Well-posedness of the Navier - Stokes - Maxwell equations

Pierre Germain, Slim Ibrahim, Nader Masmoudi

Research output: Contribution to journalArticlepeer-review


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 priori L t 2 (L x ) estimate for solutions of the forced Navier - Stokes equations.

Original languageEnglish (US)
Pages (from-to)71-86
Number of pages16
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
Issue number1
StatePublished - Feb 2014

ASJC Scopus subject areas

  • General Mathematics


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