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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 $u\left(x,y,t\right)$ be the velocity field, $p\left(x,y,t\right)$ the pressure and $b\left(x,y,t\right)$ the magnetic field. The functions $u,\phantom{\rule{4pt}{0ex}}p,\phantom{\rule{4pt}{0ex}}b$ satisfy the Cauchy problem

$\begin{array}{cc}& \frac{\partial u}{\partial t}-{\nu }_{1}\frac{{\partial }^{2}u}{\partial {x}^{2}}-{\nu }_{2}\frac{{\partial }^{2}u}{\partial {y}^{2}}+u·\nabla u-b·\nabla b+\nabla p=0,\phantom{\rule{1.em}{0ex}}divu=0,\hfill \\ & \frac{\partial b}{\partial t}-{\eta }_{1}\frac{{\partial }^{2}b}{\partial {x}^{2}}-{\eta }_{2}\frac{{\partial }^{2}b}{\partial {y}^{2}}+u·\nabla b-b·\nabla u=0,\phantom{\rule{1.em}{0ex}}divb=0,\hfill \\ & u\left(x,y,0\right)={u}_{0}\left(x,y\right),\phantom{\rule{1.em}{0ex}}b\left(x,y,0\right)={b}_{0}\left(x,y\right)\hfill \end{array}$

Two main results are established.

If

${\nu }_{1}=0,\phantom{\rule{4pt}{0ex}}{\nu }_{2}>0,\phantom{\rule{4pt}{0ex}}{\eta }_{1}>0,\phantom{\rule{4pt}{0ex}}{\eta }_{2}=0\phantom{\rule{1.em}{0ex}}\text{or}\phantom{\rule{1.em}{0ex}}{\nu }_{1}·0,\phantom{\rule{4pt}{0ex}}{\nu }_{2}=0,\phantom{\rule{4pt}{0ex}}{\eta }_{1}=0,\phantom{\rule{4pt}{0ex}}{\eta }_{2}>0$

and ${u}_{0},{b}_{0}\in {H}^{2}\left({ℝ}^{2}\right)$, $\nabla ·{u}_{0}=0$, $\nabla ·{b}_{0}=0$, then the problem has a unique global classical solution.

If ${\nu }_{1}={\nu }_{2}=0$ and ${\eta }_{1}={\eta }_{2}=\eta >0$, ${u}_{0},{b}_{0}\in {H}^{1}\left({ℝ}^{2}\right)$, $\nabla ·{u}_{0}=0$, $\nabla ·{b}_{0}=0$, then the problem has a global weak solution. If ${u}_{0},{b}_{0}\in {H}^{3}\left({ℝ}^{2}\right)$ and for some $T>0$

$\underset{q\ge 2}{sup}\phantom{\rule{4pt}{0ex}}\frac{1}{q}\phantom{\rule{4pt}{0ex}}{\int }_{0}^{T}{\parallel \nabla u\left(·,t\right)\parallel }_{q}\phantom{\rule{0.166667em}{0ex}}dt<\infty$

then a weak solution is unique on $\left[0,T\right]$.

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