Nonlinear convective instability in the compressible magnetic convection problem without heat conductivity

We investigate the convective instability problem for a full compressible viscous magnetohydrodynamic (MHD) fluid with zero resistivity and zero heat conductivity in the presence of a uniform gravitational force in a bounded domain Ω∈R3. First, we establish an instability criterion for the magnetic convection problem, and construct unstable solutions of linearized magnetic convection problem that grow in time in the Sobolev space H3. Then, based on the linearly unstable solutions, we further modify the initial data of the linearly unstable solutions to be ones of the original magnetic convection problem. Finally, using the local well-posedness of classical solutions to the original magnetic convection problem, and a modified bootstrap instability method, we can construct unstable solutions for the original magnetic convection problem in the sense of Hadamard under the instability criterion.