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Preprint 2011-014

On the Rayleigh–Taylor instability for incompressible, inviscid magnetohydrodynamic flows

Ran Duan, Fei Jiang and Song Jiang

Abstract: We study the Rayleigh–Taylor instabolity for two incompressible, immiscible, inviscid magnetohydrodynamic (MHD) fluids with zero resistivity, evolving with a free interface in the presence of a uniform gravitational field. We first construct the Rayleigh–Taylor steady-state solution with a denser fluid lying above the light one. Then, we turn to an analysis of the equations obtained from linearizing around such a steady state. By solving a system of ordinary differential equations, we construct the normal mode solutions to the linearized problem that grow exponentially in time. A Fourier synthesis of these normal mode solutions allows us to construct solutions that grow arbitrarily quickly in the Sobolev space $H^k$, thus leading to an ill-posedness result for the linearized problem in the sense of Hadamard. Using these pathological solutions, we can then demonstrate the ill-posedness of the original non-linear problem in some sense.

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Ran Duan
Fei Jiang
Song Jiang
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; 2011-07-27.