Nonlinear thermal instability in the magnetohydrodynamics problem without heat conductivity

Abstract

We investigate the nonlinear thermal instability of the magnetohydrodynamic problem for a full compressible viscous fluid with zero resistivity and zero heat conductivity in the presence of a uniform gravitational force in a bounded domain Ω∈R3. We establish that under some instability conditions, the equilibrium-state is linearly unstable by constructing a suitable energy functional and employing the modified variational method. Then, on the basis of the constructed linearly unstable solutions and the local well-posedness of classical solutions to the original nonlinear problem, we reconstruct the initial data of linearly unstable solutions to be the one of the original nonlinear problem and obtain an appropriate energy estimate of Gronwall-type. Finally, we establish that the equilibrium-state is nonlinearly unstable in view of the energy estimate by a bootstrap method.