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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.
\[ \nu_1=0,\;\nu_2>0,\;\eta_1>0,\;\eta_2=0\quad \text{or}\quad \nu_1.0,\;\nu_2=0,\;\eta_1=0,\;\eta_2>0 \] 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.
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 \([0,T]\).

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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[1] Caflisch, R.; Klapper, I.; Steele, G., Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Comm. math. phys., 184, 443-455, (1997) · Zbl 0874.76092
[2] Chae, D., Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. math., 203, 497-513, (2006) · Zbl 1100.35084
[3] Danchin, R.; Paicu, M., Global existence results for the anisotropic Boussinesq system in dimension two, (19 September 2008)
[4] Duvaut, G.; Lions, J.-L., Inéquations en thermoélasticité et magnétohydrodynamique, Arch. ration. mech. anal., 46, 241-279, (1972) · Zbl 0264.73027
[5] Hmidi, T.; Keraani, S.; Rousset, F., Global well-posedness for Euler-Boussinesq system with critical dissipation, (22 March 2009)
[6] Hmidi, T.; Keraani, S.; Rousset, F., Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, (9 April 2009)
[7] Hou, T.; Li, C., Global well-posedness of the viscous Boussinesq equations, Discrete contin. dyn. syst., 12, 1-12, (2005) · Zbl 1274.76185
[8] Kenig, C.E.; Ponce, G.; Vega, L., Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. amer. math. soc., 4, 323-347, (1991) · Zbl 0737.35102
[9] Sermange, M.; Temam, R., Some mathematical questions related to the MHD equations, Comm. pure appl. math., 36, 635-664, (1983) · Zbl 0524.76099
[10] Wu, J., Viscous and inviscid magnetohydrodynamics equations, J. anal. math., 73, 251-265, (1997) · Zbl 0903.76099
[11] Zhang, P., Global smooth solutions to the 2-D nonhomogeneous Navier-Stokes equations, Int. math. res. not., 2008, 1-26, (2008), art. ID rnn098 · Zbl 1178.35297
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