Abstract
This paper is concerned with the initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. We show that the system has a unique classical solution for $H^3$ initial data, and the non-slip boundary condition for velocity field and the perfectly conducting wall condition for magnetic field. In addition, we show that the kinetic energy is uniformly bounded in time.
Highlights
This paper is concerned with nonlinear instability of a smooth Rayleigh-Taylor (RT) steady-state solution to the following three-dimensional (3D) nonhomogeneous incompressible magnetohydrodynamic (MHD) equations with zero resistivity in the presence of a uniform gravitational field: 2010 Mathematics Subject Classification
The impact of the magnetic filed on the instability will be analyzed, for example, we shall show that if the steady magnetic field is vertical and small, the magnetic field is unstable, verifying the physical phenomenon: instability of the velocity leads to the instability of the magnetic field through the induction equation
If we consider Ω = (2πLT)2 × (−l, m), to the derivation of (3.71) in [35], we can show the stability of the velocity for any classical solution of the linearized problem satisfying boundary conditions u|x3=−l = u|x3=m = 0, provided the vertical steady magnetic field is sufficiently large
Summary
This paper is concerned with nonlinear instability of a smooth Rayleigh-Taylor (RT) steady-state solution to the following three-dimensional (3D) nonhomogeneous incompressible magnetohydrodynamic (MHD) equations with zero resistivity (i.e. without magnetic diffusivity) in the presence of a uniform gravitational field (see, for example, [2, 28, 29, 31] on the derivation of the equations): 2010 Mathematics Subject Classification. If we consider Ω = (2πLT)2 × (−l, m), to the derivation of (3.71) in [35], we can show the stability of the velocity for any classical solution of the linearized problem satisfying boundary conditions u|x3=−l = u|x3=m = 0, provided the vertical steady magnetic field is sufficiently large.
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