{0})$ are prescribed.

The velocity component (2.1) is continuously differentiable everywhere but the
component (2.2) is not continuous at the origin. Dynamics are best understood
from an integrated form of the equations given below.

It is immediate from (2.1) and (2.2) that the entropy function $H(x,y)$,
defined by

$H(x,y):=\frac{(x^{2}+y^{2})}{2x},$ (2.4)

is invariant with respect to time $t$; that is

$\frac{dH(x(t),y(t))}{dt}=0,$ (2.5)

or

$H(x,\,y)=c$ (2.6)

for positive constant $c$.

Note that $H(x,y)$ is not to be confused with the (kinetic) energy. The
relationship between these quantities is presented in Section 5.

The constant entropy function (2.4) implies that the state variables traverse
the one parameter family of closed circular orbits

$x^{2}+y^{2}=2xc.$ (2.7)

Consequently,

$(x-c)^{2}+y^{2}=c^{2},$ (2.8)

and consequently $0\leq x(t)\leq 2c,\,-c\leq y(t)\leq c.$ Moreover, (2.1)
implies that $x(t)$ increases or decreases as $y(t)$ is positive or negative
so that the closed circular orbits are traversed clockwise. For varying $c\geq
1$ they touch each other at their common point of intersection located at the
origin. Since solutions trace the family of orbits (2.8), equation (2.2) may
be alternatively written as

$\dot{y}=(c-x).$ (2.9)

Artstein’s study [1, eqn. 6.1] is within the broader context of a control
problem in which (2.1) and (2.9) are generalised to the family

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle(2xy)w,$ (2.10)
$\displaystyle\dot{y}$ $\displaystyle=$ $\displaystyle(y^{2}-x^{2})w,$ (2.11)

where $w(x,y)$ is a scalar control chosen to ensure stabilisability of the
equilibrium state $x=y=0.$ It follows immediately that since

$\frac{dx}{2xy}=\frac{dy}{(y^{2}-x^{2})},$ (2.12)

the paths followed by solutions to (2.10) and (2.11) are given by the family
of circular closed orbits (2.8) or alternatively (2.7), and consequently the
function $H(x,t)$ is constant along each path. Dependent upon the choice of
$w$, the Lipschitz condition may fail which would imply lack of uniqueness for
paths with given initial data and which we now examine.

Particular choices of the control function $w(x,y)$ deserve attention.

First, set

$w=x^{-\alpha},\qquad\alpha>0.$ (2.13)

Substitution in (2.10) and (2.11) after appeal to (2.7) yields

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle
2x^{(\frac{3}{2}-\alpha)}(2c-x)^{1/2},$ (2.14) $\displaystyle\dot{y}$
$\displaystyle=$ $\displaystyle 2x^{(1-\alpha)}(c-x).$ (2.15)

Hence (2.7), (2.14) represent an integrated form of (2.10), (2.14) where
conservation of $H$ given (2.6) is extended by continuity to the origin. It is
here that lack of Lipschitz continuity of the right hand side of (2.14) makes
itself apparent and the usual loss of uniqueness occurs in $x$ for
$1\leq\alpha<3/2$ (see, for example, Burkill [5]), while $y$ is well defined
from the choice of $x$. As stated below, $\alpha=1$ is the value adopted by
Dafermos [7].

As second choice, put

$w=\frac{1}{2x^{2}},$ (2.16)

to obtain from (2.10) and (2.11) the Hamiltonian system

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle\frac{y}{x},$ (2.17)
$\displaystyle\dot{y}$ $\displaystyle=$
$\displaystyle\frac{(y^{2}-x^{2})}{2x^{2}},$ (2.18)

whose Hamiltonian is the function $H(x,y)$.

Finally, consider

$w=\frac{1}{2x}$ (2.19)

which reduces (2.10) and (2.11) to the system (2.1) and (2.2) considered by
Dafermos [7]. Explicit solutions, of interest when interpreted in Section 3 as
Lagrange trajectories, are easily derived. Details are presented in Appendix
A. Express (2.8) as

$\dot{x}(t)=y(t)=\pm\sqrt{x(2c_{0}-x)},$

where $c$ has assumed the particular value $c_{0}$ specified from initial data
according to

$c_{0}=\frac{(x_{0}^{2}+y_{0}^{2})}{2x_{0}}.$ (2.20)

It is supposed that initial data are such that $c_{0}\geq 1.$ Choose the
positive square root to obtain by integration

$x(t)-c_{0}=c_{0}\cos{(-t+\theta_{0})}.$ (2.21)

The corresponding expression for $y(t)$, obtained by differentiation of the
last relation, becomes

$y(t)=c_{0}\sin{(-t+\theta_{0})},$ (2.22)

and shows that the constant $\theta_{0}$ is determined from initial data to be

$\tan{\theta_{0}}=\frac{y_{0}}{x_{0}-c_{0}}.$ (2.23)

The circular orbits (2.21) and (2.22) may otherwise be derived by integration
of (2.1) and (2.9). They are centred at $(c_{0},0)$, where $c_{0}$ is
determined from initial conditions (2.20), and are restricted to the set
parametrized by $1\leq c_{0}\leq R$ sketched in Figure 1.

$x$$y$$1$

Figure 1: Phase portrait for (2.21) and (2.22).

Orbits tracked by the state variables, as predicted by (2.8), are centred at
$(c_{0},\,0)$ and pass through the origin at times
$(-t+\theta_{0})=-(2n+1)\pi,\,n=0,1,2,\ldots$ . Each circular orbit is
completed in the same time $2\pi$ but at speeds dependent upon the radius:

$\left(\dot{x}^{2}+\dot{y}^{2}\right)^{1/2}=c_{0},$

corresponding to rigid body rotation.

Further properties are derived upon conversion to polar coordinates
$(r,\theta)$. Set $x=r\cos{\theta},\,y=r\sin{\theta}$ so that (2.1) and (2.2)
become

$\displaystyle\dot{\theta}$ $\displaystyle=$ $\displaystyle-\frac{1}{2},$
(2.24) $\displaystyle\dot{r}$ $\displaystyle=$
$\displaystyle\frac{1}{2}r\tan{\theta},$ (2.25)

which by integration lead to

$\displaystyle\theta(t)$ $\displaystyle=$
$\displaystyle-\frac{t}{2}+\theta_{0}/2,$ (2.26) $\displaystyle
r(t)\cos{(\theta_{0}/2)}$ $\displaystyle=$ $\displaystyle
r_{0}\cos{\theta(t)},$ (2.27)

where $\theta_{0}$ satisfies (2.23) and $(r_{0},\,\theta_{0}/2)$, the initial
values of $(r(t),\,\theta(t))$, accordingly are given by

$r_{0}=\left(x^{2}_{0}+y^{2}_{0}\right)^{1/2},\qquad\tan{(\theta_{0}/2)}=\frac{y_{0}}{x_{0}},$

and

$r_{0}=2c_{0}\cos{(\theta_{0}/2)}.$

The last expression inserted into (2.27) gives

$r(t)=2c_{0}\cos{\theta(t)},$ (2.28)

which is the circular orbit previously obtained.

The state variables expressed by either (2.21) and (2.22), or by (2.26) and
(2.28), indicate the explicit manner in which orbits pass through the origin
$(0,0)$.

The calculations so far suppose that $c\geq 1$. To include the region $c<1$,
Dafermos [7] considers the modified system

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle
y,\qquad\dot{y}=\frac{(y^{2}-x^{2})}{2x},\quad\mbox{for}\qquad(x-1)^{2}+y^{2}\geq
1,$ (2.29) $\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle
y,\qquad\dot{y}=(1-x),\qquad(x-1)^{2}+y^{2}<1.$ (2.30)

Solutions to (2.30) are of the same form as (2.21) and (2.22) and indicate
that corresponding trajectories move clockwise on circles

$(x-1)^{2}+y^{2}=a^{2},\qquad a<1,$ (2.31)

of radius $a<1$ centred at $(1,0)$ when $c<1$ in (2.6).

Of special interest is the circle of unit radius for which $a=c=1$.

Irrespective of prescribed initial conditions, each member of the family of
orbits created by a sequence of different constants $c\geq 1$ possess the
common property of lying in the positive half-plane and passing through the
coordinate origin where they are mutually tangential. Dafermos [7] observes
that global uniqueness of solutions to (2.1)-(2.2) is violated on allowing
switching at the origin between orbits of different entropy levels $H(x,y)$
defined by (2.6). Specifically, a trajectory with initial conditions such that

$H(x_{0},\,y_{0}):=\frac{(x^{2}_{0}+y^{2}_{0})}{2x_{0}}=c_{0}>1,$ (2.32)

on reaching the origin switches to another trajectory for which

$H(x,y)=c_{1}>1,\qquad c_{1}\neq c_{0}.$ (2.33)

The composite trajectory is clearly continuous in the time variable $t$ and
satisfies (2.29) for almost all $t$; in fact for all $t$ for which
$(x(t),\,y(t))\neq(0,\,0)$. From (2.9) the jump in the acceleration
$\ddot{x}=\dot{y}$ at the origin is given by

$[\ddot{x}]_{(0,\,0)}=[\dot{y}]_{(0,\,0)}=c_{1}-c_{0}.$ (2.34)

Hence, $\dot{y}$ is bounded for all $t$ and $y$ is Lipschitz continuous.

Trajectories for which $c<1$ remain interior to the circle of radius $1$ and
do not pass through the origin. Consequently, switching is not possible.

The following theorem summarises for convenience these non-uniqueness results
which are also applicable to the general control problem (2.10) and (2.11). In
Section 5 the conclusion is compared to the non-uniqueness theorem of
Buckmaster and Vicol [4] for the Navier-Stokes equations.

$R$$\Omega_{2}$$\Omega_{1}$$x$$y$$1$

Figure 2: The region $\Omega=\Omega_{1}\cup\Omega_{2}$.

######  Theorem 2.1.

For $R>1$, define the plane region $\Omega=\Omega_{1}\cup\Omega_{2}$, depicted
in Figure 2, by

$\displaystyle\Omega_{1}$ $\displaystyle:=$
$\displaystyle\left\\{(x,y)\in\mathrm{I\\!R\\!}^{2}:1\leq(x-1)^{2}+y^{2},\,(x-R)^{2}+y^{2}\leq
R^{2}\right\\},$ (2.35) $\displaystyle\Omega_{2}$ $\displaystyle:=$
$\displaystyle\left\\{(x,y)\in\mathrm{I\\!R\\!}^{2}:(x-1)^{2}+y^{2}<1\right\\}$
(2.36)

and let $e(t)$ denote the piece-wise constant function

$e(t)=\begin{cases}c_{0},&0\leq t\leq t_{1},0\leq t_{1}<2\pi,\\\
c_{n},&t_{n}=t_{1}+2(n-1)\pi\leq t<t_{n}+2\pi,\end{cases}$ (2.37)

for $n=1,2,\dots$ and $0\leq t<\infty$.

Then for any sequence $c_{n}\geq 1$, there exists a Lipschitz continuous
solution to (2.1) and (2.2). Trivially, the initial value problem with initial
data $(x(0)=x_{0},\,y(0)=y_{0})\in\Omega_{1}$ possesses an infinite number of
Lipschitz continuous solutions.

Dafermos [7] regains uniqueness by suggesting two admissibility criteria. The
first asserts that the physically meaningful solution to (2.29) and (2.30) on
$\Omega=\Omega_{1}\cup\Omega_{2}$ is the limit as $\gamma\rightarrow 0^{+}$ of
solutions $(x_{\gamma},\,y_{\gamma})$ to the system with friction:

$\displaystyle\dot{x}_{\gamma}$ $\displaystyle=$ $\displaystyle y_{\gamma},$
(2.38) $\displaystyle\dot{y}_{\gamma}$ $\displaystyle=$ $\displaystyle
g(x_{\gamma},\,y_{\gamma})-\gamma y_{\gamma},$ (2.39)

where

$g(x,y)=\begin{cases}\frac{(y^{2}-x^{2})}{2x},&(x-1)^{2}+y^{2}\geq 1,\\\
(1-x),&(x-1)^{2}+y^{2}<1.\end{cases}$

Solutions $(x_{\gamma},\,y_{\gamma})$ to (2.38) and (2.39) possess the limit
$(x_{\gamma},\,y_{\gamma})\rightarrow(x,y)$ as $\gamma\rightarrow 0^{+}$ where

  * (i)

$(x_{\gamma}(0),\,y_{\gamma}(0))\rightarrow(x_{0},\,y_{0})$ and

$c_{0}=\frac{(x^{2}_{0}+y^{2}_{0})}{2x_{0}}.$

  * (ii)

The limit $(x,\,y)$ moves clockwise on the circle radius $c_{0}$ until it
reaches the origin $x=0,\,y=0$.

  * (iii)

Upon reaching the origin, the solution switches _once_ to the circle with
entropy $e=1$ of radius $1$ centred at $(x,\,y)=(1,0)$. It remains on this
circle moving clockwise.

The second criterion proposed by Dafermos is the _entropy rate admissibility
criterion_ which, although not applied directly to the present problem,
requires that physically admissible solutions should have not only non-
increasing entropy $e(t)$, but that the entropy should decrease at maximum
possible rate. A simple illustration of the criterion for the system (2.29)
and (2.30) is sketched in Figures 3-4 for possible entropy profiles (2.37).

Let $\delta>0$ and in Figure 3 select the point $\tau_{1}=t_{1}-\delta$ such
that $0<\tau_{1}=t_{1}-\delta<t_{1}$ and $\tau_{1}$ lies within the first
interval specified in (2.37) and therefore locates a point on the branch
$e=c_{0}$.

$t$$e(t)$$t_{1}$$t_{1}+2\pi$$t_{1}+4\pi$$t_{1}+6\pi$$t_{1}+8\pi$$c_{0}$$c_{1}$$c_{3}$$c_{2}$$1$$\times$$\times$$\times$$\times$$\times$

Figure 3: Discrete energy profile.

Subsequent points given by $\tau_{n}=t_{1}-\delta+2\pi n,\,n=1,2,\ldots$ are
chosen to lie in the interval $(t_{n},t_{n}+2\pi)$ so that $\tau_{n}$ is a
point on $e=c_{n}$. On joining the points $\tau_{n},\,n=1,2,\ldots$ a
piecewise linear graph is produced whose piecewise linear approximation is the
original entropy function $e(t)$. The usual entropy criterion stipulates decay
of energy and accordingly the piecewise linear graph must be non-increasing.
This, however, still permits an infinite number of profiles. A single
preferred profile follows from the entropy rate admissibility criterion which
identifies a single profile corresponding to the graph of maximal negative
slope shown in Figure 4. Thus, the entropy rate criterion leads to the single
solution derived by the limiting friction argument.

$t$$e(t)$$c_{0}$$1$$\times$$\times$

Figure 4: Energy profile with maximal negative slope.

An immediate conclusion is that while it is possible for a smooth solution to
(2.29) and (2.30) with initial data in $\Omega_{1}$ to remain on the circle of
radius $c_{0}$, the admissible unique solution given by either the limiting
friction method or entropy rate criterion, though continuous, is not smooth in
the sense of (2.34). Consequently, the nonlinear oscillator considered by
Dafermos contradicts the conjecture _that to achieve uniqueness only smooth
solutions should be admitted._

The notion of _smoothness_ is crucial not only here but for the discussion in
Section 5 of the entropy rate criterion in relation to the theorems to be
there stated of Buckmaster and Vicol [4] and of De Lellis and Szèkelyhidi [8,
9]. Consider, for example, the profile $e(t),\,e(0)\neq 0$ shown in Figure 5.

$t$$e(t)$

Figure 5: Smooth energy profile.

The entropy rate criterion selects the preferred profile as that having
maximal decreasing derivative $e^{\prime}(t)$. Consequently, for any fixed
energy profile $e(t)$ there will always be one with a more rapidly decreasing
profile. Hence all non-zero smooth profiles given by the results of Buckmaster
and Vicol [4] and by De Lellis and Szèkelyhidi [9, 10] are inadmissible
according to the entropy rate criterion. But clearly the limit of such
profiles is the maximal profile $e(t)=e(0)H(t)$ where $H(t)$ satisfies
$H(0)=1$ and $H(t)=0$ for $t>0$. This maximal profile is neither smooth nor
even a weak solution. A similar argument used by Feireisl [11] proves
inadmissibility of weak solutions to the compressible Euler equations derived
by convex integration methods. On the other hand, when $e(0)=0$, the maximal
entropy rate yields the trivial admissible profile $e(t)=0,\,t\geq 0$. This
remark indicates that the profile found by Scheffer [21] and by Shnirelmann
[22] for the Euler equations is inadmissible.

######  Remark 2.1 (Regular oscillations).

As noted by Dafermos [7], the entropy rate admissibility criterion applied to
the nonlinear oscillator implies that starting from any initial data, the
oscillation after once having passed through the origin subsequently is
regularly periodic for all time.

The next task is to interpret the solution to (2.1) and (2.2) as Lagrange
trajectories appropriate to certain steady plane fluid flows satisfying either
the Euler or Navier-Stokes equations.

##  3 Plane Euler and Navier-Stokes equations

This section examines implications of Theorem 2.1 for compressible steady
flows governed by Euler and Navier-Stokes equations on the plane region
$\Omega_{1}$ exterior to the unit circle defined by (2.35) and on the region
$\Omega=\Omega_{1}\cup\Omega_{2}$. (See Figure 2.) It is supposed that (2.29)
represents Lagrange trajectories of fluid particles and that for both the
Euler and Navier-Stokes equations the velocity vector has cartesian components

$\displaystyle u(x,y)$ $\displaystyle=$ $\displaystyle y,$ (3.1)
$\displaystyle v(x,y)$ $\displaystyle=$
$\displaystyle\frac{(y^{2}-x^{2})}{2x},$ (3.2)

for which the entropy $H(x,y)$ defined by (2.6) is constant.

######  Remark 3.1 (Lagrange trajectories and Euler’s equation.).

Throughout this Remark, vector quantities are denoted in bold type. Lagrange
trajectories and the Euler equations represent two distinct methods for
describing fluid motion. Lagrange trajectories trace the motion of a fluid
particle in space and time and consequently the vector velocity field v is a
function of initial spatial position $\textbf{x}_{0}$ and time $t$. The
position vector $\textbf{x}(t)=\textbf{x}(\textbf{x}_{0},t)$ and velocitiy
vector $\textbf{v}(\textbf{x},t)=\textbf{v}(\textbf{x}_{0},t)$ are related by
the system of ordinary differential equations:

$\dot{\textbf{x}}=\textbf{v}(\textbf{x}_{0},t),\qquad\textbf{x}(0)=\textbf{x}_{0},$
(3.3)

and for a solution to exist and to be unique it is sufficient that v is
Lipschitz continuous. There are, however, conditions under which the solution
may neither exist nor be unique.

The same velocity field v occurs in the Euler description except that the
motion is considered at fixed spatial position x and at varying time. The
governing equations are the Euler system of partial differential equations
subject to prescribed initial and boundary data. Existence and uniqueness of a
solution follow from theorems in partial differential equations and depend
upon function spaces and definition of solution. The solution, however
defined, may be non-unique, but can be used as the velocity in the ordinary
differential equations (3.3) to derive the Lagrange trajectories which
likewise may be non-unique for given initial data. The procedure may be
reversed and the possible non-unique Lagrange trajectories first computed for
specified velocity field. This velocity is then substituted in the Euler
equations and the corresponding pressure, density, and initial and boundary
data calculated. The initial boundary problem obtained by this semi-inverse
method may be non-unique and solutions may exist additional to the velocity
used in the system (3.3).

The interrelation between solutions for Lagrange trajectories and for the
Euler system is remarked upon by Robinson and Sadowky [19, 20]. The semi-
inverse procedure is adopted in what follows.

Similar remarks apply to the connexion between Lagrange trajectories and the
Navier-Stokes equations examined later in this Section.

It is noted that Lagrange trajectories have recently found applications in
oceanography, atmospherics and biology.

The following lemmas establish that the particular component velocities on the
right of (3.1) and (3.2) satisfy the compressible Euler equations with mass
density

$\rho(x)=x^{-1}.$ (3.4)

######  Remark 3.2.

The conclusion, however, is not confined to velocity components (3.1) and
(3.2), nor to the mass density (3.4). Appendix D explains how a certain
general class of velocities derived from a conservative system of Lagrange
trajectories also satisfies Euler’s equations for suitable choice of mass
density and pressure.

######  Lemma 3.1.

In the interior of $\Omega_{1}$, defined by (2.35), the velocity components
(3.1) and (3.2) satisfy the continuity equation

$\frac{\partial}{\partial x}\left(\rho u\right)+\frac{\partial}{\partial
y}\left(\rho v\right)=0.$ (3.5)

Proof: By direct substitution. Note, however, that since

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial
y}=\frac{y}{x},\qquad x\neq 0,$ (3.6)

the fluid is compressible except possibly on $y=0,\,x\neq 0.$

######  Lemma 3.2.

On $\partial\Omega_{1}\backslash\left\\{(0,0)\right\\}$, the boundary
condition

$(un_{1}+vn_{2})=0,\qquad(x,y)\in\partial\Omega_{1}\backslash(0,0),$ (3.7)

holds, where $n_{1},\,n_{2}$ are the cartesian components of any normal vector
on the boundary.

Proof: The inner boundary given by

$(x-1)^{2}+y^{2}=1,\qquad(x,y)\neq(0,0),$ (3.8)

has unit normal whose components are $\left((x-1),\,y\right)$. Moreover, on
(3.8)

$u(x,y)=y,\qquad v(x,y)=(1-x),$

and (3.7) is immediate when $(x,y)\neq(0,0)$. The same argument applies to the
outer boundary

$(x-R)^{2}+y^{2}=R^{2},\qquad R>1.$ (3.9)

Condition (3.7) trivially implies tangential flow at the boundary.

######  Lemma 3.3.

The given velocity and density on the interior of $\Omega_{1}$ satisfy the
balance of steady linear momentum

$\displaystyle\frac{\partial}{\partial x}\left(\rho
u^{2}\right)+\frac{\partial}{\partial y}\left(\rho uv\right)+\frac{\partial
p}{\partial x}$ $\displaystyle=$ $\displaystyle 0,$ (3.10)
$\displaystyle\frac{\partial}{\partial x}\left(\rho
uv\right)+\frac{\partial}{\partial y}\left(\rho v^{2}\right)+\frac{\partial
p}{\partial y}$ $\displaystyle=$ $\displaystyle 0,$ (3.11)

subject to the specified pressure

$p(x,y):=\frac{(x^{2}+y^{2})}{2x}-1=H-1.$ (3.12)

As $H$ is constant along trajectories, we hence have the pressure $p$ constant
along trajectories.

Proof: Insertion of velocity and density into the balance of steady linear
momentum (3.10) after integration yields

$p(x,y)=\frac{(x^{2}+y^{2})}{2x}+f(y),$

where $f$ is an arbitrary function. On the other hand, (3.11) leads to

$p(x,y)=\frac{y^{2}}{2x}+g(x),$

for arbitrary function $g$. Set $f\equiv 0$ and $g=(x/2+b)$ for arbitrary
constant $b$ taken to be $b=-1$ to ensure that $p(x,y)$ vanishes on the inner
boundary (3.8).

Lemma 3.1, Lemma 3.7 and Lemma 3.3 establish the following theorem,

######  Theorem 3.1.

When the velocity, density and pressure satisfy (3.1), (3.2), (3.4), and
(3.12), the fluid flow with Lagrange particle trajectories (2.29) and (2.30)
in $\Omega_{1}$ satisfy the continuity equation (3.5), balance of steady
linear momentum (3.10) and (3.11), and the boundary condition (3.7).

Note that the velocity field (3.1) and (3.2) generates Lagrange trajectories
that are non-unique for initial data specified in $\Omega_{1}$. A unique
Lagrange trajectory is obtained from the entropy rate admissibility criterion
employed in accordance with the construction of the previous section.

The region $\Omega_{1}$ occupied by the fluid may be enlarged to include the
interior of the unit circle $(x-1)^{2}+y^{2}=1$ as proposed by Dafermos; see
(2.30). As before (see (2.36)), denote the unit disc by

$\Omega_{2}:=\left\\{(x,y):(x-1)^{2}+y^{2}\leq 1\right\\}.$ (3.13)

Then for $(x,y)\in\Omega_{2}$ put

$\displaystyle\rho(x,y)$ $\displaystyle=$ $\displaystyle 1,$ (3.14)
$\displaystyle u(x,y)$ $\displaystyle=$ $\displaystyle y,$ (3.15)
$\displaystyle v$ $\displaystyle=$ $\displaystyle(1-x).$ (3.16)

It follows from mass conservation, or directly, that

$\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial
y}\right)=0,\qquad(x,y)\in\Omega_{2},$ (3.17)

and the flow is incompressible. Expressions for balance of steady linear
momentum become

$\displaystyle\frac{\partial}{\partial x}(u^{2})+\frac{\partial}{\partial
y}(uv)+\frac{\partial p}{\partial x}$ $\displaystyle=$ $\displaystyle
0,\qquad(x,y)\in\Omega_{2},$ $\displaystyle\frac{\partial}{\partial
x}(uv)+\frac{\partial}{\partial y}(v^{2})+\frac{\partial p}{\partial y}$
$\displaystyle=$ $\displaystyle 0,\qquad(x,y)\in\Omega_{2},$

and are satisfied for pressure $p(x,y)$ given by

$p(x,y)=\frac{(x^{2}+y^{2}-2x)}{2},\qquad(x,y)\in\Omega_{2},$ (3.18)

which vanishes on $\partial\Omega_{2}$. In consequence, the pressure in
$\Omega=\Omega_{1}\cup\Omega_{2}$ is continuous across the inner boundary
$\partial\Omega_{2}$ (the unit circle (3.8)) except at the origin $(0,\,0)$.
In this respect, an arbitrary constant can be added to the pressure (3.12)
inside $\Omega_{1}$ provided the same constant is added to the pressure (3.18)
in $\Omega_{2}$.

The vector field (3.15) and (3.16) on the unit circle satisfies the boundary
condition

$(un_{1}+vn_{2})=0,\qquad(x,y)\in\partial\Omega_{2},$ (3.19)

and fluid particles inside $\Omega_{2}$ flow tangential to
$\partial\Omega_{2}$ and cannot penetrate into $\Omega_{1}$. Equally,
particles flowing inside $\Omega_{1}$ can never reach inside $\Omega_{2}$, but
as already shown, flow tangential to $\partial\Omega_{2}$. It may be easily
checked by direct computation that the Rankine-Hugoniot jump conditions are
satisfied across the inner boundary $\partial\Omega_{2}.$

The Lagrange trajectories in $\Omega_{2}$ are circles centred at $(1,0)$ and
of radius $c,\,0\leq c\leq 1$ and are uniquely defined by initial data. An
admissibility criterion is therefore required only for trajectories in the
region $\Omega_{1}$ and as previously shown leads to the unique trajectory in
which particles steadily traverse the unit cirle (3.8). Fluid behaviour in the
composite region $\Omega=\Omega_{1}\cup\Omega_{2}$ subject respectively to the
velocities (3.1),(3.2), (3.15), (3.16) and pressures (3.12) and (3.18) in
$\Omega_{1}$ and $\Omega_{2}$ results in the fluid in $\Omega_{1}$ rapidly
becoming a steady swirling motion entirely confined clockwise to the unit
circle (3.8). The inner region $\Omega_{2}$ remains fully occupied by
incompressible fluid moving with velocity (3.15) and (3.16) subject to
pressure (3.18).

Similar conclusions to Theorem 3.1 are valid for the plane compressible steady
flow satisfying the Navier-Stokes equations.

######  Theorem 3.2.

Let the velocity, density and pressure satisfy (3.1), (3.2), (3.4), and (3.12)
in the region $\Omega_{1}$ . Then for viscosity coefficients $\mu,\,\lambda$
given by

$\displaystyle\mu$ $\displaystyle=$ $\displaystyle 4\cos^{2}{\theta},$ (3.20)
$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle-4\theta\cot{\theta}-4\cos^{2}{\theta},$ (3.21)

where $x=r\cos{\theta},\,y=r\sin{\theta}$, the particle trajectories (2.29) in
$\Omega_{1}$ are solutions to the Navier-Stokes equations specified by the
continuity equation (3.5), the balance of steady linear momentum

$\displaystyle\frac{\partial(\rho u^{2})}{\partial x}+\frac{\partial(\rho
uv)}{\partial y}$ $\displaystyle=$
$\displaystyle\frac{\partial\Sigma_{11}}{\partial
x}+\frac{\partial\Sigma_{21}}{\partial y},$ (3.22)
$\displaystyle\frac{\partial(\rho uv)}{\partial x}+\frac{\partial(\rho
v^{2})}{\partial y}$ $\displaystyle=$
$\displaystyle\frac{\partial\Sigma_{21}}{\partial
x}+\frac{\partial\Sigma_{22}}{\partial y},$ (3.23)

where

$\displaystyle\Sigma_{\alpha\beta}$ $\displaystyle=$
$\displaystyle-p\delta_{\alpha\beta}+\sigma_{\alpha\beta},\qquad\alpha,\,\beta=1,2,$
(3.24) $\displaystyle\sigma_{11}$ $\displaystyle=$
$\displaystyle(\lambda+2\mu)\frac{\partial u}{\partial
x}+\lambda\frac{\partial v}{\partial y},$ (3.25) $\displaystyle\sigma_{22}$
$\displaystyle=$ $\displaystyle(\lambda+2\mu)\frac{\partial v}{\partial
y}+\lambda\frac{\partial u}{\partial x},$ (3.26) $\displaystyle\sigma_{12}$
$\displaystyle=$ $\displaystyle\mu\left(\frac{\partial u}{\partial
y}+\frac{\partial v}{\partial x}\right),$ (3.27)

and the boundary condition (3.7). Here, $\delta_{\alpha\beta}$ denotes the
standard Kronecker delta.

Proof: By direct substitution of the stated expressions.

Theorem 3.2 has similar implications to Theorem 3.1 for the relationship
between Lagrange trajectories and the Navier-Stokes equations. A given
velocity that solves the compressible Navier-Stokes equations in $\Omega_{1}$
leads to corresponding non-unique Lagrange trajectories for the same initial
data. The entropy rate admissibility criterion, however, distinguishes a
unique trajectory.

By comparison, it is shown by Robertson and Sadowkski [19, 20] for the
incompressible Navier-Stokes equations on a bounded three-dimensional region
subject to Dirichlet boundary data, that for sufficiently regular velocities
the Lagrange trajectories are unique for almost all initial data.

As before, the analysis may be extended to the enlarged region
$\Omega=\Omega_{1}\cup\Omega_{2}$ employing (3.15) and (3.16) as velocity
components in the unit disc $\Omega_{2}$. Besides the incompressibility
condition (3.17), the gradient of the velocity components (3.15) and (3.16)
for $(x,y)\in\Omega_{2}$ satisfy the relations

$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}=0,$ (3.28)

and therefore $\sigma_{\alpha\beta}=0$ for $\alpha,\,\beta=1,2.$ That is, the
viscous contribution vanishes identically, and the Navier-Stokes equations are
satisfied for all viscosity coefficients $\lambda,\,\mu$ inside the unit disc
$\Omega_{2}$. Accordingly, Lagrange trajectories in $\Omega_{2}$ exhibit the
same properties as those noted for the Euler equations implying that the fluid
particles in $\Omega_{1}$ and $\Omega_{2}$ cannot interpenetrate. The entropy
rate admissibility criterion establishes that the unique motion in
$\Omega_{1}$ is again concentrated solely on the unit circle around which it
continuously swirls clockwise.

A final important observation is that in the problems of this section the
density $\rho(x)=x^{-1}$ in $\Omega_{1}$ becomes singular on the $y-$axis
leading to density blow-up. Fluid particle aggregation is therefore to be
expected and occurs for the entropy rate admissible trajectories but not for
the “wild” non-admissible ones. In fact, this expectation is reflected in
numerical computations presented in the next section.

##  4 Numerical approximation

This section discusses the numerical approximation of solutions to the system
(2.29) and (2.30) relevant for Lagrange trajectories in the combined region
$\Omega=\Omega_{1}\cup\Omega_{2}$. Of special interest are numerical methods
which preserve the qualitative behaviour of the flow in phase space and induce
convergence to the solution selected by the entropy rate admissibility
criterion. The basic example of such geometric numerical integrators (see
[13]) is the symplectic Euler method applied here.A constant step size $h>0$
is used to compute approximations $(x_{n},\,y_{n})$ of the solution
$(x(nh),\,y(nh)),\,n\in N$ to (2.29) and (2.30). As before, denote initial
conditions by $(x_{0},\,y_{0}):=(x(0),\,y(0))$ and define $x_{n},\,y_{n}$ by

$\begin{rcases}y_{n+1}=y_{n}+h\frac{y_{n}^{2}-x_{n}^{2}}{2x_{n}},\qquad
x_{n+1}=x_{n}+hy_{n+1},&\text{for }(x_{n}-1)^{2}+y_{n}^{2}\geq 1,\\\
y_{n+1}=y_{n}+h(1-x_{n}),\quad x_{n+1}=x_{n}+hy_{n+1},&\text{for
}(x_{n}-1)^{2}+y_{n}^{2}<1.\end{rcases}$ (4.1)

As described in [13, Chapt. IX], the symplectic Euler method determines a flow
that preserves an approximation to the entropy function $H(x,y)$, defined in
(2.4), to an error of order $h$.

Figure 6: Trajectories of symplectic Euler discretisation for different
initial conditions.

Figure 6 illustrates trajectories of (4.1) corresponding to four initial
conditions $(x_{0},\,y_{0})$ satisfying $(x_{0}-1)^{2}+y_{0}^{2}>1$, when
$h=10^{-4}$. In all cases, the numerical trajectories traverse the level sets
of $H$ to high accuracy. Near the origin $(0,\,0)$, the trajectories switch to
the unit circle $(x-1)^{2}+y^{2}=1$ as expected from the entropy rate
admissibility criterion. The black trajectory, which corresponds to the
initial condition $(x_{0},\,y_{0})=(\frac{1}{2},\,\frac{9}{8})$, is further
analysed below.

Figure 7: Trajectories for symplectic Euler discretisation near $(0,\,0)$ for
$h=2.5\times 10^{-4}$(blue); $h=5\times 10^{-4}$(red); $h=10^{-3}$(black). The
unit circle $(x-1)^{2}+y^{2}=1$ in green.

Figure 7 shows the behaviour near the origin in varying detail depending upon
step size: $h=2.5\times 10^{-4}$(blue), $h=5\times 10^{-4}$(red),
$h=10^{-3}$(black). All trajectories start from the same initial condition
$(x_{0},\,y_{0})$, cross the $x-$axis at a positive value $x\sim h^{2}$ and
then follow the unique solution close to the unit circle. The well-known
analysis of the symplectic Euler method of the linear equation in the domain
$(x-1)+y^{2}<1$ shows that the trajectory remains in a neighbourhood of the
unit circle of order $h$. The rigorous analysis presented in [13, Chapt. 11],
however, is required to establish that trajectories always cross the $x-$axis
at points $x>0,$ and thereafter enter and remain near the unit circle. In
fact, for small $h$, the level sets of $H$ are perturbed to the right near the
origin. The geometry of the discrete flow explains the selection of the
solution obtained by the entropy rate admissibility criterion.

Figure 8:  $H(x,y)$ as function of $t$ for initial condition
$(x_{0},\,y_{0})=(\frac{1}{2},\frac{9}{8})$.

Figure 9: Error $(H(x,y)-1)$ as function of $t$ for initial condition
$(x_{0},\,y_{0})=(\frac{1}{2},\,\frac{9}{8}).$

Figure 10: Trajectory of explicit Euler discretisation for initial condition
$(x_{0},\,y_{0})=(\frac{1}{2},\,\frac{9}{8}).$

Figure 8 depicts the numerically computed entropy profile $t\mapsto
H(x(t),\,y(t))$ for initial condition
$(x_{0},\,y_{0})=(\frac{1}{2},\,\frac{9}{8})$ introduced above. Again take
$h=10^{-4}$. When the trajectory first passes through the origin, $H$ drops to
a value near $1$ and then remains approximately constant.

Figure 9 indicates the discretisation error in the entropy profile
$t\mapsto\left[H(x(t),\,y(t))-1\right].$ The error is of size less than
$10^{-4}=O(h)$ except for floating point errors when $(x,\,y)$ is near the
origin where $H$ possesses a singularity.

It has thus been shown how geometric properties in phase space of the
symplectic Euler method select the solution to the system (2.29) and (2.30)
according to the entropy rate admissibility criterion. Numerical methods
without such properties are not expected to select these solutions.

Figure 10 shows the trajectory computed by the explicit Euler method again for
initial condition $(x_{0},\,y_{0})=(\frac{1}{2},\,\frac{9}{8})$. For this
discretisation, the numerical solution moves to a larger circle with increased
entropy $H$, even with smaller step size $h=10^{-5}.$

##  5 Relevant convex integration results

Section 2 describes how the entropy rate admissibility criterion applied to
the constant functions (2.6) selects a unique orbit from the infinitely many
circular paths generated by the Dafermos equations (2.1) and (2.2). A
piecewise non-increasing linear graph of maximal negative slope leads to the
requisite single profile. A notable feature of the construction, stated in
Theorem 2.1, is the uncountable number of profiles that can be arbitrarily
chosen for prescribed initial conditions.

This feature resembles results of Buckmaster and Vicol [4] and of De Lellis
and Szèkelyhidi [10] which for convenience are recalled in the following
theorem.

######  Theorem 5.1.

There exists $\beta>0$ such that for any non-negative smooth function
$E(t):[0,T]\rightarrow\mathrm{I\\!R\\!}_{\geq 0}$ there exists a weak vector
solution $\textbf{v}\in
C^{0}_{t}\left([0,T];H^{\beta}_{x}(\mathrm{I\\!T\\!}^{3})\right)$ of the
Navier-Stokes equations such that

$\int_{\mathrm{I\\!T\\!}^{3}}|\textbf{v}(x,t)|^{2}\,dx=E(t),$

for all $t\in[0,T]$. Moreover, the associated vorticity
$\nabla\times\textbf{v}$ lies in
$C^{0}_{t}\left([0,T];L^{1}_{x}(\mathrm{I\\!T\\!}^{3})\right),$ where
$\mathrm{I\\!T\\!}^{3}=\mathrm{I\\!R\\!}^{3}\backslash 2\pi Z^{3}$ denotes the
unit cube with periodic boundary conditions.

It is apparent that arbitrary prescription of the energy $E(t)$ produces an
infinite number of solutions in the given class for the same initial
conditions. An admissibility criterion is required to identify a single
physically relevant velocity. Whether an entropy rate admissibility criterion
of the type considered here is appropriate in this respect remains an
important open question.

A detailed comparison with the analysis of Section 3 reveals distinct
differences and any analogy between the two treatments is likely to be
superficial. Indeed, the entropy $H(x,\,y)$ used in Section 3 is not a local
(kinetic) energy function $\eta(t)$ defined by

$\eta(t):=\frac{\rho}{2}\left(\dot{x}^{2}+\dot{y}^{2}\right)$ (5.1)

where $\rho(x,y)$ is a specified density. The relation to $H(x,y)$ is obtained
by substitution of (2.10) and (2.11) in (5.1) and is given by

$\eta(t)=2x^{2}w^{2}(x,y)\rho(x,y)H^{2}(x,y).$ (5.2)

Consequently, functions $\eta$ and $H^{2}$ are identical only when
$2x^{2}w^{2}\rho=1$.

Finally, note that uniqueness properties arising in convex integration studies
refer to solutions of the respective equations and not to the associated
Lagrange trajectories. Whether there are non-unique trajectories for certain
non-unique convex integration solutions remains a second open question.

### Acknowledgements:

_Valuable discussions during the preparation of this paper with Professors Z.
Artstein, E. Feireisl, A. Ostermann and E. Titi are gratefully acknowledged.
Topics treated in this paper first arose during the 2021 Workshop on _Convex
Integration and Partial Differential Equations_ organised by ICMS, Edinburgh._

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## Appendices

##  Appendix A State variables.

Let state variables $(x(t),\,y(t))$ satisfy (2.1) and (2.2) in region
$\Omega_{1}$ defined by (2.35). Consider for $x\neq 0,\,y\neq 0,$ the
expression

$\displaystyle\frac{d}{dt}\left(\frac{y^{2}}{x}\right)$ $\displaystyle=$
$\displaystyle\frac{y}{x^{2}}\left(2\dot{y}x-y\dot{x}\right)$ $\displaystyle=$
$\displaystyle\frac{y}{x^{2}}\left(y^{2}-x^{2}-y\dot{x}\right)$
$\displaystyle=$
$\displaystyle\frac{y}{x^{2}}\left(y\dot{x}-x^{2}-y\dot{x}\right)$
$\displaystyle=$ $\displaystyle-y$ $\displaystyle=$ $\displaystyle-\dot{x},$

which on integration gives (cp., (2.20))

$\frac{y^{2}}{x}+x=\frac{y^{2}_{0}}{x_{0}}+x_{0}\equiv 2c_{0}$ (A.1)

or

$(x-c_{0})^{2}+y^{2}=c_{0}^{2},$

which may be rewritten as

$y(t)=\pm\sqrt{x(2d-x)}\qquad(=\dot{x}).$

On setting $c_{0}z(t)=(x(t)-c_{0})$ the last expression becomes

$\dot{z}(t)=\pm\sqrt{(1-z^{2})},$

so that

$z(t)=\pm\cos{(\mp t+\sin^{-1}{z_{0}})},$

and therefore

$x(t)=c_{0}\pm c_{0}\cos{(\mp t+\sin^{-1}{z_{0}})}.$ (A.2)

A corresponding expression for $y(t)$ obtained on differentiation of the last
relation is

$y(t)=\pm c_{0}\sin{(\mp t+\sin^{-1}{z_{0}})}.$ (A.3)

The positive root in (A.2) and (A.3) yields (2.21) and (2.22) with
$\theta_{0}=\sin^{-1}{z_{0}}$ given by (cp., (2.23))

$\tan{\theta_{0}}=\frac{y_{0}}{(x_{0}-c_{0})}.$

Alternatively, elimination of the dependent variable $y$ between (2.1) and
(2.9) leads to a differential equation for $x$ whose solution is

$x(t)-c_{0}=A\cos{(t)}+B\sin{(t)},$ (A.4)

and consequently

$y(t)=-A\sin{(t)}+B\cos{(t)},$ (A.5)

where $A,\,B,$ constants determined by initial conditions $(x_{0},\,y_{0})$,
are given by

$\displaystyle A$ $\displaystyle=$ $\displaystyle x_{0}-c_{0},$ (A.6)
$\displaystyle B$ $\displaystyle=$ $\displaystyle y_{0},$ (A.7)

and $c_{0}$ is specified by (A.1). Expressions (A.4) and (A.5) are equivalent
to (A.2) and (A.3).

##  Appendix B Navier-Stokes equations.

Throughout Appendix B and Appendix C a suffix notation and summation
convention are adopted together with a subscript comma to denote partial
differentiation. Components of vectors, unless otherwise stated, are with
respect to a given Cartesian coordinate system.

Consider a compressible fluid (of variable density) in plane steady motion
that satisfies the isotropic Navier-Stokes equation for a given velocity field
having Cartesian components $(v_{1},\,v_{2})$. The aim, using a semi-inverse
method, is to determine the viscosity coefficients $\lambda(x_{1},\,x_{2})$
and $\mu(x_{1},\,x_{2})$ that ensure the corresponding stress distribution is
in equilibrium subject to zero body force. The region occupied by the fluid is
not defined at this stage. Nor are boundary conditions considered.

The compatible strain rate components, given by

$e_{\alpha\beta}=\frac{1}{2}\left(v_{\alpha,\beta}+v_{\beta,\alpha}\right),\qquad\alpha,\,\beta=1,2,$
(B.1)

are related to the stress components by (3.25)-(3.27) concisely written as

$\sigma_{\alpha\beta}=\lambda e_{\gamma\gamma}\delta_{\alpha\beta}+2\mu
e_{\alpha\beta},$ (B.2)

where $\delta_{\alpha\beta}$ denotes the Kronecker delta. The viscosity
coefficients $\lambda(x_{1},\,x_{2}),\,\mu(x_{1},\,x_{2})$ are functions of
position to be determined such that the stress is in equilibrium under zero
body force; that is

$\sigma_{\alpha\beta,\beta}=0.$ (B.3)

In consequence, the Navier-Stokes equations (3.25) and (3.26) reduce to the
Euler equations (3.10) and (3.11).

Easy deductions from (B.2) are the trace relation

$\sigma_{\alpha\alpha}=2(\lambda+\mu)e_{\alpha\alpha},$ (B.4)

and

$2\mu=\frac{\left(\sigma_{11}-\sigma_{22}\right)}{\left(e_{11}-e_{22}\right)}=\frac{\sigma_{12}}{e_{12}},$
(B.5)

which represents an additional fundamental constraint between stress and
strain rate explicitly independent of viscosity coefficients. Trivial
rearrangement of (B.5) gives

$\frac{\left(\sigma_{11}-\sigma_{22}\right)}{\sigma_{12}}=\frac{\left(e_{11}-e_{22}\right)}{e_{12}}.$
(B.6)

It is well-known that a solution to the system (B.3) expressed in terms of the
Airy stress function $\chi(x,y)$ is represented by:

$\sigma_{11}=-\chi_{,22},\qquad\sigma_{22}=-\chi_{,11},\qquad\sigma_{12}=\chi_{,12}.$
(B.7)

Substitution of (B.7) in (B.6) yields

$\chi_{,11}-\chi_{,22}-\Lambda(x_{1},\,x_{2})\chi_{,12}=0,\qquad\Lambda:=\frac{(e_{11}-e_{22})}{e_{12}},$
(B.8)

which is the partial differential equation satisfied by the Airy stress
function $\chi(x,y)$ explicitly independent of viscosity coefficients
$\lambda,\,\mu$.

On the assumption that (B.8) can be solved for general $(v_{1},\,v_{2})$ and
therefore $\Lambda$, the solution may be used to derive the equilibrium stress
components and consequently the viscosity coefficient $\mu$ from (B.5)3.
Furthermore, it follows from (B.7) and (B.4) that

$\chi_{,\alpha\alpha}=-2(\lambda+\mu)(e_{11}+e_{22}),$ (B.9)

which may be used to determine $(\lambda+\mu)$ and therefore
$\lambda(x_{1},x_{2})$.

##  Appendix C Derivation of viscosity coefficients in Theorem 3.2

Derivation of expressions (3.20) and (3.21) for the viscosity coefficients
stipulated in Theorem 3.2 is conveniently described in terms of the complex
variable $z$ and its conjugate $\bar{z}$ defined by

$z=(x_{1}+ix_{2}),\qquad\bar{z}=(x_{1}-ix_{2}).$ (C.1)

Details of the following computations are contained in standard texts; e.g.,
Muskhelishvili [17].

Differentiation with respect to $z$ and $\bar{z}$ is defined to be

$\displaystyle\frac{\partial}{\partial z}$ $\displaystyle:=$
$\displaystyle\frac{1}{2}\left(\frac{\partial}{\partial
x}-i\frac{\partial}{\partial y}\right),$ (C.2)
$\displaystyle\frac{\partial}{\partial\bar{z}}$ $\displaystyle:=$
$\displaystyle\frac{1}{2}\left(\frac{\partial}{\partial
x}+i\frac{\partial}{\partial y}\right).$ (C.3)

The particular velocity components (2.1) and (2.2) of present concern,
repeated for convenience

$v_{1}=x_{2},\qquad
v_{2}=\frac{(x^{2}_{2}-x_{1}^{2})}{2x_{1}},\qquad(x_{1},\,x_{2})\in\Omega_{1},$
(C.4)

are the real and imaginary parts of the (non-analytic) velocity field
$v(z,\,\bar{z})$ represented by

$v=v_{1}+iv_{2}=-i\frac{z^{2}}{\left(z+\bar{z}\right)}.$ (C.5)

The corresponding strain rate components become

$\displaystyle e_{11}$ $\displaystyle=$ $\displaystyle 0,$ (C.6)
$\displaystyle e_{22}$ $\displaystyle=$ $\displaystyle
i\frac{\left(\bar{z}-z\right)}{\left(z+\bar{z}\right)},$ (C.7) $\displaystyle
e_{12}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{\left(z^{2}+\bar{z}^{2}\right)}{\left(z+\bar{z}\right)^{2}}.$
(C.8)

Insertion into the formula for $\Lambda(z,\bar{z})$ (see (B.8)2) gives

$\displaystyle\Lambda=2i\frac{\left(z^{2}-\bar{z}^{2}\right)}{\left(z^{2}+\bar{z}^{2}\right)}.$
(C.9)

Consequently, (B.8) assumes the form

$\left(\chi_{,zz}+\chi_{,\bar{z}\bar{z}}\right)\left(z^{2}+\bar{z}^{2}\right)+\left(\chi_{,zz}-\chi_{,\bar{z}\bar{z}}\right)\left(z^{2}-\bar{z}^{2}\right)=0,$
(C.10)

which upon rearrangement simplifies to

$z^{2}\chi_{,zz}+\bar{z}^{2}\chi_{,\bar{z}\bar{z}}=0.$ (C.11)

Among the possible solutions to (C.11), select that obtained from setting

$z^{2}\chi_{,zz}=k=-\bar{z}^{2}\chi_{,\bar{z}\bar{z}},$ (C.12)

for real constant $k$. Integration of the equation on the left gives

$\chi(z,\bar{z})=-k\log{z}+zf(\bar{z})+q(\bar{z}),$ (C.13)

where $f(.),\,q(.)$ are arbitrary functions. Similarly, integration of the
equation on the right of (C.12) yields

$\chi(z,\bar{z})=k\log{\bar{z}}+\bar{z}g(z)+p(z),$ (C.14)

where $g(.),\,p(.)$ are arbitrary functions. Choose

$\displaystyle p(z)$ $\displaystyle=$ $\displaystyle-k\log{z},\qquad
q(\bar{z})=k\log{\bar{z}},$ $\displaystyle g(z)$ $\displaystyle=$
$\displaystyle 2z\log{z},\qquad f(\bar{z})=-2\bar{z}\log{\bar{z}}.$

to derive respectively from (C.13) and (C.14) the equivalent expressions

$\displaystyle\chi(z,\bar{z})$ $\displaystyle=$
$\displaystyle-k\log{\frac{z}{\bar{z}}}-2z\bar{z}\log{\bar{z}},$
$\displaystyle\chi(z,\bar{z})$ $\displaystyle=$
$\displaystyle-k\log{\frac{z}{\bar{z}}}+2z\bar{z}\log{z},$

and consequently the final form for the Airy stress function $\chi(z,\bar{z})$
is:

$\chi(z,\bar{z})=\left(z\bar{z}-k\right)\log{\frac{z}{\bar{z}}}.$ (C.15)

Substitution in (B.7) determines the equilibrium stress components as

$\displaystyle\sigma_{11}(z,\bar{z})$ $\displaystyle=$ $\displaystyle
i\left[(z\bar{z}+k)\frac{(z^{2}-\bar{z}^{2})}{(z\bar{z})^{2}}+2\log{\frac{z}{\bar{z}}}\right],$
(C.16) $\displaystyle\sigma_{22}(z,\bar{z})$ $\displaystyle=$ $\displaystyle
i\left[(z\bar{z}+k)\frac{(\bar{z}^{2}-z^{2})}{(z\bar{z})^{2}}+2\log{\frac{z}{\bar{z}}}\right],$
(C.17) $\displaystyle\sigma_{12}(z,\bar{z})$ $\displaystyle=$
$\displaystyle\frac{(z\bar{z}+k)(z^{2}+\bar{z}^{2})}{(z\bar{z})^{2}},$ (C.18)

while the viscosity coefficient $\mu(z,\,\bar{z})$ from (B.5)3 and (C.8) is
given by

$\mu=\frac{(z\bar{z}+k)(z+\bar{z})^{2}}{(z\bar{z})^{2}}.$ (C.19)

In view of (C.6), (C.6),(C.7) and (C.16), the viscosity coefficient
$\lambda(z,\,\bar{z})$ is

$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\frac{\sigma_{11}}{e_{22}}$ $\displaystyle=$
$\displaystyle\frac{2(z+\bar{z})}{(\bar{z}-z)}\log{\frac{z}{\bar{z}}}-\frac{(z\bar{z}+k)(z+\bar{z})^{2}}{(z\bar{z})^{2}}.$

In terms of polar coordinates $(r,\theta)$, where

$z=r\exp{(i\theta)},\qquad\bar{z}=r\exp{(-i\theta)},$

these expressions are written as

$\displaystyle\sigma_{11}$ $\displaystyle=$
$\displaystyle-2\left[2\theta+\left(\frac{(r^{2}+k)}{r^{2}}\sin{2\theta}\right)\right],$
$\displaystyle\sigma_{22}$ $\displaystyle=$ $\displaystyle
2\left[-2\theta+\left(\frac{(r^{2}+k)}{r^{2}}\sin{2\theta}\right)\right],$
$\displaystyle\sigma_{12}$ $\displaystyle=$ $\displaystyle
2\left(\frac{(r^{2}+k)}{r^{2}}\right)\cos{2\theta},$ $\displaystyle e_{11}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle e_{22}$ $\displaystyle=$
$\displaystyle\tan{\theta},$ $\displaystyle e_{12}$ $\displaystyle=$
$\displaystyle\frac{\cos{2\theta}}{4\cos^{2}{\theta}},$ $\displaystyle\mu$
$\displaystyle=$ $\displaystyle 4\frac{(r^{2}+k)}{r^{2}}\cos^{2}{\theta},$
$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle-4\theta\cot{\theta}-4\left(\frac{(r^{2}+k)}{r^{2}}\right)\cos^{2}{\theta}.$

It may easily be checked by direct substitution that the stress is in
equilibrium under zero body force and that

$v_{\alpha}n_{\alpha}=0$

at all points on $\partial\Omega_{1}\backslash(0,0)$.

The proof of (3.20) and (3.21) is complete on taking $k=0$ in the above
expressions.

##  Appendix D Conservative Lagrange trajectories

Let $(x,y)$ be the rectangular coordinates of a point moving with respect to
time and suppose that the differentiable function $G(x,y)$ is conserved so
that $G(x(t),\,y(t))$ is constant. Examples are the entropy in an adiabatic
system or a one parameter family of plane closed curves discussed in Section
2. Define velocities $(u,\,v)$ by

$\displaystyle u(x,t)$ $\displaystyle:=$ $\displaystyle\dot{x},$ (D.1)
$\displaystyle v(x,t)$ $\displaystyle:=$ $\displaystyle\dot{y},$ (D.2)

to obtain from

$\frac{dG(x(t),\,y(t))}{dt}=0$

the relation

$\frac{\partial G}{\partial x}u+\frac{\partial G}{\partial y}v=0.$ (D.3)

Consequently, _any_ $u$ determines $v$ from the expression

$v=-\frac{\partial G}{\partial x}\left(\frac{\partial G}{\partial
y}\right)^{-1}u$ (D.4)

such that the system (D.1) and (D.2) is conservative with $G(x,y)$ as first
integral.

The particular choice

$u(x,y)=\frac{\partial G}{\partial y}$ (D.5)

yields a Hamiltonian system for which

$v=-\frac{\partial G}{\partial x}.$ (D.6)

It follows that the incompressibity condition

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (D.7)

holds provided the order of the second partial derivatives of $G$ can be
reversed.

For fluids with variable mass density, the continuity equation is satisfied by
modifying the definition of $u$. Thus, suppose

$u:=\frac{\partial G}{\partial y}w(x,\,y),$ (D.8)

where $w(x,\,y)$ is some sufficiently smooth function, and $G(x,\,y)$
continues to be the first integral of the conservative system (D.1) and (D.2).
Accordingly, $v(x,\,y)$ is modified to

$v:=-\frac{\partial G}{\partial x}w(x,\,y),$ (D.9)

and the continuity equation

$\frac{\partial}{\partial x}(\rho u)+\frac{\partial}{\partial y}(\rho v)=0,$
(D.10)

is satisfied provided when $(i)$ the mass density $\rho(x,y)$ is chosen to be
$\rho=Dw^{-1}$, for constant $D$, and $(ii)$ the order of the second partial
derivatives of $G$ can be reversed.

######  Remark D.1.

Apart from differentiability of $G$, no further assumptions have been
introduced on either $G(x,\,y)$ or $w(x,\,y)$, whose choice clearly determines
the smoothness of the mass density $\rho(x,\,y)$.

######  Remark D.2.

Lagrange trajectories corresponding to the system (3.1) and (3.2) are a
special case of (D.1) and (D.2) upon selecting $w(x,y)=x$ and setting

$G(x,\,y)=\frac{(x^{2}+y^{2})}{2x}\quad\left(=H(x,y)\right).$ (D.11)

The mass density becomes $\rho=x^{-1}$ and

$\frac{\partial G}{\partial
x}=\frac{(x^{2}-y^{2})}{2x^{2}},\qquad\frac{\partial G}{\partial
y}=\frac{y}{x}.$ (D.12)

which on insertion into (D.8) and (D.9) yields the required velocity
components. Note that

$\frac{\partial^{2}G}{\partial x\partial y}\neq\frac{\partial^{2}G}{\partial
y\partial x}\qquad\mbox{ for }(x,\,y)=(0,\,0).$

######  Remark D.3.

Substitution of (D.8) and (D.9) in the Euler equations yields equations for
the partial derivatives of the pressure which can be solved for the pressure.
Consequently, for any conserved quantity $G(x,y)$ and any choice of $w(x,\,y)$
the Euler equations can be solved for appropriate pressure determined semi-
inversely.

When $w=x$ the pressure is given by (3.12) which to within a constant is the
conserved quantity $G(x,y)$ given by (D.11).

######  Remark D.4.

For the choice $w(x,\,y)=x^{\alpha},\,0<\alpha<1$, the density becomes
$\rho=x^{-\alpha}$ and though singular at the origin, is integrable on the
region $\Omega$ and computations involving mass density are valid.