Communications on Pure and Applied Analysis (CPAA)

Global solutions to the incompressible magnetohydrodynamic equations

Pages: 763 - 783, Volume 11, Issue 2, March 2012      doi:10.3934/cpaa.2012.11.763

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Xiaoli Li - Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China (email)
Dehua Wang - Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, United States (email)

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.

Keywords:  Magnetohydrodynamic (MHD), incompressible flow, existence and uniqueness, strong solutions, weak-strong uniqueness.
Mathematics Subject Classification:  Primary: 35Q36, 35D05; Secondary: 76W05.

Received: August 2010;      Revised: June 2011;      Available Online: October 2011.