Analysis Seminar

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On the Global Dynamics of Three Dimensional Incompressible Magnetohydrodynamics

Speaker: Pin Yu, YMSC, Tsinghua University and CMSA, Harvard Univesity

Location: Warren Weaver Hall 1302

Date: Thursday, November 19, 2015, 11 a.m.

Synopsis:

We construct and study global solutions for the 3-dimensional incompressible 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).