Long Time Well-Posedness and Dynamics for the Plasma-Vacuum Interface Problem in Magnetohydrodynamics (MHD) Nonlinear Partial Differential Equations.
DescriptionIn this project, the PI proposes to study the long time well-posedness and dynamics for he plasma-vacuum interface problem of ideal incompressible MHD equations. The interface is a free surface which cannot be determined a priori, but is a part of solutions. The specific issues to be explored in this project include: 1) For the initial data being small perturbations of the background solution and satisfying the stability condition, determine whether the plasma-vacuum interface problem of ideal incompressible MHD equations is well-posed global-in-time in Sobolev spaces. If the global-in-time solution is not available, identify the life-span of the solutions in terms of the size of the initial perturbation; 2) determine the long time dynamics and behavior of solutions related to the celebrated Alfven wave of MHD equations, achieve some understanding of the interaction of Alfven-type waves and the surface waves of the interface. Compared with topic of the long time well-posedness and dynamics for fluid equations without taking into account of magnetic effects, few results are available for ideal MHD equations with free surfaces.The problems proposed to study in this project are challenging and difficult. They involve complicated wave interactions such as the interaction of Alfven-type waves and the surface waves of the interface, an issue not addressed in literature. New analytic andgeometric ideas and techniques need to be developed to deal with those issues. Besides dealing with the evolution of the free interface, ideas and techniques of higher order hyperbolic weighted quasilinear energy energy estimates in the spirit of those used inthe proof of global nonlinear stability of Minkowski space-time for vacuum Einstein equations of Christodoulou and Klainerman in domains with free surfaces will be developed in this project.
|Effective start/end date||1/01/21 → …|