Global solutions to the  incompressible magnetohydrodynamic equations

Xiaoli Li; Dehua Wang

doi:10.3934/cpaa.2012.11.763

Article Contents

2012, Volume 11, Issue 2: 763-783. Doi: 10.3934/cpaa.2012.11.763

This issue Previous Article On the regularity of solutions to the Navier-Stokes equations Next Article Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings

Global solutions to the incompressible magnetohydrodynamic equations

Xiaoli Li^1, and
Dehua Wang^2,

1.
Department of Mathematical Sciences, Tsinghua University, Beijing 100084
2.
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260

Received: July 31, 2010

Revised: May 31, 2011

Published: February 2012

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Abstract

An initial-boundary value problem of the three-dimensional incompressible magnetohydrodynamic (MHD) equations is considered in a bounded domain. The homogeneous Dirichlet boundary condition is prescribed on the velocity, and the perfectly conducting wall condition is prescribed on the magnetic field. The existence and uniqueness is established for both the local strong solution with large initial data and the global strong solution with small initial data. Furthermore, the weak-strong uniqueness of solutions is also proved, which shows that the weak solution is equal to the strong solution with certain initial data.

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Mathematics Subject Classification: Primary: 35Q36, 35D05; Secondary: 76W05.

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