Global well-posedness to the Cauchy problem of 2D inhomogeneous incompressible magnetic Bénard equations with large initial data and vacuum

1.   Introduction

In this paper, we consider the Cauchy problem of the following 2D density-dependent magnetic Bénard equations

which is equipped the following initial conditions and far-field behavior:

where $\rho$, $u$, $\theta$, $b$ and $p$ denote the density, velocity, temperature, magnetic field, and pressure of the fluid, respectively. $\mu > 0$ is the viscosity coefficient, $\kappa > 0$ is the heat conductivity coefficient, and $\nu > 0$ is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field. $e_2 = (0, 1)^T$, where $T$ is the transpose.

The magnetic Bénard equations (1.1) illuminates the heat convection phenomenon under the dynamics of the velocity and magnetic fields in electrically conducting fluids such as plasmas (see ^[10,11] for details). If we ignore the Rayleigh-Bénard convection term $u\cdot e_2$, system (1.1) recovers the inhomogeneous incompressible MHD equations (i.e., $\theta\equiv 0$). Let us review some previous works about the standard incompressible MHD equations. In the absence of vacuum, Abidi-Paicu ^[1] established the local and global (with small initial data) existence of strong solutions in the framework of Besov spaces. Chen et al. ^[2] proved a global solution for the global well-posedness to the 3D Cauchy problem for the bounded density. In the presence of vacuum, imposing the following compatibility condition,

for some $(p_0, g)\in H^1\times L^2$. Chen et al. ^[3] obtained the unique local strong solutions to the 3D Cauchy problem with general initial data. Song ^[13] studied the local well-posedness of strong solutions without additional compatibility condition (1.3), which extended the main result of ^[3]. Recently, Gao-Li ^[4] shown the global strong solutions with vacuum in bounded domain, provided that initial data is suitable small. Later on, Zhang-Yu ^[15] extended this result to the whole space. For the 2D case, Huang-Wang ^[5] investigated the global existence of strong solution with general large data in bounded domain provided that the compatibility condition (1.3) holds. Recently, Lv et al. ^[8] showed the global existence of strong solutions to the 2D Cauchy problem with the large data and vacuum.

Let us go back to the system (1.1). Very recently, by weighted energy method, Zhong ^[16] showed the local existence of strong solutions to the Cauchy problem of (1.1) in $\Bbb R^2$. However, the global existence of strong solution to the 2D Cauchy problem of (1.1) with vacuum and general initial data is not addressed. In fact, this is the main aim of this paper.

Before stating the main results, we first explain the notations and conventions used throughout this paper. For $R > 0$. Set

Moreover, for $1\le r\le \infty$ and $k\ge 1$, the standard Sobolev spaces are defined as follows:

Without loss of generality, we assume that initial density $\rho_0$ satisfies

which implies that there exists a positive constant $N_0$ such that

Throughout this paper, always denote

with $\sigma_0 > 0$ fixed. The main result of this paper is stated as the following theorem:

Theorem 1.1. In addition to (1.4) and (1.5), assume that the initial data $(\rho_0, u_0, \theta_0, b_0)$ satisfies for any given numbers $a > 1$ and $q > 2$,

Then the problems (1.1) and (1.2) has a unique global strong solution $(\rho\ge 0, u, \theta, b, p)$ satisfying that for any $0 < T < \infty$,

and

for some positive constant $N_1$ depending only on $\sqrt{\rho_0}u_0$, $N_0$, and $T$.

Remark 1.1. We remark that Theorem 1.1 is proved without any smallness on the initial data. Moreover, the initial density can contain vacuum states and even has compact support. We also point out that Theorem 1.1 extends the result of Zhong ^[16] to the global one. In particular, when $b = 0$, the incompressible magnetic Bénard equations (1.1) reduces to the incompressible Bénard equations, Theorem 1.1 also extends Zhong ^[17] to the global one.

We now make some comments on the key ingredients of the analysis in this paper. For the initial data satisfying (1.7), Zhong ^[16] recently established the local existence and uniqueness of strong solutions to the Cauchy problems (1.1) and (1.2) (see Lemma 2.1). Thus, to extend the local strong solution to be a global one, we need to obtain global a priori estimates on strong solutions to (1.1) and (1.2) in suitable higher norms. However, due to critically of Sobolev's inequality in $\Bbb R^2$, it seems difficult to bound $\|u\|_{L^p}$ just in term of $\|\sqrt{\rho}u\|_{L^2}$ and $\|\nabla u\|_{L^2}$ for any $p\ge 2$. Moreover, compared with ^[9], for the systems (1.1) and (1.2) here, the strong coupling terms and Rayleigh-Bénard convection terms, such as $u\cdot\nabla b$, $\rho u\cdot e_2$, and $\rho\theta e_2$, will bring out some new difficulties.

To overcome these difficulties mentioned above, some new ideas are needed. First, using the structure of the 2D magnetic equations, we multiply $(1.1)_4$ by $4|b|^2b$ and thus obtain the useful a priori estimate on $L^2(\Bbb R^2\times(0, T))$-norm of $|b||\nabla b|$ (see (3.5)), which is crucial in deriving the $L^\infty(0, T; L^2(\Bbb R^2))$-norm of $\nabla u$, $\nabla\theta$ and $\nabla b$. Next, in order to derive the estimates on $L^\infty(0, T; L^2(\Bbb R^2))$-norm of $\nabla u$, $\nabla\theta$, motivated by ^[9], multiplying $(1.1)_2$ and $(1.1)_3$ by $\dot{u}: = u_t+u\cdot\nabla u$ and $\dot{\theta}: = \theta_t+u\cdot\nabla\theta$ instead of usual $u_t$ and $\theta_t$ respectively, we deduce that the key point to obtain the estimate on the $L^\infty(0, T; L^2(\Bbb R^2))$-norm of the gradient of the velocity $u$ and temperature $\theta$ is to bound the terms

We find $I_2$ in fact can be bounded by $\|\nabla p\|_{L^2}\|\nabla u\|_{L^2}^2$ (see (3.8)), since $\partial_ju^i\partial_iu^j\in \mathcal{H}^1$ due to the fact that ${\rm div}u = 0$ and $\nabla^\bot\cdot\nabla u = 0$ (see Lemma 2.4). Moreover, the usual $L^2(\Bbb R^2\times(0, T))$-norm of $b_t$ cannot be directly estimated due to the strong coupled term $u\cdot\nabla b$. Thus, we multiplying $(1.1)_4$ by $\Delta b$ instead of usual $b_t$, the coupled term can be controlled after integration by parts. Thirdly, to tackle the difficulty caused by the lack of the Sobolev inequality, motivated by ^[8,16,17], by introducing a weighted function to the density, as well as a Hardy-type inequality in ^[7] by Lions, the $\|\rho^\eta v\|_{\sigma}$ ($\eta > 0$, $\sigma > \max\{2, \frac{2}{\eta}\}$) is controlled in term of $\|\sqrt{\rho}v\|_{L^2}$ and $\|\nabla v\|_{L^2}$ (see (3.18)), which plays an important role in bounding the Rayleigh-Bénard convection terms $\rho u\cdot e_2$ and $\rho\theta e_2$, and deriving the estimates on the $L^\infty(0, T; L^2(\Bbb R^2))$ of $\sqrt{t}\sqrt{\rho}u_t$ and $\sqrt{t}\sqrt{\rho}\theta_t$. Finally, with these a priori estimates on the velocity, temperature and magnetic field at hand, some useful spatial weighted estimates on both $b$, $\nabla u$ and $\nabla\theta$ are derived, which yields the bounded of $L^\infty(0, T; L^2(\Bbb R^2))$-norm of $\sqrt{t}\nabla^2b$ (see Lemma 3.7).

The rest of the paper is organized as follows. In Section 2, we collect some elementary facts and inequalities which will be needed in later analysis. Section 3 is devoted to the a priori estimates. Finally, we give the proof of Theorem 1.1 in Section 4.

2.   Preliminaries

In this section, we will recall some known facts and elementary inequalities which will be used frequently later.

We start with the local existence of strong solutions whose proof can be found in ^[16].

Lemma 2.1. Assume that $(\rho_0, u_0, \theta_0, b_0)$ satisfies (1.7). Then there exists a small time $T > 0$ and a unique strong solution $(\rho, u, \theta, b, p)$ to the problems (1.1) and (1.2) in $\Bbb R^2\times T$ satisfying (1.8) and (1.9).

Next, the following Gagliardo-Nirenberg inequalities will be stated, which see ^[12] for the detailed proof.

Lemma 2.2. For all $v\in C_0^\infty(\Bbb R^n)$, integer $j$, $0\le j < m$, $1\le r, q\le \infty$, and $\frac{j}{m}\le \vartheta\le 1$, there exists a positive constant $C$ depending only on $j$, $m$, $n$, $p'$, $q$, and $r$ such that

where

and $m-j-\frac{n}{r}$ is not a nonnegative integer. If $1 < r < \infty$ and $m-j-\frac{n}{r}$ is a nonnegative integer, (2.1) holds with $\vartheta\in [\frac{j}{m}, 1)$.

As a key technical ingredient for our approach, we need the following weighted bounds for functions in the space $\tilde{D}^{1, 2}(\Bbb R^2)\triangleq\{v\in H_{{\rm loc}}^1(\Bbb R^2):\nabla v\in L^2(\Bbb R^2)\}$, whose proof can be found in [6, Lemma 2.4].

Lemma 2.3. Let $\bar{x}$ be as in (1.6). Assume that $\rho\in L^1\cap L^\infty$ be a non-negative function satisfying

with $M_1, M_2 > 0$ and $B_{N_1}\subset\Bbb R^2$ ($N_1\ge1$). Then there exists $C = C(M_1, M_2, N_1) > 0$ such that

Moreover, for any $\eta > 0$ and $\sigma > \max\{2, \frac{2}{\eta}\}$, there exits $C = C(\sigma, < /italic > < italic > \eta, M_1, M_2, N_1) > 0$ such that

Finally, let $\mathcal{H}^1$ and BMO stand for the usual Hardy and BMO spaces (see [14, Section 4]). Then the following well-known facts play a key role in the proof of Lemma 3.2, whose proof can be found in ^[9].

Lemma 2.4. (i) There is a positive constant $C$ such that

for all $E\in L^2$ and $B\in L^2$ with

(ii) There is a positive constant $C$ such that for all $v\in\widetilde{D}^{1, 2}(\Bbb R^2)$, it holds

3.   A priori estimates

In this section, we will establish some necessary a priori bounds for strong solutions $(\rho, u, \theta, b, p)$ to the Cauchy problems (1.1) and (1.2) to extend the local strong solution. Thus, let $T > 0$ be a fixed time and $(\rho, u, \theta, b, p)$ be the strong solution to (1.1) and (1.2) on $\Bbb R^2\times(0, T]$ with initial data $(\rho_0, u_0, \theta_0, b_0)$ satisfying (1.4)–(1.6). In what follows, we will use the convention that $C$ denotes a generic positive constant depending on initial data and $T$.

We begin with the following standard energy estimate and the estimate on the $L^\infty(0, T; L^1\cap L^\infty)$-norm of the density.

Lemma 3.1. Under the assumption of Theorem 1.1, it holds that

Proof. First, it follows from the transport equation $(1.1)_1$ and making use of $(1.1)_4$ (see Lions [7, Theorem 2.1]) that

Next, multiplying the Eqs $(1.1)_{2, 3, 4}$ by $(u, \theta, b)$ and integrating by parts over $\Bbb R^2$, one obtains by using ${\rm div}u = {\rm div}b = 0$,

which together with Gronwall's inequality yields (3.1) and completes the proof of lemma.

Lemma 3.2. Under the assumption of Theorem 1.1, it holds that

Proof. (1). Multiplying $(1.1)_4$ by $4|b|^2b$ and integrating the resulting equation over $\Bbb R^2$, one has

which together with Gronwall's inequality and (3.1) yields that

(2). Multiplying $(1.1)_2$ by $\dot{u}: = u_t+u\cdot\nabla u$ and integrating by parts over $\Bbb R^2$, we find that

It follows from integration by parts and Hölder's inequality that

Notice that ${\rm div}(\partial_ju) = \partial_j{\rm div}u = 0$ and $\nabla^\bot\cdot(\nabla u^j) = 0$, we infer from Lemma 2.4 that

In view of $(1.1)_4$, Hölder's and Gagliardo-Nirenberg inequalities, we deduce after integrating by parts that

For the estimates of $I_4$, we derive

Combining the above estimates yields

where $N(t): = 2\int b\cdot\nabla u\cdot b dx$. We get by direct computations

(3). Multiplying $(1.1)_3$ by $\dot{\theta}: = \theta_t+u\cdot\nabla\theta$ and integrating by parts over $\Bbb R^2$, one has

(4). Multiplying $(1.1)_4$ by $\Delta b$ and integrating by parts over $\Bbb R^2$, we infer from Hölder's inequality, Gagliardo-Nirenberg inequality, and (3.5) that

(5). It follows from the standard $L^{p'}$-estimates of Stokes equations that for any $p'\in[2, \infty)$,

which combined with (3.1) gives

On the other hand, in view of the standard estimate of elliptic system, one obtains

Adding $(3.10)+(c_1+\frac{1}{2})\times(3.12)+(3.11)$ altogether for enough large constant $c_1 > 0$, it follows from (3.14) and (3.15) that

which together with (3.1), (3.5), Gronwall's inequality, and choosing $\varepsilon$, $\kappa$, $\delta$ small enough, one obtains

(6). It follows from [9, Lemma 3.4] and (1.5) that

which along with (3.1) and (2.3) entails that for any $\eta > 0$ and $\sigma > \max\{2, \frac{2}{\eta}\}$, there is a constant $\bar{C}(\sigma, \eta) > 0$ such that

In particular, this together with (2.3) and (3.1) yields

Thus, we infer from (3.14)–(3.16), (3.5) and (3.1), Hölder's and and Garliardo-Nirenberg inequalities that

(7). We infer from (3.14), (3.15), (3.19), (3.1), (3.4), and Gagliardo-Nirenberg inequality that

which yields to

This together with (3.16) and (3.20) leads to

Thus, it follows from (3.16), (3.20), and (3.22) that (3.4) holds. The proof of Lemma 3.2 is completed.

Lemma 3.3. Under the assumption of Theorem 1.1, it holds that

Proof. Multiplying $(1.1)_4$ by $\bar{x}^ab$ and integrating by parts over $\Bbb R^2$, one has

It follows from (3.19), (3.4), Hölder's and Gagliardo-Nirenberg inequalities that

where we use the fact that $\bar{x}^{-1}\nabla\bar{x}$ and $\bar{x}^{-1}\nabla^2\bar{x}$ are uniformly bounded on $\Bbb R^2$ since $|\bar{x}^{-1}\nabla\bar{x}|\le C/(1+|x|)$ and $|\bar{x}^{-1}\nabla^2\bar{x}|\le C/(1+|x|^2)$, and $(e+y)^{-\alpha}\log(e+y)\le \alpha^{-1}$ for $\alpha > 0$ and $y\ge 0$.

Substituting $N_1-N_3$ into (3.24), we obtain that

which together with Gronwall's inequality yields (3.23). The proof of Lemma 3.3 is completed.

Lemma 3.4. Under the assumption of Theorem 1.1, it holds that

Proof. Firstly, differentiating $(1.1)_2$, $(1.1)_3$ with respect to $t$ respectively, we have

Multiplying (3.27), (3.28) by $u_t$, $\theta_t$ respectively, and integrating it by parts over $\Bbb R^2$, it implies

By using Hölder's, Gagliardo-Nirenberg inequalities, (3.1), (3.4), (3.18), and (3.19), one gets

Putting all above estimates into (3.29), we thus obtain

Next, differentiating $(1.1)_4$ with respect to $t$ gives

Multiplying (3.31) by $b_t$, and integrating it by parts over $\Bbb R^2$, one has

which leads to

for enough large constant $c_2 > 0$.

Moreover, multiplying (3.30) by $c_2+1$ and adding the resulting inequality with (3.32), and choosing $\delta$ suitable small, one obtains

Multiplying (3.33) by $t$, we obtain (3.26) after using Gronwall's inequality, (3.5), (3.4) and (3.16). The proof of Lemma 3.4 is completed.

Lemma 3.5. Under the assumption of Theorem 1.1, it holds that

Proof. First, it follows from the mass equation $(1.1)_1$ that $|\nabla\rho|^r$ satisfies for any $r\ge 2$,

which together with integrating by parts over $\Bbb R^2$ implies

Next, one gets from Gagliardo-Nirenberg inequality that

On the one hand, it is easy to check that

which together with (3.4) and (3.26) implies that

On the other hand, it follows from (3.4) and (3.26) that

Thanks to (3.37), (3.39) and (3.40), we immediately obtain

Thus, applying Gronwall's inequality to (3.36) gives

Finally, similar to (3.39) and (3.40), we obtain from $(1.1)_3$ by $L^q$-estimates to elliptic equations that

which together with (3.39), (3.40), and (3.42) yields (3.34) and completes the proof of lemma.

Next, the following high order weighted estimates on the density has been proven in [9, Lemma 3.6]. We omit the detailed proof here for simplicity.

Lemma 3.6. Under the assumption of Theorem 1.1, it holds that

Lemma 3.7. Under the assumption of Theorem 1.1, it holds that

Proof. First, multiplying $(1.1)_4$ by $\bar{x}^a\Delta b$ and integrating by parts over $\Bbb R^2$ lead to

By using Hölder's inequality, Gagliardo-Nirenberg inequality, (3.23), (3.4) and (3.19), one obtains

Substituting the above estimates into (3.46), we have

due to the following fact that

Thus, multiplying (3.47) by $t$, we deduce from Gronwall's inequality, (3.4) and (3.34) that

Next, it follows from $(1.1)_4$, (3.19), (3.4), and Gagliardo-Nirenberg inequality that

which together with (3.26) and (3.49) yields that (3.45) and completes the proof of lemma.

4.   Proof of Theorem 1.1

With a priori estimates in Section 3 at hand, we are ready to prove Theorem 1.1.

Proof of Theorem 1.1. By Lemma 2.1, there exists a $T_* > 0$ such that the problems 1.1 and 1.2 has a unique strong solution $(\rho, u, \theta, b, p)$ on $\Bbb R^2\times (0, T_*]$. Now, we will extend the local solution to all time.

Set

First, for any $0 < \tau < T_* < T\le T^*$ with $T$ finite, one deduces from (3.1), (3.4), (3.26), (3.34) and (3.45) that for any $q > 2$,

where one has used the standard embedding

Moreover, it follows from (3.34) and (3.44) and [7, Lemma 2.3] that

Finally, if $T^* < \infty$, it follows from (4.2), (4.3), (3.1), (3.4), (3.34) and (3.45) that

satisfies the initial condition (1.7) at $t = T^*$. Thus, taking $(\rho, u, \theta, b)(x, T^*)$ as the initial data, Lemma 2.1 implies that one can extend the strong solutions beyond $T^*$. This contradicts the assumption of $T^*$ in (4.1). The proof of Theorem 1.1 is completed.

5.   Conclusions

In this paper, we are concerned with the Cauchy problem of inhomogeneous incompressible magnetic Bénard equations with vacuum as far-field density in $\Bbb R^2$. Using the weighted function to the density, as well as the Hardy-type inequality, we have successfully established the time-uniform a priori estimates of solutions. Thus, we can extend the local strong solutions to the global one.

Acknowledgments

The authors would like to thank the Editorial Board and four reviewers for their constructive comments and suggestions, which greatly improved the final version of the paper.

The author was supported by Scientific Research Foundation of Jilin Province Education Department (JJKH20210883KJ), and Doctoral research start-up fund project of Changchun Normal University.

Conflict of interest

The author declares that there are no conflicts of interest in this paper.