Global solutions to the incompressible magneto-micropolar system in a bounded domain in 2D

Abstract

We prove the existence and uniqueness of global strong solutions to the incompressible magneto-micropolar system in a two dimensional bounded domain with Navier type boundary condition for the velocity.

Introduction

In this paper, we consider the following 2D incompressible magneto-micropolar system [1]:

$${\partial }_{t}u+u\cdot \nabla u+\nabla \pi -(\mu +\zeta )\Delta u=2\zeta \mathrm{rot}\varphi +(E+u\times b)\times b,$$$${\partial }_t\varphi +u\cdot \nabla \varphi -\tilde{\mu }\Delta \varphi +4\zeta \varphi =2\zeta \mathrm{rot}u,$$$${\partial }_{t}E-\mathrm{rot}b+E+u\times b=0,$$$${\partial }_{t}b+\mathrm{rot}E=0,$$$$\mathrm{div}u=0,\mathrm{div}b=0\text{in}\Omega \times (0,\infty ),$$$$u\cdot n=0,\mathrm{rot}u=0,\varphi =0,b\cdot n=0,E=0\text{on}\partial \Omega \times (0,\infty ),$$$$(u,\varphi ,E,b)(\cdot ,0)=(u_0,{\varphi }_0,E_0,b_0)\left(\cdot \right)\text{in}\Omega \subset {\mathbb{R}}^2.$$

Here, $u,\pi ,\varphi ,E$ and $b$ denote the velocity, pressure, micro-rotational velocity, electric field, and magnetic field, respectively. We introduce $j=E+u\times b$ as the electric current. $\Omega \subset {\mathbb{R}}^2$ is a bounded and simply connected domain with smooth boundary $\partial \Omega$, and $n$ is the unit outward normal vector to $\partial \Omega$. The viscosity coefficients $\mu ,\zeta$, and $\tilde{\mu }$ are positive constants.

Note that in 2D case, vector fields $u,w,E$, and $b$ may be understood as $u≔(u_1,u_2,0),w≔(0,0,\varphi ),E≔(0,0,E)$, and $b≔(b_1,b_2,0)$ and therefore we regard $w,E$ and $\mathrm{rot}u$ as scalar functions.

When $\zeta =\varphi =0$, the system reduces to the Maxwell–Navier–Stokes system which has been receiving much mathematical attention. In [2], global existence of regular solutions in $\Omega ={\mathbb{R}}^2$ is proved by using the Besov-type $\tilde{L}$ space technique. Kang and Lee [3] reproved the result in [2] by using energy estimates and Brezis–Gallouet inequality. Duan [4] studied large time behavior of the solutions. Fan–Zhou [5] and Kang–Lee [3] also showed some regularity criteria. In [6], [7], the local existence of mild solutions has been studied.

Very recently, Fan–Zhang–Zhou [8] showed the local well-posedness of strong solutions with vacuum.

The aim of this note is to prove the result in [2], [3] (with $\Omega ≔{\mathbb{R}}^2$) in a bounded domain $\Omega \subset {\mathbb{R}}^2$. We will prove

Theorem 1.1

Let $u_0,{\varphi }_0,E_0,b_0\in H^2\left(\Omega \right)$ satisfy $\mathrm{div}{u}_0=\mathrm{div}{b}_0=0$ in $\Omega$ and $u_0\cdot n=0,\mathrm{rot}{u}_0=0,b_0\cdot n=0$, and ${\varphi }_0=E_0=0$ on $\partial \Omega$. Then, for any $T>0$, there exists a unique solution to the problem (1.1)–(1.7) such that

$$u,\varphi ,E,b\in L^{\infty }(0,T;H^2).$$

In the following proof, we will use the well-known Brezis–Gallouet inequality (logarithmic Sobolev inequality) [9, Theorem 3.10, P.35]:

$${\Vert f\Vert }_{L^{\infty }}\le C(1+{\Vert f\Vert }_{L^2}+{\Vert \nabla f\Vert }_{L^2}{log}^{\frac{1}{2}}(e+{\Vert f\Vert }_{H^2})).$$

We will use the following two inequalities [10], [11]:

$${\Vert w\Vert }_{H^s\left(\Omega \right)}\le C({\Vert \mathrm{div}w\Vert }_{H^{s-1}\left(\Omega \right)}+{\Vert \mathrm{rot}w\Vert }_{H^{s-1}\left(\Omega \right)}+{\Vert w\Vert }_{H^{s-1}\left(\Omega \right)}+{\Vert w\cdot n\Vert }_{H^{s-\frac{1}{2}}\left(\partial \Omega \right)}),$$$${\Vert w\Vert }_{H^s\left(\Omega \right)}\le C({\Vert \mathrm{div}w\Vert }_{H^{s-1}\left(\Omega \right)}+{\Vert \mathrm{rot}w\Vert }_{H^{s-1}\left(\Omega \right)}+{\Vert w\Vert }_{H^{s-1}\left(\Omega \right)}+{\Vert w\times n\Vert }_{H^{s-\frac{1}{2}}\left(\partial \Omega \right)}),$$

We will also use the following Poincaré inequality [12]:

$${\Vert w\Vert }_{L^2\left(\Omega \right)}\le C({\Vert \mathrm{div}w\Vert }_{L^2\left(\Omega \right)}+{\Vert \mathrm{rot}w\Vert }_{L^2\left(\Omega \right)})$$

for any $w\in H^1\left(\Omega \right)$ with $w\cdot n=0$ or $w\times n=0$ on $\partial \Omega$.

Using (1.10), (1.11), (1.12), we obtain

$${\Vert u\Vert }_{H^1}\le C{\Vert \mathrm{rot}u\Vert }_{L^2},{\Vert u\Vert }_{H^2}\le C{\Vert \mathrm{rot}{}^{2}u\Vert }_{L^2}=C{\Vert \Delta u\Vert }_{L^2},$$$${\Vert b\Vert }_{H^1}\le C{\Vert \mathrm{rot}b\Vert }_{L^2},{\Vert b\Vert }_{H^2}\le C{\Vert \mathrm{rot}{}^{2}b\Vert }_{L^2}=C{\Vert \Delta b\Vert }_{L^2},$$$${\Vert E\Vert }_{H^1}\le C{\Vert \mathrm{rot}E\Vert }_{L^2},{\Vert E\Vert }_{H^2}\le C{\Vert \Delta E\Vert }_{L^2}\le C{\Vert \mathrm{rot}{}^{2}E\Vert }_{L^2},$$