Energy equality in the isentropic compressible Navier-Stokes-Maxwell equations

1.   Introduction

In this paper, the following isentropic compressible Navier-Stokes-Maxwell (CNSM) system is considered, which consists of Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism. The coupling comes from the Lorentz force in the fluid equation and the electric current in the Maxwell following equations:

with the initial data

where $\rho$ is the density, $u$ is the velocity field, and $E$ and $b$ represent electronic and magnetic fields, respectively. The fluid pressure is represented by $P(\rho)$ meets:

where $a$ is a physical constant and $\gamma$ is the adiabatic exponent. The viscosity coefficients $\mu$ and $\lambda$ are constant and satisfy the physical restrictions $\mu > 0$ and $2\mu + 3\lambda \geq 0$. $j$ is the electric current expressed by the Ohms law. The force term $j\times B$ in Navier-Stokes equations comes from Lorentz force under a quasi-neutrality assumption of the net charge carried by the fluid. If the electric current is ignored (i.e., $j = 0$), (1.1) reduces to the well-known isentropic compressible Navier-Stokes (CNS) system. Equation (1.1) is one of the most important mathematical models in continuum mechanics. Lions ^[1] and Feireisl ^[2,3,4] proved that the CNS system admits a weak solution, as long as the adiabatic exponent $\gamma > \frac{3}{2}$. Due to a lack of regularity of weak solutions, it is not known whether weak solutions satisfy the energy equality for both incompressible and compressible fluids equations. It is a nature problem: how "good" is regularity for weak solutions needed to ensure the energy equality?

For a CNS system, the appearance of $\rho$ makes $\partial_t(\rho u)$ nonlinear, and; therefore, some density regularity is required in when using commutator estimates. Yu ^[5] used the Lions's commutator estimate to show energy conservation for compressible Navier-Stokes equations with a degenerate viscosity but without vacuum. Nguyen et al. ^[6] extended Yu's result with a weaker regularity condition in a bounded domain. Liang ^[7] established a $L^pL^s$ type condition for the energy equality, in particular, there was no need regularly assume the density derivative. Recently, Ye et al. ^[8] showed that Lions' condition for energy balance is also valid for the weak solutions of isentropic compressible Navier-Stokes equations allowing for vacuum under suitable integral conditions on the density and its derivative. This is a very interesting result.

Comparing with fruitful results for either an incompressible Navier-Stokes system or a compressible case, there are a few results regarding an incompressible/compressible NSM model due to its hyperbolic structure. For the incompressible NSM system, Ma and Wu ^[9] obtained the Shinbrot type energy conservation criteria for the weak solution. In addition, for the distributional solution, he showed the Lions' energy conservation criteria ^[10]. To the best of our knowledge, there is no result concerning the energy equality for a CNSM system (1.1).

Motivated by ^[7,8,9,10], the purpose of this paper is to establish the conditional energy conservation of weak solutions to the CNSM system (1.1) allowing for vacuum. By proving an energy conservation/equality, the commutator estimates are required for treating the nonlinear terms. Furthermore, due to the special structure of parabolic hyperbolic coupling, the derivative to the velocity field $u$ needs to be transfered. We state the result in detail in the theorem below.

Theorem 1.1. Let $0\leq \rho < c < \infty$, $\nabla \sqrt{\rho}\in L^{4}(0, T;L^{4}(\mathbb{T}^{3}))$, $u\in L^{\infty}(0, T;L^{2}(\mathbb{T}^3))\cap L^{2}(0, T;H^{1}(\mathbb{T}^3))$, $(E, b)\in L^{\infty}(0, T;L^{2}(\mathbb{T}^3))$ and $j\in L^{2}(0, T;L^{2}(\mathbb{T}^3))$ be a weak solution to system (1.1). In addition, if $(u, b)\in L^{4}(0, T;L^{4}(\mathbb{T}^{3}))$ and $E\in L^{2}(0, T;L^{2}(\mathbb{T}^{3}))$, then the weak solution satisfies the following energy equality:

Remark 1.1. This theorem extends the energy equality of the incompressible NSM to the isentropic compressible equations (1.1) with vacuum.

Remark 1.2. Since $u, b \in L^{\infty}(0, T;L^{2}(\mathbb{T}^{3}))\cap L^{p}(0, T;L^{q}(\mathbb{T}^{3}))$, for any $\frac{1}{p}+\frac{1}{q}\leq\frac{1}{2}$, $q\geq4$, one can deduce that

and

for some $a\in (0, 1)$. Thus, it is also true that $L^{4}(0, T;L^{4}(\mathbb{T}^{3}))$ in Theorem 1.1 is replaced with $L^{p}(0, T;L^{q}(\mathbb{T}^{3}))$, for any $\frac{1}{p}+\frac{1}{q}\leq\frac{1}{2}$, $q\geq4$.

2.   Preliminaries

We will recall some definitions and lemmas that will be used later. First, we denote $\mathcal{D}\left(\mathbb{T}^3\right)$ as the space of indefinitely differentiable with compact support and $\mathcal{D'}\left(\mathbb{T}^3\right)$ as the space of distributions.

Definition 2.1. The $(\rho, u, E, b)$ is called a weak solution to the CNSM systems (1.1) and (1.2) if $(\rho, u, E, b)$ satisfies the following assumptions for any time $t\in[0, T]$$:$

$\bullet$ The problems (1.1) and (1.2) holds in $\mathcal{D}'(0, T;\mathbb{T}^3)$;

$\bullet$ The equation $(1.1)_{1}$ is satisfied in the sense of renormalized solutions: for any function $b\in C^{1}(\mathbb{R})$ such that $b'(x) = 0$ for $x\geq M)$, we get in $\mathcal{D}'(0, T;\mathbb{T}^3)$$:$

where $M$ is a constant that varies for different functions $b$.

$\bullet$ The weak solutions require the following properties:

$\bullet$ The energy inequality for weak solutions holds:

Let $\eta\in C^\infty_c(\mathbb{R}^d)$ ($d$ is the number of the space dimension) be a standard mollification kernel and set

We should notice that $w^\varepsilon$ is well-defined on $\Omega^\varepsilon = \{x\in\Omega: d(x, \partial\Omega) > \varepsilon\}$. Next, we recall some useful lemmas which will be frequently used throughout the paper.

Lemma 2.1. ^[8] Let $r, s, r_{1}, r_{2}, s_{1}, s_{2}\in [1, +\infty)$ with $\frac{1}{r} = \frac{1}{r_{1}}+\frac{1}{r_{2}}$ and $\frac{1}{s} = \frac{1}{s_{1}}+\frac{1}{s_{2}}$. Assume $f\in L^{r_{1}}(0, T;L^{s_{1}}(\mathbb{T}^{3}))$ and $g\in L^{r_{2}}(0, T;L^{s_{2}}(\mathbb{T}^{3}))$. Then, for any $\varepsilon > 0$, there holds

and

Lemma 2.2. ^[8,11,12] Let $1\leq r, s, r_1, s_1, r_2, s_2\leq \infty$, with $\frac{1}{r} = \frac{1}{r_1}+\frac{1}{r_2}$ and $\frac{1}{s} = \frac{1}{s_1}+\frac{1}{s_2}$. Let $\partial$ be a partial derivative in space or time; in addition, let $\partial_t f, \ \nabla f \in L^{r_1}(0, T;L^{s_1}(\mathbb{T}^3))$, $g\in L^{r_2}(0, T;L^{s_2}(\mathbb{T}^3))$. Then, there holds

or some constant $C > 0$ independent of $\varepsilon$, $f$ and $g$. Moreover, as $\varepsilon\to 0$ if $r_2, s_2 < \infty,$

Lemma 2.3. ^[13] Let $B_{0}\hookrightarrow B \hookrightarrow B_{1}$ be three Banach spaces with compact embedding $B_{0}\hookrightarrow\hookrightarrow B_{1}$, and let there exist $0 < \delta < 1$ and $C > 0$ such that

Denote for $T > 0$,

with

Assume that $s_{\delta} > 0$ and $G$ is a bounded set in $W(0, T)$, Then, we have the following:

$\bullet$ If $s_{*}\leq 0$, then G is relatively compact in $L^{p}(0, T;B)$ for all $1\leq p < p^{*}: = -\frac{1}{s^{*}}$.

$\bullet$ If $s_{*} > 0$, then G is relatively compact in $C(0, T;B)$.

3.   Proof of theorem 1.1

First, we mollify the system (1.1) and obtain

and

for any $0 < \varepsilon < 1$.

Next, let $\phi(t)$ be a smooth solution function compactly supported in $(0, +\infty)$. Multiplying (3.2)–(3.4) by $\phi(t)u^{\varepsilon}$, $\phi(t)E^{\varepsilon}$, and $\phi(t)b^{\varepsilon}$, respectively, then integrating over $(0, T)\times \mathbb{T}^{3}$, one has the following:

We use (A)–(H) and (J)–(L) to represent the terms on the left-hand side of (3.5), respectively. We will estimate them as follows.

3.1.   Estimate of (A)

By a straightforward computation, we can obtain the following:

We know that $(A_{3})$ is the desire term while $(A_{2})$ will be canceled with the term $(B_{2})$ later. By Hölder's inequality and Lemma 2.2, it gives that the following:

Based on system (1.1), $\rho_{t}$ and $\nabla \rho$ can be denoted as follows:

We will obtain the estimate of $\rho_{t}$ and $\nabla \rho$ by using $0\leq\rho < c < \infty$, $(u, \nabla\sqrt{\rho})\in{L^{4}(0, T;L^{4}(\mathbb{T}^{3}))}$ and $\nabla u\in {L^{2}(0, T;L^{2}(\mathbb{T}^{3}))}$ in Theorem 1.1, which implies that

and

Inserting (3.6) and (3.7) into $(A_{1})$ yields the following:

From Lemma 2.2, we get the estimate of $(A_{1})$ that

3.2.   Estimate of (B)

By utilizing integration by parts and the mass equation $(1.1)$, we deduce that

Taking the mass equation $(1.1)_{1}$ into consideration, we know that $(A_{2})+(B_{2}) = 0$. The (B3) is the desired term.

By Hölder's inequality and triangle inequality, we deduce the following:

Thanks to the standard properties of mollifiers, we have the following:

3.3.   Estimates of (C) and (D)

Utilizing integration by parts, we know that the following (C) and (D) are the desired terms, where

and

3.4.   Estimate of (E)

Utilizing integration by parts and applying (1.1) leads to the following:

The term $(E_{3})$ is the desired term, and the estimate of $(E_{1})$ and $(E_{2})$ will be finished as follows:

and

By the upper bounded of $\rho$ and Hölder's inequality, we have the following:

Combining the standard properties of mollifiers and (3.9), we know that

3.5.   Estimates of (F) and (J)

Next, we turn to estimate (F) and (J), of which the proof is inspired by ^[10], and we include that

We see that $(FJ)_{2}$ is desired term, while the estimates of $(FJ)_{1}$ and $(FJ)_{2}$ will be finished. By Hölder's inequality, we can conclude that

and

However, the following results are valid by using Hölder's inequality:

and

Therefore, from $(FJ)_{1}$, $(FJ)_{3}$, (3.10) and (3.11), with the help of Lemma 2.1, we obtain the following:

3.6.   Estimates of (G), (H), (L) and (K)

The remaining is to estimate (G), (H), (L) and (K). Using a straightforward computation leads to

and

and

Then, summarizing all above the aforementioned estimates, putting them into (3.5) and taking the limit as $\varepsilon\rightarrow 0$, we obtain the following:

We can express it in the following form:

Next, we study a similar method in ^[5] and shall prove the energy equality up to the initial time $t = 0$. First, we claim that the following results are valid for any $t_{0}\geq 0$:

Based on the mass equation $(1.1)$, we can write

and

which, together with the assumptions in Theorem 1.1, gives

and

Hence, due to Lemma 2.3, it yields that the following:

Consequently, for any $t_{0}\geq 0$, by the right temporal continuity of $\rho^{\gamma}$ in $L^{2}(\mathbb{T}^{3})$ and $L^{2}(\mathbb{T}^{3})\subset L^{1}(\mathbb{T}^{3})$, we deduce that the following:

Furthermore, using the momentum equation $(1.1)_{2}$, we obtain the following:

Then, because of Lemma 2.3, we have the following:

Similarly, from $(1.1)_{3}$, $(1.1)_{4}$ and $(2.2)$, we can deduce that the following:

On the other hand, the assumptions in Theorem 1.1 implies

which can be obtained that leads to the following conclusion:

Hence, for any $t_{0}\geq 0$, from (3.17), we get that the following:

Meanwhile, utilizing (2.2), (3.14), (3.16), (3.17) and the assumptions in Theorem 1.1 yields to the following:

which implies

Similarly, we can establish the right temporal continuity of $\sqrt{\rho}u$ in $L^{2}(\mathbb{T}^{3})$; hence, for any $t_{0}\geq 0$, we have the following:

Combining (3.15), (3.18) and (3.20), we have now completed the proof of (3.13).

We notice that (3.12) is valid for $\phi$ belonging to $W^{1, \infty}$ rather than $C^{1}$. Therefore, for any $t_{0} > 0$, we can use a new test function $\phi_{\tau}$ to represent $\phi$ for some positive $\tau$ and $\alpha$ such that $\tau+\alpha < t_{0}$, that is

Then, substituting this function into (3.12), we have the following:

Letting $\alpha\rightarrow 0$ and using the fact that $\int_{0}^{t}\int_{\mathbb{T}^{3}}\phi_{\tau}(\mu|\nabla u|^{2}+(\mu+\lambda)|\text{div} u|^{2}+|j|^{2})dxdt$ is continuous with respect to $t$ and the Lebesgue point Theorem, for all $\tau$ and $t_{0}\in[0, T]$, we arrive at the following:

Finally, taking $\tau\rightarrow 0$, combining the continuity of $\int_{0}^{t_{0}}\int_{\mathbb{T}^{3}}(\mu|\nabla u|^{2}+(\mu+\lambda)|\text{div} u|^{2}+|j|^{2})dxdt$ and (3.13), for all $t_{0}\in [0, T]$, we can deduce that

This now completes the proof of Theorem 1.1.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

Jie Zhang was supported by Innovation Research for the Postgraduates of Guangzhou University (No. 2022GDJC-D08), Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515010566) and National Natural Science Foundation of China (No. 12171111). Fan Wu was supported by the Science and Technology Project of Jiangxi Provincial Department of Education (No. GJJ2201524) and the Jiangxi Provincial Natural Science Foundation(No. 20224BAB211003).

Conflict of interest

The authors declare there is no conflicts of interest.