The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion
The following 2D incompressible MHD equations are studied in the paper. \[ \begin{aligned} \frac{\partial u}{\partial t}+u\cdot \nabla u+\nabla p-\frac{\partial^2 u}{\partial x^2}-b\cdot \nabla b=0, \\ \frac{\partial b}{\partial t}+u\cdot \nabla b-\frac{\partial^2 b}{\partial x^2}-b\cdot \nabla u=0,\\ \nabla\cdot u=0,\quad \nabla\cdot b=0,\\ u(x,y,0)=u_0(x,y),\quad b(x,y,0)=b_0(x,y), \end{aligned} \] where \((x,y)\in \mathbb{R}^2\), \(t\geq 0\), \(u(x,y,t)=(u_1,u_2)\) is the velocity, \(p(x,y,t)\) is the pressure, \(b(x,y,t)=(b_1,b_2)\) is the magnetic field. The paper presents several a priori estimates to the solutions of the problem. If \((u_0,b_0)\in H^2(\mathbb{R}^2)\) and \(2<r<\infty\) then the horizontal component of any solution satisfies to the inequality \[ \| (u_1,b_1)(t)\|_{2r}\leq B_0(t) \sqrt{r\log r}+B_1, \] where \(B_0\) is a smooth function of \(t\) and \(B_1\) depends only on \(\| (u_0,b_0)(t)\|_{2r}\). If \[ \int\limits_0^T\| (u_1,b_1)(t)\|_{\infty}^2dt<\infty \] for some \(T>0\), then \(\| (u,b)\|_{H^2}\) is finite on \([0,T]\). The pressure obeys the global bound for any \(T>0\) and \(0<t<T\) \[ \| p(\cdot,t)\|_q\leq C(T),\quad \int\limits_0^T\| p(\cdot,t)\|_{H^s}^2dt<C(T), \] where \(1<q\leq 3\) and \(0<s<1\). The regularized version of the problem is considered too: \[ \begin{aligned} \frac{\partial u}{\partial t}+u\cdot \nabla u+\epsilon(-\Delta)^\delta u+\nabla p-\frac{\partial^2 u}{\partial x^2}-b\cdot \nabla b=0,\\ \frac{\partial b}{\partial t}+u\cdot \nabla b+\epsilon(-\Delta)^\delta b-\frac{\partial^2 b}{\partial x^2}-b\cdot \nabla u=0, \\ \nabla\cdot u=0,\quad \nabla\cdot b=0,\\ u(x,y,0)=u_0(x,y),\quad b(x,y,0)=b_0(x,y) \end{aligned} \] with \(\epsilon>0\) and \(\delta>0\). The solution to this problem satisfies to the inequality for any \(T>0\) and \(0<t\leq T\) \[ \| (u,b)\|_{H^2}^2+\int\limits_0^t\left(\left\| (\frac{\partial u}{\partial x},\frac{\partial b}{\partial x})\right\|_{H^2}^2 +\epsilon\| (\Lambda^\delta u,\Lambda^\delta b)\|_{H^2}^2\right)d\tau\leq C, \] where \(C\) is a constant depending on \(T\) and \(\| (u_0,b_0)\|_{H^2}\) only.
scientific article; zbMATH DE number 6147000