A coupled parabolic-elliptic system arising in the theory of magnetic relaxation

A coupled parabolic-elliptic system arising in the theory of magnetic relaxation

Thu, 30/05
12:00 		James Robinson (University of Warwick) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
A coupled parabolic-elliptic system arising in the theory of magnetic relaxation
    In 1985 Moffatt suggested that stationary flows of the 3D Euler equations with non-trivial topology could be obtained as the time-asymptotic limits of certain solutions of the equations of magnetohydrodynamics. Heuristic arguments also suggest that the same is true of the system
    \[ -\Delta u+\nabla p=(B\cdot\nabla)B\qquad\nabla\cdot u=0\qquad \]
    \[ B_t-\eta\Delta B+(u\cdot\nabla)B=(B\cdot\nabla)u \]
    when $ \eta=0 $. In this talk I will discuss well posedness of this coupled elliptic-parabolic equation in the two-dimensional case when $ B(0)\in L^2 $ and $ \eta>0 $.
    Crucial to the analysis is a strengthened version of the 2D Ladyzhenskaya inequality: $ \|f\|_{L^4}\le c\|f\|_{L^{2,\infty}}^{1/2}\|\nabla f\|_{L^2}^{1/2} $, where $ L^{2,\infty} $ is the weak $ L^2 $ space. I will also discuss the problems that arise in the case $ \eta=0 $.
    This is joint work with David McCormick and Jose Rodrigo.