Seminar series
Date
Thu, 30 May 2013
12:00
12:00
Location
Gibson 1st Floor SR
Speaker
James Robinson
Organisation
University of Warwick
- 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 $\eta$ is positive.
- 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.