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The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion. (English) Zbl 1270.35143
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.

##### MSC:
 35B45 A priori estimates in context of PDEs 35Q35 PDEs in connection with fluid mechanics 76D09 Viscous-inviscid interaction 76W05 Magnetohydrodynamics and electrohydrodynamics
##### Keywords:
global regularity; horizontal dissipation
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