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Well-posedness of the Navier—Stokes—Maxwell equations

Published online by Cambridge University Press:  30 January 2014

Pierre Germain ,
Slim Ibrahim  and
Nader Masmoudi
Pierre Germain
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1185, USA (pgermain@cims.nyu.edu)
Slim Ibrahim
Affiliation:
Department of Mathematics and Statistics, University of Victoria, PO Box 3060, Station CSC, Victoria, BC V8P 5C3, Canada (ibrahim@math.uvic.ca)
Nader Masmoudi
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1185, USA (masmoudi@cims.nyu.edu)
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Abstract

We study the local and global well-posedness of a full system of magnetohydrodynamic equations. The system is a coupling of the incompressible Navier—Stokes equations with the Maxwell equations through the Lorentz force and Ohm's law for the current. We show the local existence of mild solutions for arbitrarily large data in a space similar to the scale-invariant spaces classically used for Navier—Stokes. These solutions are global if the initial data are small enough. Our results not only simplify and unify the proofs for the space dimensions 2 and 3, but also refine those in [8]. The main simplification comes from an a prioriLt2 (Lx∞) estimate for solutions of the forced Navier—Stokes equations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 144 , Issue 1 , February 2014 , pp. 71 - 86
Copyright
Copyright © Royal Society of Edinburgh 2014 

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