Nonlinear Stability and Instablity in Rayleight--Taylor Problem of Stratisfied Compressible MHD Fluids (1702.07529v1)

Abstract: We establish criteria of stability and instability for the stratified compressible magnetic Rayleigh--Taylor (RT) problem. More precisely, if under the stability condition $\Xi <1$, we show the existence of unique solution with algebraic decay in time for the (compressible) magnetic RT problem with proper initial data in Lagrangian coordinates. The stability result presents that sufficiently large vertical (base) magnetic field can inhibit the development of RT instability. On the other hand, if $\Xi >1$, there exists an unstable solution to the magnetic RT problem in the Hadamard sense. This shows that the RT instability still occurs when the strength of base magnetic field is small or the base magnetic field is horizontal with proper large horizontal period cell. Moreover, by analyzing the stability condition in magnetic RT problem for vertical magnetic fields, we can observe that the compressibility destroys the stabilizing effect of magnetic fields in the vertical direction. Fortunately, the instability in vertical direction can be inhibited by the stabilizing effect of pressure, which also plays an important role in the mathematical proof for stability of the magnetic RT problem. In addition, we will extend the results in magnetic RT problem to the (compressible) viscoelastic RT problem, and find that the stabilizing effect of elasticity is stronger than the one of magnetic fields.