Global regularity for the 3D compressible magnetohydrodynamics with general pressure

In this paper, we are concerned with the compressible magnetohydrodynamics (MHD) in three space dimensions. The fluid motion is described in the following system of partial differential equations (see Cabannes [Cab70] for a more comprehensive discussion on the system):

Here $\rho$, $u=(u^{1},u^{2},u^{3})$ and $B=(B^{1},B^{2},B^{3})$ are functions of $x\in\mathbb{R}^{3}$ and $t\geq 0$ representing density, velocity and magnetic field; $P=P(\rho)$ is the pressure; $\mu$, $\lambda$, $\nu$ are viscous constants. The system (1.1)-(1.4) is solved subjected to some given initial data:

The well-posedness of the MHD system (1.1)-(1.4) have been studied by many mathematicians in decades (see for example [CT10, HW10, HW08, Kaw84, LSW11, LSX16, LY11, Sar09, SH12, Sue20b] and the references therein), and we now give a brief review on the related results. When the initial data is taken to be close to a constant state in $H^{3}$, Kawashima [Kaw84] proved the existence of global-in-time $H^{3}$ solutions to the MHD system. Later, Hoff and Suen [SH12] generalised Kawashima’s results to obtain global smooth solutions when the initial data is taken to be $H^{3}$ but only close to a constant state in $L^{2}$. The existence of global weak solutions with large initial data was proved by Hu and Wang [HW10, HW08] and Sart [Sar09], which are extensions of Lions-type weak solutions [Lio98] for the Navier-Stokes system. With initial $L^{2}$ data close to a constant state, Hoff and Suen [SH12, Sue12, Sue20b] generalised Hoff-type intermediate weak solutions [Hof95, Hof05, Hof06, LS16, Sue13b, Sue14, CS16, Sue20a] to obtain global solutions to the MHD system.

On the other hand, the global existence of smooth solution to (1.1)-(1.4) with arbitrary smooth data is still unknown, hence it is reasonable to consider the possibilities of blowing up of smooth solution. For the corresponding Navier-Stokes system, Xin [Xin98] proved that smooth solution will blow up in finite time in the whole space when the initial density has compact support, while Rozanova [Roz08] showed similar results for rapidly decreasing initial density. Fan-Jiang-Ou [FJO10], Sun-Wang-Zhang [SWZ11] and Suen [Sue20c] established some blow-up criteria for the classical solutions to 3D compressible flows, which were further extended by Lu-Du-Yao [LDY12] for MHD system. For the isothermal case when $P(\rho)=K\rho$ for some $K>0$, it was proved in [Sue13a, Sue15] that without vacuum in the initial density, when the density and magnetic field are essential bounded, the smooth solutions to (1.1)-(1.4) can be extended globally in time.

The main goal of the present paper is to generalise and strengthen the corresponding results of [Sue13a, Sue15]. The main novelties of this current work can be summarised as follows:

• •

  We introduce some new type of estimates on functionals which are used for decoupling the velocity and magnetic field;

• •

  We obtain a blow-up criterion for (1.1)-(1.4) for general pressure term $P(\rho)$ which is not restricted to the isothermal case;

• •

  We assure that the blow-up criterion dependsds only on density, which is an improvement for the results obtained from [Sue13a, Sue15] in which the magnetic field was present in the criterion;

• •

  We do not impose any extra compatibility condition on the initial data, which is required in the work [SWZ11, HLX11, Sue20c].

We give a brief description on the analysis applied in this work. To extract the “hidden regularity” from the velocity $u$ and magnetic field $B$, we introduce an important canonical variable associated with the system (1.1)-(1.4), which is known as the effective viscous flux. To see how it works, by the Helmholtz decomposition of the mechanical forces, we can rewrite the momentum equation (1.2) as follows (summation over $k$ is understood):

where $\dot{u}^{j}=u^{j}_{t}+u\cdot u^{j}$ is the material derivative on $u^{j}$ and the effective viscous flux $F$ is defined by

Differentiating (1.6), we obtain the following Poisson equation

where $g^{j}=\rho\dot{u}^{j}+(\frac{1}{2}|B|^{2})_{x_{j}}-\text{\rm div}(B^{j}B)$. The Poisson equation (1.8) can be viewed as the analog for compressible MHD of the well-known elliptic equation for pressure in incompressible flow. By exploring the Rankine-Hugoniot condition (see [SH12]), one can deduce that the effective viscous flux $F$ is relatively more regular than $\text{\rm div}(u)$ or $P(\rho)$, and it turns out to be crucial for the overall analysis in the following ways:

1. (i)

   The equation (1.6) allows us to decompose the acceleration density $\rho\dot{u}$ as the sum of the gradient of the scalar $F$ and the divergence-free vector field $\omega^{\cdot,k}_{x_{k}}$ (we ignore those lower-order terms involving $B$). The skew-symmetry of $\omega$ insures that these two vector fields are orthogonal in $L^{2}(\mathbb{R}^{3})$, so that $L^{2}$-bounds for the terms on the left side of (1.6) immediately give $L^{2}$ bounds for the gradients of both $F$ and $\omega$. These in turn will be used for controlling $\nabla u$ in $L^{4}$ when $u(\cdot,t)\notin H^{2}$, which are crucial for estimating different functionals in $u$ and $B$; see section 3 and the proof of Theorem 3.2.

2. (ii)

   One of the key step in obtaining higher order estimates on the solutions is to bound the time integral of $\|\nabla u(\cdot,t)\|_{L^{\infty}}$. The key observation is to decompose $u$ as $u=u_{F}+u_{P}$, where $u_{F}$, $u_{P}$ satisfy

   $\displaystyle\left\{\begin{array}[]{lr}(\mu+\lambda)\Delta u_{F}^{j}=F_{x_{j}}+(\mu+\lambda)\omega^{j,k}_{x_{k}}\\ (\mu+\lambda)\Delta u_{P}^{j}=(P-P(\tilde{\rho}))_{x_{j}}.\\ \end{array}\right.$

   Using the a priori bounds on the effective viscous flux $F$, the time integral of $\|\nabla u_{F}(\cdot,t)\|_{L^{\infty}}$ can be estimated in terms of $F$, which can be further estimated in terms of some a priori bounds on $\dot{u}$ and $B$. On the other hand, to control $\int_{0}^{t}||\nabla u_{P}(\cdot,s)||_{\infty}ds$, by applying the analysis on Newtonian potentials given in [BC94], one can show that if $\Gamma$ is the fundamental solution for the Laplace operator on $\mathbb{R}^{3}$, then $u^{j}_{P}(\cdot,t)=(\mu+\lambda)^{-1}\Gamma_{x_{j}}*(P(\rho(\cdot,t))-\tilde{P})$ is log-Lipschitz provided that $P(\rho(\cdot,t))\in L^{\infty}$ holds. This is sufficient to guarantee that the integral curve $x(\cdot,t)$ of $u=u_{F}+u_{P}$ is Hölder continuous under the assumption that $u_{F}$ has enough regularity as claimed. If we assume that the initial density is Hölder continuous, then using the mass equation (1.1), it implies that the density is also Hölder continuous for positive time. Hence with such improved regularity on the density, it allows us to obtain the desired bound on $u_{P}$; see section 4 and the proof of Theorem 4.1. Such method was also exploited in [Sue20b] for proving global-in-time existence and uniqueness of weak solutions to (1.1)-(1.4).

We now give a precise formulation of our results. First concerning the assumptions on the parameters, we have:

Given $\tilde{\rho}>0$, for the initial data, we assume that

The following is the main result of this paper:

The rest of the paper is organised as follows. In section 2, we recall some known facts and inequalities which will be useful for later analysis. In section 3, we begin the proof of Theorem 1.1 with a number of a priori bounds for smooth solutions, which are summarised in Theorem 3.2. Finally in section 4, we complete the proof of Theorem 1.1 via a contradiction argument by deriving higher order $H^{3}$-bounds for smooth solutions.

In this section, we recall some known facts and useful inequalities for our analysis. We first state a local existence theorem for (1.1)-(1.4) proved by Kawashima [Kaw84, pg. 34–35 and pg. 52–53]:

We make use of the following standard facts (see Ziemer [Zie89, Theorem 2.1.4, Remark 2.4.3, and Theorem 2.4.4] for example). First, given $r\in[2,6]$ there is a constant $C(r)$ such that for $w\in H^{1}(\mathbb{R}^{3})$,

For any $r\in(1,\infty)$ there is a constant $C(r)$ such that for $w\in W^{1,r}(\mathbb{R}^{3})$,

In this section we derive a priori estimates for the local solution $(\rho-\tilde{\rho},u,B)$ on $[0,t]$ with $t\in[0,T^{*})$ as described by Theorem 1.1. Here $T^{*}$ is the maximal time of existence which is defined in the following sense:

We will prove Theorem 1.1 using a contradiction argument. Therefore, for the sake of contradiction, we assume that

To facilitate our exposition, we first define some auxiliary functionals for $t\in[0,T^{*})$:

The following is the main theorem of this section:

We prove Theorem 3.2 in a sequence of lemmas. Throughout this section, for $t\in[0,T^{*})$, $C$ always denotes a generic constant which depends only on $\mu$, $\lambda$, $a$, $\nu$, $\tilde{\rho}$, $M_{0}$, $t$ and the initial data. For simplicity, we drop the symbols $dx$, $ds$ or $dxds$ from the integrals.

We first give the following estimates on the effective vicious flux $F$ which is defined in (1.7).

Using the estimates (3.3)-(3.4) on $F$, we have the following estimates on $\nabla u$ and $\nabla\omega$:

We are now ready for giving the estimates for proving Theorem 3.2. We begin with the following $L^{2}$ estimates on $(\rho,u,B)$ for all $t\in[0,T^{*})$:

Next we derive the following $L^{6}$ bounds for $u$ and $B$. Such bounds are crucial for obtaining higher order estimates on $u$ and $B$.

With the help of (3.7) and (3.8), we can start estimating the functionals $\Phi_{1}$ and $\Phi_{2}$ appeared in (3.2). We first consider $\Phi_{1}$ and $\Phi_{4}$:

Next, we derive the following estimates for $\Phi_{2}$ in terms of $\Phi_{3}$ and $\Phi_{5}$: