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Abstract
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We shall investigate the stabilizing effect of impressed (equilibrium) magnetic fields in the Rayleigh-Taylor (RT) problem for both incompressible nonhomogeneous and compressible stratified viscous magnetohydrodynamic (MHD) fluids of zero resistivity in the presence of a uniform gravitational field in a horizontally periodic domain, in which the velocity is non-slip on both upper and lower flat boundaries. When an initial perturbation around an impressed magnetic RT equilibrium state satisfies some relations, and the strength of the impressed (vertial or horizontal) magnetic field is bigger than the critical number, we can use the Bogovskii function in the standing-wave form and adapt a two-tier/three-tier energy method in Lagrangian coordinates to show the existence of a unique global-in-time (perturbed) stability solution to the magnetic RT problem. For the case that the strength of the impressed magnetic field is smaller than the critical number, by developing new analysis techniques based on the method of bootstrap instability, we show that the nonlinear RT instability will occur. The current result reveals from the mathematical point of view that a sufficiently strong vertical impressed magnetic field has a stabilizing effect and can prevent the RT instability in MHD flows from occurring. Similar conclusions can be also verified for the horizontal magnetic field when the domain is vertically periodic. (joint work with Fei Jiang)
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