On the global dynamics of three dimensional imcompressible magnetohydrodynamics
On the global dynamics of three dimensional imcompressible magnetohydrodynamics
We construct and study global solutions for the 3-dimensionalÌýimcompressible MHD systems with arbitrary small viscosity. In particular,Ìýwe provide a rigorous justification for the following dynamical phenomenonÌýobserved in many contexts: the solution at the beginning behave likeÌýnon-dispersive waves and the shape of the solution persists for a very longÌýtime (proportional to the Reynolds number); thereafter, the solution willÌýbe damped due to the long-time accumulation of the diffusive effects;Ìýeventually, the total energy of the system becomes extremely small comparedÌýto the viscosity so that the diffusion takes over and the solutionÌýafterwards decays fast in time.ÌýWe do not assume any symmetry condition. The size of data and the a prioriÌýestimates do not depend on viscosity. The proof is built upon a novel useÌýof the basic energy identity and a geometric study of the characteristicÌýhypersurfaces.Ìý The approach is partly inspired byÌýChristodoulou-Klainerman's proof of the nonlinear stability of MinkowskiÌýspace in general relativity.ÌýThis is a joint work with Ling-Bing HE (Tsinghua University) and Li XUÌý(Chinese Academy of Sciences).