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Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion. (English) Zbl 1213.35159
The existence of global in time solutions to the 2D MHD equations is studied.
Let \(u(x,y,t)\) be the velocity field, \(p(x,y,t)\) the pressure and \(b(x,y,t)\) the magnetic field. The functions \(u,\;p,\;b\) satisfy the Cauchy problem
\[ \begin{aligned} &\frac{\partial u}{\partial t}-\nu_1\frac{\partial^2 u}{\partial x^2} -\nu_2\frac{\partial^2 u}{\partial y^2}+u\cdot\nabla u -b\cdot\nabla b+\nabla p=0,\quad \operatorname{div}u=0,\\ & \frac{\partial b}{\partial t}-\eta_1\frac{\partial^2 b}{\partial x^2} -\eta_2\frac{\partial^2 b}{\partial y^2}+u\cdot\nabla b -b\cdot\nabla u=0,\quad \operatorname{div}b=0,\\ &u(x,y,0)=u_0(x,y),\quad b(x,y,0)=b_0(x,y) \end{aligned} \] Two main results are established.
1.
If
\[ \nu_1=0,\;\nu_2>0,\;\eta_1>0,\;\eta_2=0\quad \text{or}\quad \nu_1.0,\;\nu_2=0,\;\eta_1=0,\;\eta_2>0 \] and \(u_0,b_0\in H^2({\mathbb R}^2)\), \(\nabla\cdot u_0=0\), \(\nabla\cdot b_0=0\), then the problem has a unique global classical solution.
2.
If \(\nu_1=\nu_2=0\) and \(\eta_1=\eta_2=\eta>0\), \(u_0,b_0\in H^1({\mathbb R}^2)\), \(\nabla\cdot u_0=0\), \(\nabla\cdot b_0=0\), then the problem has a global weak solution. If \(u_0,b_0\in H^3({\mathbb R}^2)\) and for some \(T>0\)
\[ \sup_{q\geq 2}\;\frac{1}{q}\;\int^T_0\|\nabla u(\cdot,t)\|_q\,dt<\infty \] then a weak solution is unique on \([0,T]\).

MSC:
35B65 Smoothness and regularity of solutions to PDEs
35Q35 PDEs in connection with fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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References:
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