On the global dynamics of three dimensional imcompressible magnetohydrodynamics

Monday, October 26, 2015 -
3:15pm to 4:30pm
We construct and study global solutions for the 3-dimensional imcompressible MHD systems with arbitrary small viscosity. In particular, we provide a rigorous justification for the following dynamical phenomenon observed in many contexts: the solution at the beginning behave like non-dispersive waves and the shape of the solution persists for a very long time (proportional to the Reynolds number); thereafter, the solution will be damped due to the long-time accumulation of the diffusive effects; eventually, the total energy of the system becomes extremely small compared to the viscosity so that the diffusion takes over and the solution afterwards decays fast in time. We do not assume any symmetry condition. The size of data and the a priori estimates do not depend on viscosity. The proof is built upon a novel use of the basic energy identity and a geometric study of the characteristic hypersurfaces.  The approach is partly inspired by Christodoulou-Klainerman's proof of the nonlinear stability of Minkowski space in general relativity. This is a joint work with Ling-Bing HE (Tsinghua University) and Li XU (Chinese Academy of Sciences).
Speaker: 
Pin Yu
YMSC, Tsinghua University & CMSA, Harvard Univesity
Event Location: 
Fine Hall 110