INTERPOLATION AND LADYZHENSKAYA INEQUALITY IN A COUPLED ELLIPITIC-PARABOLIC PROBLEM

In 1985 Moffatt proposed the method of magnetic relaxation to construct stationary Euler flows with non-trivial topology. The idea is to take an initial magnetic field and let it evolve under the dynamics of the MHD system with zero viscosity; the time asymptotic limit of the magnetic field should then yield a function that satisfies the stationary Euler equations.

Since the MHD equations are only used to produce the limiting field, the problem can be changed in a number of ways and still (at least heuristically) provide a stationary Euler flow.

In this talk I will discuss a simplification of the system in which the fluid evolution is replaced by an elliptic equation. Our aim is to show that the resulting system is locally well-posed. Despite the apparent simplicity of the equations it turns out that this requires results that are at the limit of what is available - elliptic regularity in $L^1$, the limiting case of the Aubin-Lions compactness theorem, and a strengthened form of the Ladyzehnskaya inequality derived using the theory of interpolation.