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We establish criteria of stability and instability for the stratifiedcompressible magnetic Rayleigh--Taylor (RT) problem. More precisely, if underthe stability condition $\Xi <1$, we show the existence of unique solution withalgebraic decay in time for the (compressible) magnetic RT problem with properinitial data in Lagrangian coordinates. The stability result presents thatsufficiently large vertical (base) magnetic field can inhibit the developmentof RT instability. On the other hand, if $\Xi >1$, there exists an unstablesolution to the magnetic RT problem in the Hadamard sense. This shows that theRT instability still occurs when the strength of base magnetic field is smallor the base magnetic field is horizontal with proper large horizontal periodcell. Moreover, by analyzing the stability condition in magnetic RT problem forvertical magnetic fields, we can observe that the compressibility destroys thestabilizing effect of magnetic fields in the vertical direction. Fortunately,the instability in vertical direction can be inhibited by the stabilizingeffect of pressure, which also plays an important role in the mathematicalproof for stability of the magnetic RT problem. In addition, we will extend theresults in magnetic RT problem to the (compressible) viscoelastic RT problem,and find that the stabilizing effect of elasticity is stronger than the one ofmagnetic fields.

Nonlinear Stability and Instablity in Rayleight--Taylor Problem of Stratisfied Compressible MHD Fluids