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 . The main simplification comes from an a priori L t 2 (L x ∞) estimate for solutions of the forced Navier - Stokes equations.
|Original language||English (US)|
|Number of pages||16|
|Journal||Proceedings of the Royal Society of Edinburgh Section A: Mathematics|
|State||Published - Feb 1 2014|
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