Nonlinear Rayleigh-Taylor instability for nonhomogeneous incompressible viscous magnetohydrodynamic flows

Fei  Jiang; Song Jiang; Weiwei  Wang

doi:10.3934/dcdss.2016076

Article Contents

2016, Volume 9, Issue 6: 1853-1898. Doi: 10.3934/dcdss.2016076

This issue Previous Article Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$ Next Article A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$

Nonlinear Rayleigh-Taylor instability for nonhomogeneous incompressible viscous magnetohydrodynamic flows

1.
College of Mathematics and Computer Science, Fuzhou University, Fuzhou 360108
2.
Institute of Applied Physics and Computational Mathematics, P.O.Box 8009-28, Beijing 100088

Received: February 28, 2015

Revised: July 31, 2016

Published: November 2016

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Abstract

We investigate the nonlinear instability of a smooth Rayleigh-Taylor steady-state solution (including the case of heavier density with increasing height) to the three-dimensional incompressible nonhomogeneous magnetohydrodynamic (MHD) equations of zero resistivity in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady-state solution. Then we construct solutions of the linearized problem that grow in time in the Sobolev space $H^k$, thus leading to the linear instability. With the help of the constructed unstable solutions of the linearized problem and a local well-posedness result of smooth solutions to the original nonlinear problem, we establish the instability of the density, the horizontal and vertical velocities in the nonlinear problem. Moreover, when the steady magnetic field is vertical and small, we prove the instability of the magnetic field. This verifies the physical phenomenon: instability of the velocity leads to the instability of the magnetic field through the induction equation.

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Mathematics Subject Classification: Primary: 76E25; Secondary: 35E15.

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