Non-Uniqueness in Plane Fluid Flows

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.

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

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

Initial data are specified by

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

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

or

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

Consequently,

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

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

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

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

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

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

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

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

Finally, consider

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

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

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

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

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

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.

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:

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

which by integration lead to

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

and

The last expression inserted into (2.27) gives

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

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

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

on reaching the origin switches to another trajectory for which

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

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.

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:

where

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 $e=c_{0}$.

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.

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.

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.

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.

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

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

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

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

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

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

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

It follows from mass conservation, or directly, that

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

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

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

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 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

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.

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

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 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 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 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 $h=10^{-5}.$

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.

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

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

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.