Energy conservation for the compressible ideal Hall-MHD equations

1.   Introduction

In this work, we consider the energy conservation for weak solutions of the compressible ideal Hall-magnetohydrodynamic (Hall-MHD) equations

where $\rho > 0$ is the fluid density, $u$ is the velocity field, $p$ is the pressure and $b$ is the magnetic field. $\rho$ and $p$ are scalars, $d_I$ represents the Hall coefficient and $j = \nabla \times b.$

The energy conservation of weak solutions of the Euler equations and the MHD equations is a hot topic in recent decades. For the 3D Euler equations, Onsager ^[22] put forward famous Onsager's conjecture in 1949, that is, the weak solution with Hölder continuity of exponent $\delta > \frac{1}{3}$ can guarantee the conservation of energy, but the weak solution of $\delta\leq \frac{1}{3}$ is not necessary. In 1994, Eyink ^[12] proved the first part of this conjecture by means of Fourier series expansion. Constantin et al. ^[7] considered the conservation of energy when $u \in B_{3, \infty}^\alpha, \alpha > \frac{1}{3}$ in the periodic domain. In 2008, Cheskidov et al. ^[6] improved the previous results by using Littlewood-Paley decomposition. Concerning the second part of conjecture, the first proof of the existence of a square summable weak solution that does not preserve the energy is due to Scheffer in his pioneering paper ^[24]. A different proof was later given by Shnirelman in ^[23]. Recently, non-conservation solutions for the 3D incompressible Euler equations have been constructed up to the critical $\frac{1}{3}$ regularity in ^[5,17]. For the compressible and incompressible Euler equations and Navier-Stokes equations, we can refer to Feireisl ^[14], Chen and Yu ^[9] and Akramov et al. ^[1] etc. The energy conservation for the incompressible Euler equations in bounded domains can be referred to ^[2,3,11,21].

Concerning the energy conservation for the MHD equations, there have been a few results. Recently, Gao et al. ^[16] considered the local energy equation of weak solutions for the incompressible MHD equations. Guo and Tan ^[15] studied the energy conservation equation of the 3D incompressible MHD equations from the longitudinal and transverse terms on the basis of the energy dissipation method. Wu and Tan ^[27] showed that the regularity of weak solutions of the nonhomogeneous incompressible MHD equations in Besov space is sufficient to ensure the total energy conservation. Wang and Liu ^[25] obtained the energy conservation for weak solutions of the compressible non-resistive magnetohydrodynamic flows in a bounded domain $\Omega\subset {\mathbb{R}}^3$. For the ideal MHD equations, Caflisch et al. ^[8] proved the energy conservation in a periodic domain with no boundary effect. Kang and Lee ^[18] obtained the energy and cross-helicity conservation to the ideal MHD equations in the whole space. Later on, Yu ^[28] improved the pervious results by using the special structure of the nonlinear terms in the ideal MHD equations. Wang and Zuo ^[26] and Zhang ^[29] proved the energy conservation of weak solutions to the 3D case in a bounded domain. Bie et al. ^[4] studied the energy conservation of the compressible ideal MHD equations in periodic domain by using commutator estimation.

The energy conservation for the Hall-MHD equations has also attracted the attention of many researchers. Dumas and Sueur ^[10] studied the energy identity and magneto-helicity identity for the incompressible Hall-MHD equations in the whole space $\mathbb{R}^3$. Kang et al. ^[19] obtained the energy conservation for the nonhomogeneous incompressible ideal Hall-MHD equations in the periodic domain $\mathbb{T}^d$. Recently, Kang et al. ^[20] further considered the energy conservation of impressible Hall-MHD equations in a bounded domain.

A natural question then is when the energy conservation holds true for the compressible ideal Hall-MHD Eq (1.1). Compared with the compressible Euler equations, the ideal Hall-MHD equations have higher nonlinearity in view of the coupling of the velocity field $u$ and the magnetic field $b$. Moreover, the handling of the Hall term $d_I\nabla\times\bigg(\dfrac{j\times b}{\rho}\bigg)$ brings us some difficulties. Specifically, we need to deal with the following item

(see (3.8) below for details). In this paper, by using the commutator estimation similar to that used in ^[7] and the regularization method used in ^[14], we obtain sufficient conditions for energy conservation of system (1.1). Our result establishes local energy conservation for weak solutions to system (1.1) under the additional assumption that the velocity field $u$ satisfies the condition ${\rm{div}}u\in L^1$.

Before giving the result of this paper, we define the pressure potential as

The main result of this paper is stated as follows.

Theorem 1.1. Let the space dimension $d\geq 1$ and $(\rho, u, p, b)$ be a solution of system $(1.1)$ in the sense of distributions. Assume

for some constants $\underline{\rho}, \overline{\rho}$ such that

Assume further that

Then the energy is locally conserved, that is

in the sense of distributions on $(0, T) \times {\mathbb{T}^d}$.

2.   Preliminaries

In this section, we firstly recall some properties of the Besov space $B_p^{\alpha, \infty}(\Omega)$, where $\alpha \in (0, 1)$ and $\Omega = {(0, T)\times {\mathbb{T}^d}}$. The said Besov space comprises those functions $\omega$ for which the norm

is finite (here $\Omega-\xi = \{x-\xi:x\in\Omega\}$).

Let $J\in{C_c^\infty(R^N)}$ for $N = d$ or $N = d+1$ (according to the choice of $\Omega$) be a standard mollifying kernel and set

with the notation $\omega^\epsilon = {J^\epsilon}\ast\omega$.

Next, we introduce two lemmas about the properties of mollifiers.

Lemma 2.1. (^[13]) If $1\leq p < \infty$ and $f\in L_{loc}^p(R^+ \times \Omega)$, then we get

Lemma 2.2. (^[27]) Let $\omega \in B^{\alpha, \infty}_p (\Omega)$ with $1\leq p \leq \infty$, and $\alpha \geq 0.$ Then we have

and

Finally, let us recall the following two commonly used formulas for curl, that is,

3.   Proof of Theorem 1.1

We mollify the momentum Eq (1.1)$_1$ in time and space, and the corresponding symbols are described in Section 2,

Take a sequence $p_\delta \in C^\infty[\underline{\rho}, \overline{\rho}]$ which converges uniformly to $p\in C[\underline{\rho}, \overline{\rho}]$, that is, for each $\delta > 0$,

According to the definition of $P$ in (1.2), we give the definition of $P_\delta$ as follows,

It follows from (3.1) that

Let $\varphi\in C_0^\infty ((0, T)\times {\mathbb{T}^d})$ be a test function. Multiplying with $\varphi u^\epsilon$ and integrating in time and space give

Take $\epsilon> 0$ small enough so that ${\rm supp}\: \varphi\in (\epsilon, T-\epsilon)\times {\mathbb{T}^d}$. We use an appropriate commutator, as

where

In what follows, we handle each item in (3.4). For the first integral term of the left hand side of (3.4), it follows that

For the second term, we mollify the continuity Eq (1.1)$_3$ as

and compute

For the third integral of the left hand side of (3.4), we first mollify the Eq (1.1)$_2$ and multiply $b^\epsilon$ as

Then using (2.5), we get

For the fourth integral, due to the chain rule and the mollified mass Eq (3.6), we observe that

By the definition of $P_\delta$ in (3.2), we have

After these preparations, we can compute the fourth integral as

Thus, combining (3.4), (3.5) and (3.7)–(3.9) yields

To prove Theorem 1.1, we need to show that the left side of (3.10) converges to the left side of (1.4) as first $\epsilon$ and then $\delta$ tend to zero. In fact, for each fixed $\delta > 0$, when $\epsilon \rightarrow 0$, using the standard properties of mollification, we have the following limit:

Here, we only give the proof of the following limit :

as $\epsilon \rightarrow 0.$ To prove (3.12), by the definition of $P_\delta$ in (3.2), we set

By $p_\delta(\rho)\in C^\infty[\underline{\rho}, \overline{\rho}]$, we can obtain

and

So,

we use the differential mean value theorem in the last equation, where $\xi$ is a function between $\rho$ and $\rho^\epsilon.$ According to (3.13), we get

as $\epsilon \rightarrow 0,$ which completes the proof of (3.12).

Next, we will show that the right hand side of (3.11) converges to

as $\delta \rightarrow 0$.

From the choice of $p_\delta$, we have

For the terms containing $P_\delta(\rho)$, notice that

Hence we can estimate

Combining with (3.15)–(3.17), when $\delta\rightarrow 0$, we obtain the right side of (3.11) converges to (3.14). Thus, according to (3.10), we need to show that

and

converge to zero as first $\epsilon$ and then $\delta$ tend to zero.

First it is easy to check

as $\epsilon \rightarrow 0.$ For $I_1^\epsilon$, we observe that

Therefore,

Applying (2.2), (2.3), integration by parts and Hölder inequality, we get

as $\epsilon\rightarrow 0$ for any $2\alpha +\beta > 1.$

For the second integral of the right hand side of (3.20), using Fubini theorem, (2.2) and (2.3), Hölder inequality and integration by parts, we estimate

as $\epsilon\rightarrow 0$ for any $2\alpha +\beta > 1,$ where $V = \Omega\cap (\Omega+(\tau, \xi)), |(\tau, \xi)| < \epsilon.$

The estimate for $I_2^\epsilon$ is similar,

Thus,

Thanks to (2.2), (2.3), integration by parts and Hölder inequality, we get

as $\epsilon\rightarrow 0$ for any $2\alpha +\beta > 1.$

For the second integral of the right hand side of (3.23), we have

as$\:\epsilon\rightarrow 0$ for any $2\alpha +\beta > 1.$

Next we deal with $I_3^\epsilon.$ Denote

Thus, one writes

Let us calculate $I_{31}^\epsilon$ first, and $I_{32}^\epsilon$ the same way. Similar to the estimate of $I_1^\epsilon$ and $I_2^\epsilon$, we derive

So,

Using (2.2) and Hölder inequality, we get

as $\epsilon\rightarrow 0.$

For the second integral of the right hand side of (3.26), we use (2.2), Hölder inequality and Fubini theorem to get that

as $\epsilon \rightarrow 0$.

For $I_4^\epsilon$, we observe that if $p_\delta \in C^\infty[\underline{\rho}, \overline{\rho}]$, then

for any $h, h_0\in [\underline{\rho}, \overline{\rho}]$. Note that the constant $C$ can be chosen independently of $h, h_0$. Therefore

and similarly,

Applying convolution with respect to $y$ to (3.30) we get, after invoking Jensen's inequality:

where $|x-y|\leq \epsilon.$ According to (3.29) and (3.31), one gets

We estimate

as $\epsilon \rightarrow 0$ for any $2\beta+\alpha > 1.$ Next, we show that $I_5^\epsilon$ converges to zero as first $\epsilon$ and then $\delta$ tend to zero,

as $\delta \rightarrow 0.$ Finally, let us estimate

We use (3.19) to split (3.35) into two parts, so we can estimate the first part as

when $\epsilon \rightarrow 0$ for any $2\beta+\alpha > 1.$ The second part is estimated similarly. Thus, combining (3.18)–(3.28) and (3.33)–(3.36), we have

as $\epsilon \rightarrow 0$, which completes the proof of Theorem 1.1.

4.   Conclusions

In this paper, we study the regularity and energy conservation of the weak solutions for compressible ideal Hall-MHD equations, where $(t, x)\in(0, T)\times {\mathbb{T}}^d(d\geq 1)$. Compared with the compressible Euler equations, the ideal Hall-MHD equations have higher nonlinearity in view of the coupling of the velocity field $u$ and the magnetic field $b$. Then, by exploring the special structure of the nonlinear terms in the model, we obtain the sufficient conditions for energy conservation under the additional assumption that the velocity field $u$ satisfies the condition ${\rm{div}}u\in L^1$. Our main strategy relies on commutator estimates.

Acknowledgments

This work was sponsored by the NNSF of China (Nos. 11871305, 11901346) and Research Fund for Excellent Dissertation of China Three Gorges University. The authors thank the referees for careful reading and valuable suggestions.

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.