Relative entropy method for the relaxation limit of hydrodynamic models

1.   Introduction

In this work, we consider the following compressible Euler-type systems of equations of the form

in the time-spatial domain $(0, T)\times\Omega$, where $\rho(t):\Omega\rightarrow\mathbb{R}_{+}$ for $t\geq0$ is the density obeying the equation of conservation of mass, $\textbf{u}(t):\Omega\rightarrow\mathbb{R}^d$ for $t\geq0$ is the velocity of fluid and the product $\rho\textbf{u}$ denotes the momentum flux. Here the functional $\mathcal{E}(\rho): L^1_+(\mathbb{R}^d)\rightarrow\mathbb{R}$ is the free energy functional defined on mass densities by

with $h(\rho)$ describing the entropy part or internal energy of the system, and $\frac{\delta\mathcal{E}(\rho)}{\delta\rho}$ stands for its variational derivative, given by

Here, $C_k$ is a positive constant measuring the strength of the interaction, $K(x):\mathbb{R}^d\rightarrow\mathbb{R}$ is the interaction potential depicting the nonlocal forces which usually manifest as repulsion or attraction between particles, which is assumed to be symmetric, and $\Phi(x):\Omega\rightarrow\mathbb{R}$ is a confinement potential. We refer to [10,11,38] for a general introduction to these free energies, to [5] for their applications in Keller-Segel type models, and more general models in Density Functional Theory as discussed in [25]. Finally, the term $-\frac{1}{\varepsilon}\rho\textbf{u}$ on the left-hand-side of (1.1) is responsible for a damping force with frictional coefficient $\frac{1}{\varepsilon}$ in order to look at the so-called overdamped limit.

In this work, we consider $\Omega\subset\mathbb{R}^d$ to be any smooth, connected, open set. The no-flux boundary condition for $\textbf{u}$ (i.e. $\textbf{u}\cdot\nu = 0$, $\nu$ denotes an outer normal vector to $\partial \Omega$)) or periodic boundary condition are assumed if $\Omega$ is a bounded domain or $\Omega = \mathbb{T}^d$ is periodic domain. We also extend $\rho$ by zero when $\Omega$ is bounded in order that we are able to define properly $K\ast\rho$ on $\mathbb{R}^d$. The main objective of this work is to deduce the following equilibrium equation

by taking the overdamped limit $\varepsilon\rightarrow0$ in system (1.1) under the framework of relative entropy method. This method is an efficient mathematical tool for establishing the limiting processes and stabilities among thermomechanical theories, see [6,7,16,17,19,24,31,32] for instance. With the various choices of the functional $\mathcal{E}(\rho)$, the corresponding models spanned from the system of isentropic gas dynamics and variants of the Euler-Poisson system [29,33,35] leading to the porous medium equation and nonlinear aggregation-diffusion equations in the overdamped limit, see [15,26,27,28,30,34] and references therein. More general forms of free energies with higher order terms in derivatives have also been used in the literature leading to the equations of quantum hydrodynamics [1,2], the models for phase transitions [4,36], and the dispersive Euler-Korteweg equations [21].

In this work, we only consider the functional $\mathcal{E}(\rho)$ defined by (1.2) with variation given by (1.3) where $h(\rho)$ and a pressure function denoted by $p(\rho)$ are linked by the thermodynamic consistency relations

In this case, we observe that (1.1) reduces to

and (1.4) is equivalent to

consequently, our goal concerning the relaxation limit from (1.1) to (1.4) is equivalent to considering the relaxation limit from (1.6) to (1.7). In particular, for the power-law pressure $p(\rho) = \rho^m$, the internal energy $h(\rho)$ takes the form

We will deal with slightly more general internal energy functions. For this reason, we introduce the notation

for some positive constants $k_1$, $k_2$ and $k_3$. For $m>2$, we assume that the function $o(\rho^m)$ is chosen to satisfy that $\ h_m \in C[0, +\infty)\cap C^2(0, +\infty), \ h_m^{"}(\rho)>0$ and for some constant $A>0$,

where $p(\rho)$ is determined by $h_m(\rho)$ via (1.5). For simplicity, we will drop the dependence on $m$ of $h(\rho)$ in the sequel.

We can formally obtain that weak solutions $(\rho, \rho\textbf{u})$ of the system (1.6) satisfy a standard weak form of total energy dissipation. Indeed, multiplying (1.6)$_2$ with $\textbf{u}$, using the first relation in (1.5) and (1.6)$_1$ and integrating the resulting equation over $\Omega$, provided no-flux boundary condition for $\textbf{u}$ (i.e. $\textbf{u}\cdot\nu = 0$) is valid when $\Omega\subset\mathbb{R}^d$ is a bounded domain, one derives

in the sense of distributions, where we have used the first relation in (1.5).

In order to obtain the free energy dissipation for (1.7) and further to compare its strong solution with the weak solution of (1.6), we define

and rewrite (1.7) as

where $\bar{e}: = \partial_t(\bar{\rho}\bar{\textbf{u}}) + {\rm div}_x(\bar{\rho}\bar{\textbf{u}}\otimes\bar{\textbf{u}})$. In a similar way as for (1.10), we obtain the free energy dissipation for $(\bar{\rho}, \bar{\rho}\bar{\textbf{u}})$ in the following form

where we have also assumed that no-flux boundary condition for $\bar{\textbf{u}}$ (i.e. $\bar{\textbf{u}}\cdot\nu = 0$) holds, when $\Omega\subset\mathbb{R}^d$ is a bounded domain. Notice that

and so the relation in (1.13) is essentially the well-known dissipation property for gradient flows of the form (1.4), see [10,11,38] for instance.

For notational simplicity, we define the relative quantity $h(\rho|\bar{\rho})$ here by the difference between $h(\rho)$ and the linear part of the Taylor expansion around $\bar{\rho}$ as $h(\rho|\bar{\rho}): = h(\rho) - h(\bar{\rho}) - h'(\bar{\rho})(\rho-\bar{\rho})$, and denote

which potentially measures the distance between the two solutions $(\rho, \rho\textbf{u})$ and $(\bar{\rho}, \bar{\rho}\bar{\textbf{u}})$. Indeed, assuming that the exponent of the pressure function satisfies

then the function $\Theta(t)$ provides a measure to the distance between $(\rho, \rho\textbf{u})$ and $(\bar{\rho}, \bar{\rho}\bar{\textbf{u}})$ in the relaxation limit as we will show below. The restrictions in (1.15) are due to the use of Hardy-Littlewood-Sobolev-type (HLS) inequalities. HLS inequalities are also essential for establishing the existence of global-in-time weak solutions to Keller-Segel systems for general initial data, see [3,5,9,12,37] and references therein.

Remark 1. We should always keep in mind that whenever we deal with the equality case in (1.15), the mass of our system (1.7) should be suitably smaller than a threshold value, called the critical mass, in order to deal without finite time blow-up problems, otherwise we can assume that time is small enough and deal with local in time solutions before the blow-up happens. For strict inequalities, we do not have any restrictions on the mass.

We now recall the definition of weak solutions to (1.6) we deal with in this work.

Definition 1.1. $(\rho, \rho\textbf{u})$ with $\rho\in C([0, T); L^1(\Omega)\cap L^m(\Omega))$, $\rho\geq0$ and $\rho|\textbf{u}|^2\in L^{\infty}(0, T;L^1(\Omega))$ is a weak solution of (1.6) if

● $(\rho, \rho\textbf{u})$ satisfies the weak form of (1.6);

● $(\rho, \rho\textbf{u})$ satisfies (1.10) in the sense of distributions:

for any non-negative $\theta\in W^{1, \infty}[0, \infty)$ compactly supported on $[0, \infty)$;

● $(\rho, \rho\textbf{u})$ satisfies the properties:

Notice that, for the periodic case i.e. $\Omega = \mathbb{T}^d$, we need to assume that the test functions in the weak formulations in the above Definition 1.1 satisfy the periodic boundary conditions.

Our main result is stated as follows.

Theorem 1.2. Let $T>0$ and $m\geq1$ be fixed. Let the confinement potential $\Phi(x)$ be bounded from below in $\Omega$ and $p(\rho)$ be defined through (1.5) and (1.8) and let the interaction potential be symmetric. Suppose that $C_k$ is suitably small and $(\rho, \rho\textbf{u})$ is a weak solution of (1.6) in the sense of Definition 1.1 with $\rho>0$, and $(\bar{\rho}, \bar{\rho}\bar{\textbf{u}})$ is a smooth solution of (1.7) with $\bar{\rho}>0$, $\bar{\textbf{u}} \in L^\infty(0, T;W^{1, \infty}(\Omega))\cap L^\infty(0, T; L^{\frac{m}{2(m-1)}}(\Omega))$, and $\bar{e}$ bounded. Let $\Omega$ be any smooth, connected, open subset in $\mathbb{R}^d$. Assume one of the following conditions hold:

(i) $2-\frac{2}{d}\le m \leq 2$ with $d\geq2$ and the interaction potential $K$ satisfies $K\in L^{\frac{m}{2(m-1)}}(\Omega)\cap W^{1, \infty}(\Omega)$,

(ii) $\Omega = \mathbb{T}^d$ or $\Omega$ is a bounded domain in $\mathbb{R}^d$, $m\ge 2-\frac{2}{d}$ with $d\geq2$, $\bar{\rho}\in I = [\underline{\delta}, \overline{\delta}]$ with $\underline{\delta}>0$ and $\overline{\delta}<\infty$ and the interaction potential $K$ satisfies $K\in L^p(\Omega)\cap W^{1, \infty}(\Omega)\, \, (1<p<\infty)$.

Then the following stability estimate

holds, where $C$ is a positive constant depending only on $T$, possibly $I$, $\bar{\rho}$ and its derivatives. Moreover, if $\Theta(0)\rightarrow0$ as $\varepsilon\rightarrow0$, then

Let us point out that the strictly positive assumptions on $\rho$ and $\bar{\rho}$ are vitally important for our computations in the sequel. Especially, when $\Omega = \mathbb{T}^d$ or $\Omega$ is a bounded domain in $\mathbb{R}^d$, we need to assume that $0<\underline{\delta}\leq\bar{\rho}\leq\bar{\delta}<\infty$ for getting the results on the more general range of $m$, we also need to assume the periodic boundary condition or no-flux boundary condition for $\bar{\rho}$ in these cases. Moreover we may need more regular assumptions on the interaction potential $K$ and the confinement potential $\Phi$ in order to prove the existence of solutions to our systems. We will point out, in Section 3, the specific restrictions on $K$ and $\Phi$ when we show the existence of weak solutions to the system (1.6) on two or three dimensional bounded domains. Otherwise, we just assume that $K$ and $\Phi$ are as regular as we need.

The outline of this paper is as follows. In Section 2, we first review how to obtain the relative entropy inequality for our system using the notion of weak solution in Definition 1.1. We also show our main result in Theorem 1.2 by using the assumptions on the interaction potential and relative entropy estimates. Here, we follow the blueprint of [31] being the most novel aspects how to deal with the case $m = 1$ and the interaction potential. Finally, the last section is to remind the reader of the existence of weak solutions satisfying the needed properties for Theorem 1.2 under suitable assumptions on the confinement potential. This part relies heavily on previous results in [8] being the most novel aspect how to deal with the confinement potential term.

2.   Relaxation limit: Relative entropy & convergence

In this part, we devote ourselves to compare a weak solution $(\rho, \rho\textbf{u})$ of (1.6) with a smooth solution $(\bar\rho, \bar{\rho}\bar{\textbf{u}})$ of (1.12) by using a relative entropy method. To this end, we firstly propose the following Proposition which can be seen as a first step towards our main result.

Proposition 1. Let $\Omega$ be any smooth, connected, open subset of $\mathbb{R}^d$. Let $(\rho, \rho\textbf{u})$ be a weak solution of (1.6) as in Definition 1.1 and $(\bar{\rho}, \bar{\rho}\bar{\textbf{u}})$ be a smooth solution of (1.12). Then

Proof. Firstly, we introduce the standard choice of test function in (1.16)

and we have

Letting $\kappa$ tend to $0^{+}$, one has

Moreover, integrating (1.13) over time interval $[0, t]$, one obtains

Next, we deduce from systems (1.6) and (1.12) that the differences $\rho-\bar{\rho}$ and $\rho\textbf{u}-\bar{\rho}\bar{\textbf{u}}$ are given by the following equations

Thus, the weak formulation for the equations satisfied by the differences $\rho-\bar{\rho}$ and $\rho\textbf{u}-\bar{\rho}\bar{\textbf{u}}$ in (2.5) reads

where $\varphi$ and $\tilde{\varphi}$ are Lipschitz test functions compactly supported in $[0, \infty)$ in time and $\tilde{\varphi}\cdot\nu = 0$ on $\partial\Omega$ when $\Omega\neq\mathbb{R}^d$. Using the definition of $\theta(\tau)$ in (2.2), we introduce the test functions in the above relations

and then we have by letting $\kappa\rightarrow0^{+}$ after substituting $\varphi$, $\tilde{\varphi}$ into (2.6) and (2.7)

and

We can deduce from the computation (2.3) $-$ (2.4) $-$ ((2.8) $+$ (2.9)) that

Deducing from (1.12) by using $\bar{\rho}>0$, one can obtain the equation satisfied by $\bar{\textbf{u}}$

where we have used (1.5). Furthermore, multiplying (2.11) with $\rho(\textbf{u}-\bar{\textbf{u}})$ leads to

Substituting (2.12) into (2.10) and using (1.12)$_1$, one gets

Due to the fact that $K$ is symmetric, one can deduce that

consequently,

Hence, one can finally obtain by substituting (2.14) into (2.13) that

This exactly completes the proof of the Proposition 1.

2.1.   Convergence in the relaxation limit

In this subsection, we will establish the convergence property in the relaxation limit from (1.6) to (1.12) based on Proposition 1.

With the relative relation (2.1) between solutions to (1.6) and (1.12) at hand, we can prove Theorem 1.2 by showing that terms on the right-hand-side of (2.1) can be absorbed or are $O( \varepsilon)$.

Before getting into the proof of our main theorem, we need firstly to have some auxiliary lemmas which essentially indicate that the relative potential energy can be bounded from below by some positive functions.

Lemma 2.1. Let $h(\rho)$ be defined by (1.5) and (1.8). Then for any $\bar{\rho}>0$, we have the following estimates:

and

Proof. For the case of $m = 1$, the Taylor expansion of $h(\rho)$ at $\bar{\rho}$ reads

which implies

For the case of $1<m\leq2$, similarly, the Taylor expansion of $h(\rho)$ at $\bar{\rho}$ entails that

This completes the proof of (2.15) and (2.16).

We remind the readers a result proved in [32,Lemma 2.4].

Lemma 2.2. Let $h(\rho)$ be defined by (1.5) and (1.8). If $\bar{\rho}\in I = [\underline{\delta}, \overline{\delta}]$ with $\underline{\delta}>0$ and $\overline{\delta}<+\infty$, $m>1$, then there exist positive constants $R_0$ (depending on $I$) and $C_1$, $C_2$ (depending on $I$ and $R_0$) such that

Given $h(\rho)$ defined by (1.5) and (1.8), we can verify by using a similar way as in [32,Lemma 2.3] that

Lemma 2.3. Let $\Omega$ be any smooth, connected, open subset of $\mathbb{R}^d$. Let the confinement potential $\Phi(x)$ be bounded from blow in $\Omega$ and $h(\rho)$ be defined by (1.5) and (1.8). Assume one of the following conditions hold:

(i) If $2-\frac{2}{d}\leq m \le 2$ with $d\geq2$ and the interaction potential $K$ satisfies $K\in L^{\frac{m}{2(m-1)}}(\Omega)\cap W^{1, \infty}(\Omega)$,

(ii) If $\Omega = \mathbb{T}^d$ or $\Omega$ is a bounded domain in $\mathbb{R}^d$, $m\geq2-\frac{2}{d}$ with $d\geq2$, $\bar{\rho}\in[\underline{\delta}, \overline{\delta}]$ with $\underline{\delta}>0$ and $\overline{\delta}<\infty$ and $K$ satisfies $K\in L^p(\Omega) \quad (1<p\le\infty)$.

Then there exists a positive constant $C_{*}$ such that

Proof. Firstly, let us work with the case $m = 1$ and $d = 2$. By using Hölder's inequality and Young's inequality, we obtain

Due to

where we have used the mass conservation property of $\rho$ and $\bar{\rho}$ in the last inequality. We can claim by substituting (2.20) into (2.19) and using (2.15) that (2.18) is valid for $m = 1$, $d = 2$.

For the case of $1< m\leq2$ with $d = 2$ and $2-\frac{2}{d}\leq m\leq2$ with $d\geq3$, we have

Since $\Phi$ is bounded from blow and $\int_{\Omega}(K\ast\rho)\rho\, dx \leq \|K\|_{L^{\infty}(\Omega)}\|\rho\|^2_{L^1(\Omega)}$, one can deduce from the energy estimates (1.10) and (1.13) that $\int_{\Omega}\rho^mdx$ and $\int_{\Omega}\bar{\rho}^mdx$ are bounded. Thus we have

which implies that

Substituting (2.22) into (2.21) and using (2.16), then, for $1<m\le2$ with $d = 2$ and $2-\frac{2}{d}\leq m\leq2$ with $d\geq3$, the proof of (2.18) is completed.

It remains to prove the case of $m>2$ with any $d\geq 2$ when $\Omega = \mathbb{T}^d$ or $\Omega$ is a bounded domain. In Lemma 2.2, by enlarging if necessary $R_0$ so that $|\rho-\bar{\rho}|\geq1$ for $\rho>R_0$ and $\bar{\rho}\in[\underline{\delta}, \overline{\delta}]$, then we have

Thus, one deduce that

where $1\leq r<2$ and we have used the fact that $\Omega = \mathbb{T}^d$ or $\Omega$ is a bounded domain in the second last inequality. The proof of (2.18) is completed.

Corollary 1. Let the assumptions in Lemma 2.3 hold and the parameter $C_k$ is such that $C_{k}<\frac{2}{C_{*}}$, where $C_{*}$ is defined in (2.18), then for $\lambda: = 1-\frac{C_kC_{*}}{2}>0$

So far, all the preparations have been done, we now start to prove our main result.

Proof of Theorem 1.2. Firstly, one can easily see from the definition of $\Theta(t)$ in (1.14) and the relative entropy identity (2.1) that

Now, we estimate $J_1$, $J_2$, $J_3$, and $J_4$ one by one. Using the relation between $p$ and $h$ in (1.5) and the definition of $\bar{\textbf{u}}$ in (1.11), then we deduce that $\bar{\textbf{u}} = -\nabla_x h'(\bar{\rho}) - C_k(\nabla_x K\ast\bar{\rho}) - \nabla_x\Phi$ and $\nabla_x\bar{\textbf{u}}$ are bounded functions due to the smoothness assumption on $\bar{\rho}$.

For $J_1$, one obtains

We will estimate $J_2$ for three different cases. The first case is for $m = 1$ and $d = 2$, the second case is for $2-\frac{2}{d}<m\leq2$ with $d\geq2$ or $2-\frac{2}{d}\leq m\leq 2$ with $d\geq 3$ and the third case is for $m>2$ for any $d\geq 2$. For $m = 1$ and $d = 2$, using Hölder's inequality and Young's inequality, one deduces by using integration by parts that

where we have used (2.20) in the third last inequality and Lemma 2.1 in the second last inequality.

For the case $1<m\leq2$ with $d = 2$ and $2-\frac{2}{d}\leq m\leq 2$ with $d\geq 3$, we obtain by using interpolation inequality and Young's inequality that

Substituting (2.22) into (2.26), we have by Lemma 2.1

Finally, for the case $m>2$ and any $d\geq 2$, we have by applying Young's inequality that

where $\frac{1}{p}+\frac{1}{q} = 1$, due to Lemma 2.2 used in the second last inequality.

For $J_3$, by (2.17), one has

For $J_4$, we similarly have that

where we have used the fact that $\bar{e}$ is bounded and the mass conservation of $\rho$ in the last inequality. Substituting (2.24), (2.25), (2.27), (2.28), (2.29) and (2.30) into (2.23), one can see that

Hence, Gronwall's inequality leads to

for any $t\in(0, T]$, where $\tilde{C}$ is a positive constant depending on $T$. This completes the proof of Theorem 1.2.

Recalling the definition of $\Theta(t)$ in (1.14) and the properties of $h(\rho|\bar{\rho})$ showed in Lemma 2.1 and Lemma 2.2, we can easily conclude the following result.

Corollary 2. Let all conditions in Theorem 1.2 hold, then we can conclude that the weak solution of (1.1) converges to the solution $(\bar{\rho}, \bar{\rho}\bar{\textbf{u}})$ of (1.4) in the sense that

and

where $\bar{\textbf{u}} = -\nabla_x h(\bar{\rho}) - C_k(\nabla_x K \ast\bar{\rho}) - \nabla_x\Phi$.

3.   Weak solutions to the hydrodynamic system

Our goal in this section is to prove existence of weak solutions to the system (1.6) by using the methods of convex integration and oscillatory lemma shown in the seminal work by C. De Lellis and L. Székelyhidi [18]. Similar methods are later applied to deal with the compressible Euler system by E. Chiodaroli [13], the Euler systems with non-local interactions by J. A. Carrillo et al. [8] and some more general "variable coefficients" problems in [20,14,22,23].

The proof of the existence theory for the weak solutions of Euler flow (1.6) on any bounded domain $\Omega\subset\mathbb{R}^d$, $d = 2, 3$ with smooth boundary can be done by adapting the method of convex integration in [8]. Solvability for other cases mentioned in this paper, i.e. $\Omega\subset\mathbb{R}^d$ ($d\geq2$) unbounded or $\Omega\subset\mathbb{R}^d$ ($d\geq4$) bounded with smooth boundary, are left open.

For simplicity, we take the coefficients $\varepsilon = C_k = 1$ in (1.6) and restrict ourselves to the spatially periodic boundary conditions, i.e. $x\in\Omega$, where

is the "flat" torus. One should notice that this method is applicable for the general connected bounded domains $\Omega\subset\mathbb{R}^d$ with smooth boundary endowed with the no-flux boundary conditions $\textbf{u}\cdot\nu|_{\partial\Omega} = 0$. Thus, we consider the solvability of the following system

with initial data

Theorem 3.1. Let $T\geq0$ and $\Omega$ be given as in (3.1). Suppose that

Let the initial data $\rho_0$, $\textbf{u}_0$ satisfy $\rho_0\in C^2(\Omega)$, $\rho_0\ge\underline{\rho}>0$ in $\Omega$, $\textbf{u}_0\in C^3(\Omega;\mathbb{R}^d)$. Then the system (3.2), (3.3), (3.1) admits infinitely many solutions in the space-time cylinder $(0, T)\times\Omega$ belonging to the class

For the reader's convenience and completeness of this paper, we give a sketch of the proof of Theorem 3.1 following the blueprint of [8].

3.1.   Solvability of the abstract Euler system

Firstly, we introduce the notations

where $\textbf{v}, \textbf{w}\in\mathbb{R}^d$ are two vectors, $\mathbb{R}_{ {\rm sym}}^{d\times d}$ denotes the space of $d\times d$ symmetric matrices over the Euclidean space $\mathbb{R}^d$, $d = 2, 3$, $\mathbb{R}_{ {\rm sym, 0}}^{d\times d}$ means its subspace of those with zero trace. Recalling the abstract result in [18,22] which will be used later to prove our existence result, we consider the following abstract Euler form:

Find a vector field $\textbf{v}\in C_{ {\rm weak}}([0, T];L^2(\Omega;\mathbb{R}^d))$ satisfying

in $\mathcal{D}'((0, T)\times\Omega;\mathbb{R}^d)$,

where $\textbf{h}[\textbf{v}]$, $r[\textbf{v}]$, $\mathbb{H}[\textbf{v}]$, and $e[\textbf{v}]$ are given (nonlinear) operators.

Definition 3.2. Let $Q\subset(0, T)\times\Omega$ be an open set such that

An operator

is $Q$-continuous if

● b maps bounded sets in $L^{\infty}((0, T)\times\Omega;\mathbb{R}^d)$ on bounded sets in $C_b(Q, \mathbb{R}^m)$;

● b is continuous, specifically,

● b is causal (non-anticipative), meaning

Before quoting the solvability results in [8,22] for system (3.4)-(3.6), we need to further introduce the set of subsolutions:

where $\lambda_{ {\rm max}}[A]$ denotes the maximal eigenvalue of a symmetric matrix $A$. Now, we can state the following existence result for (3.4)-(3.6), see [8,22]:

Proposition 2. Let the operators $\textbf{h}$, $r$, $\mathbb{H}$ and $e$ be $Q$-continuous, where $Q\subset [(0, T)\times\Omega]$ is an open set satisfying $|Q| = |(0, T)\times\Omega|$. In addition, suppose that $r[\textbf{v}]>0$ and that the mapping $\textbf{v}\mapsto 1/r[\textbf{v}]$ is continuous in the sense specified in Definition 3.2. Finally, assume that the set of subsolutions $X_0$ is non-empty and bounded in $L^{\infty}((0, T)\times\Omega;\mathbb{R}^d)$. Then, problem (3.4)-(3.6) admits infinitely many solutions.

3.2.   Recast (3.2)-(3.3) into the abstract Euler form

In order to apply Proposition 2 to prove the solvability of (3.2)-(3.3), we need to firstly recast them into the form of (3.4)-(3.6). If we can further verify that assumptions in Proposition 2 hold, then existence of solutions for the system (3.2)-(3.3) is proven. To this end, we take $Q = (0, T)\times\Omega$.

3.2.1.   Momentum decomposition and kinetic energy

Following [8] one can write the momentum $\rho\textbf{u}$ in the form

where

Similarly, we write the initial momentum $\rho_0\textbf{u}_0$ as

Accordingly, we may fix $\rho\in C^2([0, T]\times\Omega)$ such that for a certain potential $\Psi$,

Hence, in the sequel, we assume that that

are fixed functions. Based on the above decomposition, equation (3.2) reduces to

In order to match (3.5), we fix the "kinetic energy" so that

where $\Pi = \Pi(t)$ is a spatially homogeneous function to be determined later. Substituting (3.8) into (3.7), one can therefore rewrite (3.7) as

3.2.2.   Fix $\textbf{V}$ and recast (3.9) into abstract form

One can easily notice from (3.9) that there are still two unknowns $\textbf{v}$ and $\textbf{V}$. So our first goal in this subsubsection is to fix $\textbf{V}$ so that (3.9) can be converted to a "balance law" with a source term of zero mean. To this end, solving the following ODE:

with initial data $\textbf{V}(0) = \textbf{V}_0$, one can see that $\textbf{V} = \textbf{V}[\textbf{v}]$ depends linearly on the fixed function $\rho$. Thus, we can therefore rewrite (3.9) as

Obviously, the expression on the right-hand-side of (3.10) has zero integral mean at any time $t$. Hence, referring [8] for more details, we can find a vector $\textbf{w} = \textbf{w}[\textbf{v}]$ satisfying

Denoting

one can thus transform system (3.2)-(3.3) to the form coincide with (3.4)-(3.6), namely:

Find a vector field $\textbf{v}\in C_{ {\rm weak}}([0, T];L^2(\Omega;\mathbb{R}^d))$ satisfying

in $\mathcal{D}'((0, T)\times\Omega;\mathbb{R}^d)$,

3.3.   Proof of Theorem 3.1

Proof of Theorem 3.1. Taking $r[\textbf{v}] = \rho$, $\textbf{h}[\textbf{v}] = \textbf{V}[\textbf{v}] + \nabla_x\Psi$, $\mathbb{H}[\textbf{v}]$ defined by (3.11), and $e[\textbf{v}]$ defined by (3.12), one can easily verify that they are $Q$-continuous. Then Theorem 3.1 can be proved if we are able to show that $X_0$ is non-empty and bounded in $L^{\infty}((0, T)\times\Omega;\mathbb{R}^d)$.

To this end, taking $\textbf{v}_T = \textbf{v}_0$, $\textbf{v} = \textbf{v}_0$, and $\mathbb{F} = 0$ in the definition of $X_0$ and choosing $\Pi = \Pi(t)$ to be large enough satisfying

for any $0<\tau<T$, one can claim that there exists $\Pi_0>0$ such that the above inequality holds whenever $\Pi(t)\geq\Pi_0$ for all $t\in[0, T]$. Consequently, $\textbf{v}_0\in X_0$ and thus $X_0$ is non-empty.

In order to prove $X_0$ is bounded in $L^{\infty}((0, T)\times\Omega;\mathbb{R}^d)$, we firstly recall the purely algebraic inequality [18],

Fixing $\Pi(t)$ according to the above discussions, for any $\textbf{v}\in X_0$, we have by using the definition of $X_0$

By the definition of $\mathbb{H}$ in (3.11), one can obtain that $\mathbb{H}[\textbf{v}]\in \mathbb{R}_{ {\rm sym}, 0}^{d\times d}$. Applying the inequality (3.13), we have

which implies that $X_0$ is bounded in $L^{\infty}((0, T)\times\Omega;\mathbb{R}^d)$. So far, all the assumptions in Proposition 2 hold, and the proof of Theorem 3.1 directly follows now by using Proposition 2.

Acknowledgments

JAC was partially supported by the EPSRC grant number EP/P031587/1. JAC acknowledges partial support through the Changjiang Visiting Professorship Scheme of the Chinese Ministry of Education. YP was supported by the CSC (China Scholarship Council) during her visit at Imperial College London. AWK was partially supported by the grant Iuventus Plus no. 0871/IP3/2016/74 of Ministry of Sciences and Higher Education Republic of Poland. YP's visit at Institute of Mathematics, Polish Academy of Sciences was supported by the programme "Guests of IMPAS".