Energy equality for the multi-dimensional nonhomogeneous incompressible Hall-MHD equations in a bounded domain

1.   Introduction

The goal of this paper is to study the energy equality for weak solutions of the nonhomogeneous incompressible Hall-magnetohydrodynamics (Hall-MHD) equations in $\Omega\times(0, T)$,

with initial data

and homogeneous Dirichlet boundary conditions

where $\Omega \subset \mathbb{R}^n$ $(n\geqslant2)$ is bounded with $\partial\Omega \in C^2$. $\rho, u, b$ and $P$ are the density, velocity field, magnetic field and scalar pressure, respectively. The constant $\mu > 0$ represents the viscosity coefficient of the flow. The term $d_I \nabla\times \left(\frac{(\nabla\times b)\times b}{\rho}\right)$ denotes the Hall effect, and $d_I$ is the Hall coefficient.

In $1960$, Lighthill ^[1] said that the MHD equations cannot provide a precise description of physical phenomena and introduced first the Hall term into the MHD equations, which constitute the so-called Hall-MHD equations. Subsequently, Arichetogaray et al. ^[2] analyzed the Hall-MHD equations from either a two-fluid Euler-Maxwell system for electrons and ions or kinetic models, which are used for describing many physical phenomena in geophysics and astrophysics, such as the magnetic reconnection of space plasma, star formation and neutron stars. From the mathematical viewpoint, the Hall-MHD equations have been widely studied, involving the local and global well-posedness ^[2,3,4,5], blow-up criteria ^[4,6], large time behavior ^[7,8,9,10], Liouville-type theorems ^{[11,12,13,14]} and energy conservation ^[15,16]. It is worth pointing out that Kang et al. ^[16] recently studied the energy conservation for systems (1.1)–(1.4) in a bounded domain $\Omega \subset\mathbb{R}^3$ and obtained that if weak solutions $(\rho, u, b, P)$ satisfy

then the following energy equality is valid:

Concerning the energy equality for the MHD equations, one can refer to the works ^{[17,18,19,20]}, etc.

When $b = 0$, systems (1.1)–(1.4) reduce to the incompressible Navier-Stokes system. For the energy equality of the incompressible Navier-Stokes system, the Lions-Shinbrot type criterion on the velocity was obtained by Lions ^[21], Shinbrot ^[22], Da Veiga and Yang ^[23] and Yu ^[24]. Later, Yu ^[25] extended Shinbrot's result to the bounded domain, with an additional Besov regularity imposed on the velocity, which is essential to deal with the boundary effects, and Nguyen et al. ^[26] handled the boundary effects without requiring extra conditions of velocity field $u$ near the boundary.

For the energy equality of the compressible Navier-Stokes equations, Yu ^[27] proved the energy equality holds true if $u\in L_t^pL_x^q$, $\rho$ is bounded, and $\sqrt{\rho} \in L^\infty(0, T; H^1(\Omega))$. Later on, Chen et al. ^[28] extended the result of ^[27] to the bounded domain by performing global mollification. In addition, it is worth mentioning that Berselli and Chiodaroli ^[29], Liang ^[30] and Wang and Ye ^[31] derived the energy equality criteria in terms of the velocity and its gradient.

Inspired by the works ^[16,26,31], we provide sufficient conditions on the regularity of solutions for systems (1.1)–(1.4) to ensure the energy equality holds. Compared with the results of ^[16], we obtain the sufficient conditions concerning $\nabla u$ and $\nabla b$, rather than $u$ and $b$, to guarantee that the energy equality is valid.

Before stating our main results, we give the definition of weak solutions to systems (1.1)–(1.4).

Definition 1.1. A couple $(\rho, u, b, P)$ is called a weak solution to systems (1.1)–(1.4) with initial data (1.5) if $(\rho, u, b, P)$ satisfies the following,

$1)$ Equations (1.1)–(1.4) hold in $D'(0, T; \Omega)$, and

$2)$ $\rho(\cdot, t)\rightharpoonup \rho_{0}$ in $D'(\Omega)$ as $t\rightarrow0$, i.e.,

for every test function $\varphi \in C^{\infty}_0(\Omega)$.

$3)$ $(\rho u)(\cdot, t)\rightharpoonup \rho_{0}u_0$ in $D'(\Omega)$ as $t\rightarrow0$, i.e.,

for every test vector field $\psi\in C^{\infty}_0(\Omega)^{n}$.

$4)$ The energy inequality holds, i.e.,

Next, we state our main results as follows.

Theorem 1.1. Let $\Omega\subset \mathbb{R}^n (n\geqslant 2)$ be a bounded domain with $C^2$ boundary $\partial\Omega$. The energy equality (1.8) of weak solutions $(\rho, u, b, P)$ to systems (1.1)–(1.4) with initial data (1.5) and Dirichlet boundary conditions (1.6) is valid provided

where $1 < p, q < \infty$.

Remark 1.1. Thanks to the embeddings

the conditions $u, b \in L^{\frac{2p}{p-1}}(0, T; L^\frac{2q}{q-1}(\Omega))\cap L^p(0, T; {L^q}(\Omega))$ in (1.13) could be replaced by the following:

Remark 1.2. Setting $n = 3$, $p = q = 2$ in Theorem 1.1, we obtain the sufficient conditions in (1.7) due to the work ^[16].

On the other hand, when $n = 3$, we get the following corollary by exploiting the Gagliardo-Nirenberg inequality.

Corollary 1.1. Let $\Omega\subset \mathbb{R}^3$ be a bounded domain with $C^2$ boundary $\partial\Omega$ and $(\rho, u, b, P)$ be a weak solution of systems (1.1)–(1.4) with initial data (1.5) and Dirichlet boundary conditions (1.6). Assume that one of the following conditions is satisfied:

$1)$ $0 < c_1\leqslant\rho \left(x, t\right)\leqslant c_2 < \infty, \, \, P\in L^2(0, T; L^2(\Omega)), \, \, \nabla \times b \in L^4(0, T; L^4(\Omega))$,

$2)$ $0 < c_1\leqslant\rho \left(x, t\right)\leqslant c_2 < \infty, \, P\in L^2(0, T; L^2(\Omega)), \, \nabla \times b \in L^{\frac{2p}{p-1}}(0, T; L^\frac{2q}{q-1}(\Omega))$,

Then, the energy equality (1.8) holds.

Remark 1.3. The conditions (1.16) show that we can get the regularity involving $\nabla u$ and $\nabla b$, rather than $u$ and $b$, to ensure that the energy equality (1.8) is valid.

2.   Preliminaries

Let $\eta_\varepsilon:\mathbb{R}^n\rightarrow \mathbb{R}$ be a standard mollifier, i.e., $\eta(x) = C_0 {e^{-\frac{1}{1-|x|^2}}}$ for $|x| < 1$ and $\eta(x) = 0$ for $|x|\geqslant1$, where $C_0$ is a constant such that ${\int_{\mathbb{R}^n}\eta(x) dx = 1}$. For any $\varepsilon > 0$, we define the re-scaled mollifier $\eta_\varepsilon(x) = \frac{1}{\varepsilon^n}\eta(\frac{x}{\varepsilon})$. For any function $f\in L^{1}_{loc}(\Omega)$, its mollified version is defined as

where $\Omega_\varepsilon = :\{x\in\Omega:d(x, \partial\Omega) > \varepsilon\}$.

Next, we recall the results involving the mollifier established in ^[26].

Lemma 2.1. (^[26]) Let $2\leqslant n \in \mathbb{N}$, $\Omega\subset\mathbb{R}^n$ be a bounded domain with $C^2$ boundary $\partial\Omega$, $1\leqslant p, q\leqslant \infty$ and $f:\Omega \times (0, T)\rightarrow \mathbb{R}$.

$1)$ Suppose that $f\in L^p(0, T; L^q(\Omega))$. Then, for any $0 < \varepsilon < \delta$, there holds

Moreover, if $p, q < \infty$, then

$2)$ Assume $f\in L^p(0, T; L^q(\Omega))$ with $p, q < \infty$ and $g:\Omega \times (0, T)\rightarrow \mathbb{R}$ with $0 < c_1\leqslant g \leqslant c_2 < \infty$. Then, for any $0 < \varepsilon < \delta$, there holds

Lemma 2.2. Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with $C^2$ boundary $\partial\Omega$, $1\leqslant p, q, p_1, q_1, p_2, q_2\leqslant \infty$ with $\frac{1}{p} = \frac{1}{p_1}+\frac{1}{p_2}$ and $\frac{1}{q} = \frac{1}{q_1}+\frac{1}{q_2}$. Assume $f \in L^{p_1}(0, T; W^{1, q_{1}}(\Omega))$ and $g\in L^{p_2}(0, T; L^{q_2}(\Omega))$. Then, for any $0 < \varepsilon < \delta$ small, there holds

Moreover, if $p_2, q_2 < \infty$, then

Remark 2.1. The above lemma with $p = q, \, \, p_1 = q_1$ and $p_2 = q_2$ was proved in ^[26].

Proof. The proof is similar to that of [19,Lemma 2.2]. For any $(x, s) \in \Omega_{2\delta} \times(0, T)$, $\Omega_{2\delta}\subset\Omega_\delta\subset\Omega$, we know that

and

where $z = \frac{y-x}{\varepsilon}$ and $B(x, \varepsilon)$ is an open ball centered at $x$ with radius $\varepsilon$. The triangle inequality yields

Next, we will handle the term $\left\|R^\varepsilon\right\|_{L^p(0, T; L^q(\Omega_{2\delta}))}$. By utilizing the Minkowski inequality and Hölder inequality, we get

Due to $\left|z\right|\leqslant 1$, we have

and

On the other hand, in view of Leibniz's formula, we conclude that

Taking the norm for both sides of the above formula, we deduce

where we have used the fact that $|z|\leqslant 1$, $\theta \in [0, 1]$, and the constant $C\geqslant1$ does not rely on $\theta, \, z, \, \varepsilon, \, f$ and $g$.

Plugging (2.8)–(2.10) into (2.7) yields

For the second term of the right hand side of (2.6), one has

where $z = \frac{y-x}{\varepsilon}$. Similar to (2.7), we arrive at

Combining (2.11) and (2.13), we get

which yields (2.3), while (2.4) follows from (2.3) by the density arguments.

Taking into account the boundary terms, we need to use the following coarea formula for $0 < \kappa_1 < \kappa_2$ established in ^[26]:

Lemma 2.3. Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with $C^2$ boundary. If $f\in L^p(0, T; W^{1, q}_{0}(\Omega))$, $p, q \in[1, \infty)$, then for $\varepsilon > 0$ small,

Proof. Let $\varepsilon > 0$ be small. For any $x \in \Omega\backslash\Omega_\varepsilon$, there exists a unique $x_{\partial\Omega}$ such that $\left|x_{\partial\Omega}-x\right| = d(x, \partial\Omega)$. We define the projection mapping as

Then, we know that

where $\bf{id}$ represents the identity matrix.

Thanks to $f = 0$ on $\partial\Omega \times(0, T)$, from the coarea formula and Leibniz's rule, we deduce

Set $\mathcal{T}_\rho(x) = :x+\rho(\mathcal{T}(x)-x)$. For $\varepsilon > 0$ small and $\rho\in(0, 1)$, we conclude from (2.17) that

Therefore,

which combined with (2.18) yields (2.16).

3.   Proof of Theorem 1.1 and Corollary 1.1

In this part, we first give the proof of Theorem 1.1. Unlike the periodic region, we are concerned with the boundary terms produced by using integration by parts. Then, making use of the Gagliardo-Nirenberg inequality, we prove Corollary 1.1 by the results of Theorem 1.1.

Proof of Theorem 1.1. First of all, we denote $m = :\nabla \times b$ and mollify the systems (1.1)–(1.4) in space to obtain

in $\Omega_\varepsilon\times (0, T)$. Next, selecting $0 < \varepsilon < \varepsilon_1/10 < \varepsilon_2/10 < r_0/100$, we multiply (3.1) by $(\rho u)^\varepsilon/ \rho^{\varepsilon}$ and integrate on $\left(\tau, t \right) \times \Omega_{\varepsilon_2}$ with $0 < \tau < t < T$. Choosing $\varepsilon_3 > 0$ small, by integrating with respect to $\varepsilon_2$ on $(\varepsilon_1, \varepsilon_1+\varepsilon_3)$, we have

The terms $F, \, G, \, H, \, K$ and $L$ will be estimated, separately, as follows.

Estimate of $F$. By direct calculations and (3.3), we arrive at

where $F_1$ is the desired term, while $F_2$ will be canceled with $G_4$ later.

Estimate of $G$. Taking advantage of integration by parts and free divergence condition (3.4) implies

where the superscript "bdr" in $G_2^{bdr}$ indicates that the term includes a boundary layer, and it is clear that $G_4+F_2 = 0$. Thus, we only consider the terms $G_1, G_2^{bdr}$ and $G_3$. First of all, for $G_1$ and $G_3$, by exploiting integration by parts, we have

and

For $G_{11}$, by Hölder's inequality, $\rm Lemma \, \, 2.1\, (ii)$ and ${\rm Lemma \, \, 2.2}$, we get

which implies

Similarly, we deduce

As $\rho \in L^\infty(0, T; L^\infty(\Omega))$, $u \in L^p(0, T; W^{1, q}(\Omega))$, we obtain

Next, we estimate the boundary terms $\, G_2^{bdr}, G_{12}^{bdr}$ and $G_{32}^{bdr}$. For $G_2^{bdr}$, we employ coarea Eq (2.15), Hölder's inequality and Lemma 2.3 to conclude

Since $\nabla u \in L^p(0, T; L^q(\Omega))$ and $u \in {L^{\frac{2p}{p-1}}(0, T; L^{\frac{2q}{q-1}}(\Omega))}$, by letting $\varepsilon_3\rightarrow0$, we get

Likewise, for $G_{12}^{bdr}$ and $G_{32}^{bdr}$, we have

and

Owing to the assumption (1.13), by letting $\varepsilon_3\rightarrow0$, we derive

Estimate of $H$. We see that

As $H_3$ is the desired term, we only need to handle terms $H_1$ and $H^{bdr}_2$. For $H_1$, taking advantage of Hölder's inequality and ${\rm { Lemmas \, \, 2.1\, (i) }}$ and ${\rm { 2.2}}$, we deduce

According to $u \in {L^2(0, T; H^1(\Omega))}$ and $\rho \in {L^\infty(0, T; L^\infty(\Omega))}$, we get

The boundary term $H_{2}^{bdr}$ is estimated by using coarea Eq (2.15) and Lemma 2.3 as

Since $u \in {L^2(0, T; H^1(\Omega))}$, by $\varepsilon_3\rightarrow0$, we obtain

Estimate of $K$. In light of the free divergence condition (3.4), we rewrite $K$ as

For $K_1$, by making use of Hölder's inequality, we have

Due to $\rm Lemmas \, \, 2.1\, (i)$ and ${\rm\, \, 2.2}$, one has

Exploiting Hölder's inequality, coarea Eq (2.15) and ${\rm Lemma \, \, 2.3}$ again, we deduce

Owing to $u \in {L^2(0, T; H^1(\Omega))}$ and $P \in {L^2(0, T; L^2(\Omega))}$, by $\varepsilon_3\rightarrow0$, we derive

Estimate of $L$. First, it is clear that

where $\mathrm{div}\, (b^\varepsilon \times (u\times b)^\varepsilon) = (u \times b)^\varepsilon \cdot \nabla \times b^\varepsilon-b^\varepsilon \cdot \nabla \times (u \times b)^\varepsilon$.

Next, multiplying (3.2) by $b^\varepsilon$ yields

Plugging (3.7) into (3.6), we infer that

where $L_{41}, L_{43}$ are our expected terms, and we just need to deal with the rest of the above items.

For $L_1$, applying Hölder's inequality and $\rm Lemma \, \, 2.2$, we have

Under the hypothesis (1.13), by $\varepsilon\rightarrow0, \tau\rightarrow0$, we obtain

The term $L_2$ is estimated by utilizing Hölder's inequality and $\rm Lemma \, \, 2.2$ as

As $\rho \in L^\infty(0, T; L^\infty(\Omega))$, $u \in L^p(0, T; W^{1, q}(\Omega))$, we get

Similarly, we deal with $L_3$ and $L_{44}$ as

and

Then, we deduce that

Next, we treat the terms about the boundary $L^{bdr}_{42}$, $L^{bdr}_{45}$ and $L^{bdr}_{5}$, separately. For $L^{bdr}_{42}$, by using coarea Eq (2.15) and ${\rm Lemma}\, \, 2.3$, we have

which, together with $b\in L^2(0, T; H^1(\Omega))$, implies

Similarly, for $L^{bdr}_{45}$ and $L^{bdr}_{5}$, we get

and

Under the assumption (1.13), by letting $\varepsilon_3\rightarrow0$, we obtain

Then, we collect all the above estimates $F_1$, $H_3$, $L_{41}$, $L_{43}$ and put them into the right side of equation (3.5) to conclude

Using the weak continuity of $\rho, \, \rho u$ in (1.10) and (1.11) and limits

and

we finish the proof of Theorem 1.1.

In what follows, we will prove Corollary 1.1 based on Theorem 1.1, where $n = 3$.

Proof of Corollary 1.1. First, we prove the first case of (1.15). The basic regularity of weak solutions gives $u, b \in L^{\infty}(0, T; L^2(\Omega))\cap L^2(0, T; H^1(\Omega))$. Choosing $p = q = 2$ in (1.13), we know from Theorem 1.1 that the conditions $u, b \in L^4(0, T; L^4(\Omega))$ and $\nabla\times b\in L^4(0, T; L^4(\Omega))$ could guarantee the energy equality (1.8). Thus, to prove the first case of (1.15), it suffices to prove that $u, b\in L^s(0, T; L^t(\Omega))$ with $\frac{2}{s}+\frac{2}{t} = 1, t\geqslant4$ could yield $u, b \in L^4(0, T; L^4(\Omega))$. To this end, with the help of the Gagliardo-Nirenberg inequality, we have

which finishes the proof of the first case of (1.15).

Next, we consider the second case of (1.15) with $\frac{1}{s}+\frac{3}{t} = 1, \, 3 < t < 4$. Using the Gagliardo-Nirenberg inequality, we deduce

Thus, by Theorem 1.1, we get that (1.15) could ensure the energy equality (1.8).

In what follows, we give the proof of the first case of (1.16). According to Theorem 1.1, for $1 < p, q \leqslant3$, one knows that the conditions $\nabla u, \nabla b \in L^p(0, T; L^q(\Omega))$ and $u, b\in L^{\frac{2p}{p-1}}(0, T; L^\frac{2q}{q-1}(\Omega))$ can ensure that energy equality (1.8) is valid. Therefore, to prove the first case of (1.16), we need to show that the conditions $\nabla u, \nabla b \in L^p(0, T; L^q(\Omega))$ can yield $u, b\in L^{\frac{2p}{p-1}}(0, T; L^\frac{2q}{q-1}(\Omega))$. To this end, for ${\frac{9}{5}} \leqslant q \leqslant3$, by virtue of the Gagliardo-Nirenberg inequality, we know that

Furthermore, in view of $\frac{1}{p} +\frac{6}{5q} = 1$, we deduce that

Then, we complete the proof of the first case of (1.16).

Next, we treat the remaining case of (1.16). For ${\frac{3}{2}} < q < {\frac{9}{5}}$, it follows from the Gagliardo-Nirenberg inequality that

Thanks to $\frac{1}{p} +\frac{3}{q} = 2$, we further infer that

Then, from Theorem 1.1, we know that (1.16) could guarantee the energy equality (1.8), and the proof of Corollary 1.1 is finished.

4.   Conclusions

This paper is dedicated to the energy equality of nonhomogeneous incompressible Hall-MHD equations in a bounded domain $\Omega \subset \mathbb{R}^n$ $(n\geqslant2)$. Through the special structure of the nonlinear terms, and using the coarea formula, we get some types of regularity conditions to guarantee that the energy equality is valid. It is worth noting that among them are the regularity conditions concerning $\nabla u$ and $\nabla b$, rather than $u$ and $b$.

Acknowledgments

This work was sponsored by the NNSF of China (No. 11871305). The authors thank the referees for careful reading and valuable suggestions.

Conflict of interest

The authors declare there are no conflicts of interest.