Higher Order Regularity, Long Term Dynamics, and Data Assimilation for Magnetohydrodynamic Flows

摘要

First we consider various inviscid equations of fluid dynamics and show that if the initial data is analytic in the space variables, then the resulting flows extend as analytic functions of both space and time variables, with explicit estimates of the analyticity radius. We then consider higher order regularity of the viscous magnetohydrodynamic equations for an incompressible conductive fluid. We establish the Gevrey regularity of solutions when the initial data is in a Sobolev class, of possibly negative order, in two and three spatial dimensions. In particular, we show that solutions evolving from singular initial data instantaneously become analytic, with the analyticity radius eventually expanding in time. This in turn allows us to establish decay in higher order Sobolev norms. Finally, using a recently developed data assimilation algorithm based on linear feedback control, we show that when the initial data is unknown, sparse measurement data is sufficient for accurate reconstruction of magnetohydrodynamic flows. This algorithm convergences exponentially in time to the reference solution and moreover, the reconstruction is exact on the attractor.