Global Strong Solution of a 2D coupled Parabolic-Hyperbolic Magnetohydrodynamic System

TL;DR

This paper proves the existence of global strong solutions for a 2D coupled parabolic-hyperbolic magnetohydrodynamic system, advancing mathematical understanding of MHD models in fluid dynamics.

Contribution

It establishes the global strong solution existence for a 2D coupled parabolic-hyperbolic MHD system using advanced PDE estimates, which was previously unresolved.

Findings

Existence of global strong solutions in 2D MHD system

Application of Agmon-Douglis-Nirenberg estimates

Use of Solonnikov's $L^p$-$L^q$ estimates for evolution Stokes equations

Abstract

The main objective of this paper is to study the global strong solution of the parabolic-hyperbolic incompressible magnetohydrodynamic (MHD) model in two dimensional space. Based on Agmon, Douglis and Nirenberg's estimates for the stationary Stokes equation and the Solonnikov's theorem of $L^p$-$L^q$-estimates for the evolution Stokes equation, it is shown that the mixed-type MHD equations exist a global strong solution.

Equations

$$\{\begin{array}{rlr}& \frac{\partial u}{\partial t}+(u\cdot \nabla )u=\nu \Delta u-\frac{1}{{\rho }_0}\nabla p+\frac{{\rho }_e}{{\rho }_0}u\times \text{rot}A+f(x),& \text{in}\Omega \times [0,T),\\ & \frac{{\partial }^{2}A}{\partial t^2}=\frac{1}{{ϵ}_0{\mu }_0}\Delta A+\frac{{\rho }_e}{{ϵ}_0}u-\nabla \Phi ,& \text{in}\Omega \times [0,T),\\ & \nabla \cdot u=0,& \text{in}\Omega \times [0,T),\\ & \nabla \cdot A=0,& \text{in}\Omega \times [0,T).\end{array}$$

$$u(0,x)=\varphi (x),A(0,x)=\psi (x),A_t(0,x)=\eta (x),\text{in}\Omega ,$$

$$u(t,x)=0,A(t,x)=0,\text{on}\partial \Omega \times [0,T).$$

$$\begin{array}{rl}X& =\{u\in C_0^{\infty }(\Omega ,{\mathbb{R}}^2)|\text{div}u=0\text{in}\Omega \},\\ Y& =\text{the closure of}X\text{in}{L}^2(\Omega ,{\mathbb{R}}^2)\\ & =\{u\in L^2(\Omega ,{\mathbb{R}}^2)|\text{div}u=0\text{in}\Omega \},\\ Z& =\text{the closure of}X\text{in}{H}^1(\Omega ,{\mathbb{R}}^2)\\ & =\{u\in H_0^1(\Omega ,{\mathbb{R}}^2)|\text{div}u=0\text{in}\Omega \}.\end{array}$$

$${\int }_{\Omega }u\cdot v\text{d}x+{\int }_0^t{\int }_{\Omega }(u\cdot \nabla )u\cdot v$$

$$={\int }_0^t{\int }_{\Omega }f\cdot v\text{d}x\text{d}t+{\int }_{\Omega }\varphi \cdot v\text{d}x$$

$${\int }_{\Omega }\frac{\partial A}{\partial t}\cdot v\text{d}x+{\int }_0^t{\int }_{\Omega }\frac{1}{{ϵ}_0{\mu }_0}\nabla u\cdot \nabla v+\frac{{\rho }_e}{{ϵ}_0}u\cdot v\text{d}x\text{d}t={\int }_{\Omega }\eta v\text{d}x.$$

$$u\in L_{\text{loc}}^{\infty }(0,\infty ;H^2(\Omega ,{\mathbb{R}}^2)\cap Z),u_t\in L_{\text{loc}}^{\infty }(0,\infty ;Y)\cap L_{\text{loc}}^2(0,\infty ;Z)$$

$$p\in L_{\text{loc}}^{\infty }(0,\infty ;H^1(\Omega )),$$

$$A\in L_{\text{loc}}^{\infty }(0,\infty ;H^2(\Omega ,{\mathbb{R}}^2)\cap Z),A_t\in L_{\text{loc}}^{\infty }(0,\infty ;Z),A_{tt}\in L_{\text{loc}}^{\infty }(0,\infty ;Y).$$

$$L^p((0,T),W^{k+1,p}(\Omega ))\cap L^{\infty }(0,T;L^r(\Omega ))\subset L^q(0,T;W^{k,q}(\Omega )),$$

$$L^p(0,T;W^{1,p}(\Omega ))\cap L^{\infty }(0,T;L^r(\Omega ))\subset L^q((\Omega )\times (0,T)),$$

$$||u|{|}_{W^{k,q}}\le C||u|{|}_{W^{m,p}}^{\theta }||u|{|}_{W^{j,r}}^{1-\theta },0\le \theta \le 1,$$

$$\theta\bigg{(}m-\frac{n}{p}\bigg{)}+(1-\theta)\bigg{(}j-\frac{n}{r}\bigg{)}\geq k-\frac{n}{q},$$

$$\bigg{(}\int_{\Omega}|D^{k}u|^{q}\text{d}x\bigg{)}^{\frac{1}{q}}\leq C\bigg{(}\int_{\Omega}|D^{k+1}u|^{p}\text{d}x\bigg{)}^{\frac{1}{q}}\bigg{(}\int_{\Omega}|u|^{r}\text{d}x\bigg{)}^{\frac{1}{r}(1-p/q)},$$

$${\int }_0^T{\int }_{\Omega }|D^{k}u{|}^q\text{d}x\text{d}t\le C\underset{0\le t\le T}{\sup }||u|{|}_{L^r}^{(q-p)r}{\int }_0^T{\int }_{\Omega }|D^{k+1}u{|}^p\text{d}x\text{d}t,$$

$$\{\begin{array}{rl}& \frac{\partial u}{\partial t}=\nu \Delta u-\nabla p+f(x,t),\\ & \nabla \cdot u=0,\\ & u{|}_{\partial \Omega }=0,\\ & u(0)=u_0,\end{array}$$

$$u\in L^r(0,T;W^{2,q}(\Omega ,{\mathbb{R}}^n)),u_t\in L^r(0,T;L^q(\Omega ,{\mathbb{R}}^n)),$$

$$p\in L^r(0,T;W^{1,q}(\Omega ))$$

$$||u|{|}_{L^r(0,T;W^{2,q})}+||u_t|{|}_{L^r(0,T;L^q)}+||p|{|}_{L^r(0,T;W^{1,q})}$$

$$\le C(||f|{|}_{L^r(0,T;L^q)}+||u_0|{|}_{W^{2,q}}),$$

$$\{\begin{array}{rlr}& -\nu \Delta v-\nabla p=F(x),& \text{in}\Omega ,\\ & \nabla \cdot v=0,& \text{in}\Omega ,\\ & v{|}_{\partial \Omega }=0,& \text{on}\partial \Omega ,\end{array}$$

$$(v,p)\in W^{k+2,q}(\Omega ,{\mathbb{R}}^n)\times W^{k+1,q}(\Omega )$$

$$||v|{|}_{W^{k+2,q}}+||p|{|}_{W^{k+1,q}}\le C(||F|{|}_{W^{k,q}}+||(u,p)|{|}_{L^q})$$

$$u\in L^{\infty }(0,T;H^2(\Omega ,{\mathbb{R}}^2)\cap Z),u_t\in L^{\infty }(0,T;Y)\cap L^2(0,T;Z)$$

$$p\in L^{\infty }(0,T;H^1(\Omega )),$$

$$A\in L^{\infty }(0,T;H^2(\Omega ,{\mathbb{R}}^2)\cap Z),A_t\in L^{\infty }(0,T;Z),A_{tt}\in L^{\infty }(0,T;Y).$$

$$u\in L^2(0,T;Z)\cap L^{\infty }(0,T;Y),$$

$$A\in L^{\infty }(0,T;Z),A_t\in L^{\infty }(0,T;Y).$$

$$\frac{1}{2}\frac{\text{d}}{\text{d}t}\bigg{(}\int_{\Omega}|\nabla A_{t}|^{2}+\frac{1}{\epsilon_{0}\mu_{0}}|\Delta A|^{2}\text{d}x\bigg{)}=\frac{\rho_{e}}{\epsilon_{0}}\int_{\Omega}\nabla u\nabla A_{t}\text{d}x$$

$$\displaystyle\frac{\text{d}}{\text{d}t}\bigg{(}||\nabla A_{t}||^{2}_{L^{2}}+\frac{1}{\epsilon_{0}\mu_{0}}||\Delta A||^{2}_{L^{2}}\bigg{)}\leq 2\bigg{(}||\nabla A_{t}||^{2}_{L^{2}}+\frac{1}{\epsilon_{0}\mu_{0}}||\Delta A||^{2}_{L^{2}}+\frac{\rho^{2}_{e}}{\epsilon^{2}_{0}}||\nabla u||^{2}_{L^{2}}\bigg{)}.$$

$$\displaystyle||\nabla A_{t}||^{2}_{L^{2}}+||\Delta A||^{2}_{L^{2}}\leq e^{2T}\bigg{(}||\Delta\psi||_{L^{2}}+||\nabla\eta||_{L^{2}}+2\frac{\rho^{2}_{0}}{\epsilon^{2}_{0}}\int_{0}^{T}||\nabla u||^{2}_{L^{2}}\text{d}s\bigg{)},$$

$$\nabla A_t\in L^{\infty }(0,T;Y),\Delta A\in L^{\infty }(0,T;Y).$$

$${\int }_{\Omega }|A_{tt}{|}^2\text{d}x=\frac{1}{{ϵ}_0{\mu }_0}{\int }_{\Omega }\Delta A{A}_{tt}\text{d}x+\frac{{\rho }_0}{{ϵ}_0}{\int }_{\Omega }u{A}_{tt}\text{d}x$$

Full text

keywords

Global strong solution, Magnetohydrodynamics, Stokes equation, $L^{p}$-$L^{q}$-estimates.

1 Introduction

We consider the following 2-D incompressible magnetohydrodynamic (MHD) model, which describes the interaction between moving conductive fluids and electromagnetic fields in [10],

[TABLE]

Here $\Omega\subset\mathbb{R}^{2}$ is a bounded smooth domain, $T$ is any fixed time. $u(x,t)$, $A(x,t)$, $p(x,t)$ are the velocity field, the magnetic potential and the pressure function, respectively. $\Phi=\frac{\partial A_{0}}{\partial t}$ represents the magnetic pressure with the scalar electromagnetic potential $A_{0}$. The constants $\nu$, $\rho_{0}$, $\rho_{e}$, $\epsilon_{0}$, $\mu_{0}$ denote kinetic viscosity, mass density, equivalent charge density, electric permittivity and magnetic permeability of free space.

In this paper, we focus on the system (1.1) with the initial-boundary conditions

[TABLE]

[TABLE]

Note that the MHD model (1.1) is established based on the the Newton’s second law and the Maxwell equations for the electromagnetic fields in [10]. In addition, the global weak solutions of the corresponding 3-D MHD model(1.1) with (1.2)-(1.3) has been obtained by using the Galerkin technique and standard energy estimates in [10]. In this paper, what we are concerned is the global strong solution of the 2-D MHD model (1.1) with the initial-boundary conditions (1.2)$-$(1.3).

It is known that there have been huge mathematical studies on the existence of solutions to the N-dimension($N\geq 2$) classical MHD model established by Chandrasekhar [4]. In particular, Duvaut and Lions [5] constructed a global weak solution and the local strong solution to the 3-D classical MHD equations the initial boundary value problem, and properties of such solutions have been investigated by Sermange and Temam in [15]. Furthermore, some sufficient conditions for smoothness were presented for the weak solution to the 3-D classical MHD equations in [7] and some sufficient conditions of local regularity of suitable weak solutions to the 3-D classical MHD system for the points belonging to a $C^{3}$-smooth part of the boundary were obtained in [18]. Also, the global strong solutions for heat conducting 3-D classical magnetohydrodynamic flows with non-negative density were proved in [21].

Moreover, let’s recall some known results for the 2-D classical and generalized MHD equations. It is noticed that the 2D classical MHD equations admits a unique global strong solution in [5, 15]. Furthermore, Ren, Wu, et.al [14] have proved the global existence and the decay estimates of small smooth solution for the 2-D classical MHD equations without magnetic diffusion and Cao, Regmi and Wu [3] have obtained the global regularity for the 2-D classical MHD equations with mixed partial dissipation and magnetic diffusion. Besides, Regmi [13] established the global weak solution for 2-D classical MHD equations with partial dissipation and vertical diffusion. There are also very interesting investigations about the existence of strong solutions to the 2-D classical and generalized MHD equations, see [8, 9, 12, 15, 19, 20, 22] and references therein.

However, it is worth pointing out that the incompressible MHD system (1.1) is a mixed-type differential difference equation, which is combined with the parabolic equation (1.1)1 and the hyperbolic equation (1.1)2. The main challenge in obtaining global strong solution of 2-D MHD model(1.1) with (1.2)-(1.3) is the estimate for $||u\times\text{rot}A||_{L^{\infty}(0,T;L^{2})}$ and $||(u\cdot\nabla)u||_{L^{\infty}(0,T;L^{2})}$. The difficulty is overcome by applying the Solonnikov’s theorem [6, 11, 16] of $L^{p}-L^{q}$-estimates for the non-stationary Stokes equations and Agmon, Douglis and Nirenberg’s estimates [2, 1, 11] for the stationary Stokes equations. As is known, Solonnikov [16] first gave the proof of Maximal $L^{p}$-$L^{q}$-estimates for the Stokes equation (2.4) using potential theoretic arguments. Recently, Geissert, Hess, Hieber et.al [6] provided a short proof of the corresponding Solonnikov’s theorem in [16].

The rest of this article is organized as follows. In Section 2, we introduce some elementary function spaces, a vital embedding theorem and some regularity results of both the non-stationary Stokes equations and stationary Stokes equations. Section 3 is mainly devoted to the proof of global strong solution of (1.1)$-$(1.3).

2 Preliminaries

2.1 Notations and definitions

First, we introduce some notations and conventions used throughout this paper.

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded sufficiently smooth domain. Let $H^{\tau}(\Omega)(\tau=1,2)$ be the general Sobolev space on $\Omega$ with the norm $||\cdot||_{H^{\tau}}$ and $L^{2}(\Omega)$ be the Hilbert space with the usual norm $||\cdot||$. The space $H^{1}_{0}(\Omega)$ we mean that the completion of $C^{\infty}_{0}(\Omega)$ under the norm $||\cdot||_{H^{1}}$. If $\digamma$ is a Banach space, we denote by $L^{p}(0,T;\digamma)(1<p<\infty)$ the Banach space of the $\digamma$-valued functions defined in the interval $(0,T)$ that are $L^{p}$-integrable.

We also consider the following spaces of divergence-free functions (see Temam [17])

[TABLE]

Definition 2.1**.**

Suppose that $\phi,\eta\in Y$, $\psi\in Z$. For any $T>0$, a vector function $(u,A)$ is called a global weak solution of problem (1.1)$-$(1.3) on $(0,T)\times\Omega$ if it satisfies the following conditions:

$u\in L^{2}(0,T;Z)\cap L^{\infty}(0,T;Y),$** 2. 2.

$A\in L^{\infty}(0,T;Z),\ A_{t}\in L^{\infty}(0,T;Y),$** 3. 3.

For any function $v\in X$, there hold

[TABLE]

and

[TABLE]

Now, we define strong solution of the problem (1.1)$-$(1.3).

Definition 2.2**.**

Suppose that $\phi,\psi\in H^{2}(\Omega,\mathbb{R}^{2})\cap Z$, $\eta\in Z$, $\Phi\in L^{2}(0,T;$ $H^{1}_{0}(\Omega))$. $(u,A)$ is called a global strong solution to (1.1)$-$(1.3), if $(u,A)$ satisfy

[TABLE]

Furthermore, both (1.1) and (1.3) hold almost everywhere in $\Omega\times(0,T)$.

2.2 Lemmas

Some more lemmas will be frequently used later. One is the following embedding result in [11].

Lemma 2.3**.**

For any $k\geq 0$, the following hold

[TABLE]

where $q=(r(k+1)p+np)/(rk+n)$. In the special case of $k=0$, (2.1) equals to

[TABLE]

provided that $q=(n+r)p/n$.

Proof.

From Gagliardo-Nirenberg interpolation inequality, we have

[TABLE]

provided that

[TABLE]

where $C$ is a constant independent of $u$.

Inserting $j=0,\ q\geq p$, $m=k+1$ and $\theta=\frac{p}{q}$ into (2.3), it is easy to see that

[TABLE]

where $q=\frac{(r(k+1)p+np)}{rk+n}$.

Then we get

[TABLE]

which implies (2.1) and (2.2). ∎

The other lemma is responsible for the estimates for $u,p,u_{t}$ and follows from the $L^{p}$-$L^{q}$-estimates [6, 16] for non-stationary Stokes equations. For its proof, refer to [6, 16].

Let us consider the following Stokes equations

[TABLE]

where $\nu>0$ is a constant.

Lemma 2.4**.**

Let $\Omega\subset\mathbb{R}^{n}(n=2,3)$ be a domain with compact $C^{3}$-boundary, $1<r,p<\infty,\ 0<T<\infty$. Then for any $f\in L^{r}(0,T;L^{q}(\Omega,\mathbb{R}^{n}))$ and $u_{0}\in W^{2,q}(\Omega,\mathbb{R}^{n}))$ there exists a unique solution $(u,p)$ of (2.4) satisfying

[TABLE]

such that

[TABLE]

where $C>0$ is a constant.

Finally, we give some regularity results for the stationary Stokes system. For its proof, refer to [2, 1, 11].

Lemma 2.5**.**

Assume that $(v,p)\in W^{2,p}(\Omega,\mathbb{R}^{n})\times W^{1,p}(\Omega)(1<p<\infty)$ is a weak solution of the stationary Stokes equations

[TABLE]

and $F\in W^{k,q}(\Omega,\mathbb{R}^{n})(k\geq 0,1<q<\infty)$. Then it holds that

[TABLE]

and

[TABLE]

with some constant $C$ depending on $n$, $\Omega$ and $q$.

3 Main Results

In this section, we state the global weak solution existence theorem and the global strong solution existence one for the problem (1.1)$-$(1.3), and also prove them.

Theorem 3.1**.**

Let the initial value $\phi,\eta\in Y$, $\psi\in Z$. If $f\in Y,\Phi\in L^{2}(0,T;H^{1}_{0}(\Omega))$, then there exists a global weak solution for the problem (1.1)$-$(1.3).

Proof.

By the standard Galerkin method and the similar estimates in [10], the existence of global weak solution of (1.1)$-$(1.3) is also valid, we omit it. ∎

Theorem 3.2**.**

Let $\Omega$ be a bounded domain with compact $C^{3}$-boundary. If $\phi,\psi\in H^{2}(\Omega,\mathbb{R}^{2})\cap Z$, $\eta\in Z$, for any $f\in Y,\Phi\in L^{2}(0,T;H^{1}_{0}(\Omega))$, then there exists a global strong solution for the problem (1.1)$-$(1.3), i.e., for any $\ 0<T<\infty$

[TABLE]

Proof.

The proof can be divided into 3 steps. We will use the same generic constant $C$ to denote various constants that depend on $\mu_{0},\rho_{0},\rho_{e},\epsilon_{0}$ and $T$ only.

Step 1 The estimates and regularity for $A$.

From Theorem 3.1, for any $0<T<\infty$, we get the global weak solution

[TABLE]

Multiplying both sides of (1.1)2 by $-\Delta A_{t}$ and integrating over $\Omega$, we have

[TABLE]

since $\text{div}A=0$ and (1.3).

Using the Hölder inequality, it is easy to see that

[TABLE]

Then, by the Gronwall inequality, (3.3) implies

[TABLE]

for $\forall~{}0<T<\infty$.

Therefore, we conclude that

[TABLE]

Next, we need to derive an estimate on $||A_{tt}||_{L^{\infty}(0,T;Y)}$.

Multiplying both sides of Eqs. (1.1)2 by $A_{tt}$ integrating over $\Omega$ lead to

[TABLE]

since $-\int_{\Omega}\nabla\Phi A_{tt}\text{d}x=\int_{\Omega}\Phi\text{div}A_{tt}\text{d}x=0$.

Using the Höder inequality and Young inequality, we deduce from (3.6) that

[TABLE]

It is easy to see that

[TABLE]

Putting the estimates (3.1), (3.5) and (3.8) together, we have

[TABLE]

Hence, (3.5) and (3.9) imply the regularity for $A$.

Step 2 The $L^{\frac{4}{3}}$-$L^{\frac{4}{3}}$-estimates for $u\cdot\nabla u$ and $u\times A$.

From (3.1) and Lemma 2.3(the case that k=0), it is easy to check that

[TABLE]

Note that

[TABLE]

which implies that

[TABLE]

Combining (3.1) and (3.10), we get

[TABLE]

which in turn implies

[TABLE]

Recall that $(u,p)$ satisfying the following Stokes system

[TABLE]

where $F(x,t)=f-(u\cdot\nabla)u+\frac{\rho_{e}}{\rho_{0}}(u\times\text{rot}A)$.

By (3.12) and (3.14), we get $F\in L^{\frac{4}{3}}(0,T;L^{\frac{4}{3}}(\Omega,\mathbb{R}^{2}))$. Applying this into Lemma 2.4, we obtain that

[TABLE]

In the next step, the Lemma 2.5 will be used, since (3.15) can be rewritten as the following stationary Stokes equations

[TABLE]

where $\widetilde{F}(x,t)=f-(u\cdot\nabla)u+\frac{\rho_{e}}{\rho_{0}}(u\times\text{rot}A)-u_{t}$.

Step 3 The estimate for $||\widetilde{F}||_{L^{\infty}(\Omega,L^{2}(\Omega,\mathbb{R}^{2}))}$.

(i) The estimate for $||\nabla u||_{L^{\infty}(0,T;L^{2})}$.

Multiplying Eq. (1.1)1 by $u_{t}$ and integrating over $\Omega$, we have

[TABLE]

Note that the following continuous embeddings

[TABLE]

Combining (3.19), Hölder inequality and $\epsilon-$Young inequality, we derive that

[TABLE]

and

[TABLE]

which together with Gronwall’s inequality implies

[TABLE]

(ii) The estimate for $||u_{t}||_{L^{\infty}(0,T;L^{2})}$.

Taking $t$-derivative of Eq. (1.1)1, then one gets that

[TABLE]

Multiplying (3.23) by $u_{t}$ and integrating over $\Omega$, we obtain

[TABLE]

since

[TABLE]

Next, we estimate the two terms on the right hand of (3.24). By (3.19) and integrating by parts yield

[TABLE]

And similarly,

[TABLE]

Hence, by (3.24), (3.25) and (3.26), we get that

[TABLE]

which together with Gronwall’s inequality completes the estimate

[TABLE]

(iii) The estimates for $||(u\cdot\nabla u)||_{L^{\infty}(0,T;L^{2})}$ and $||u\times A||_{L^{\infty}(0,T;L^{2})}$.

From (3.22), it is easy to see that

[TABLE]

Hence

[TABLE]

It is known that $H^{1}\hookrightarrow L^{q}(1<q<\infty)$ when $n=2$. Note that

[TABLE]

provided that $1<r<2$. Hence

[TABLE]

By using the Hölder inequality and the Sobolev embedding theorem, it follows that

[TABLE]

Together (3.5) with (3.32), we have

[TABLE]

According to (3.28), (3.31), (3.33) and the assumption, $\widetilde{F}$ in (3.17) satisfies

[TABLE]

Applying (3.34) into Lemma 2.5, we get

[TABLE]

Using the Sobolev embedding theorem $W^{2,r}\hookrightarrow C^{\alpha}\hookrightarrow C^{0}(0<\alpha<1,n=2,)$, we deduce from (3.29) and (3.35) that

[TABLE]

By (3.28), (3.33) and (3.36), we get that

[TABLE]

Applying (3.37) into Lemma 2.5, we obtain that for any $T>0$

[TABLE]

Therefore, (3.1), (3.5), (3.9), (3.28) and (3.38) complete the proof.

∎