On A Free Boundary Problem of Compressible Euler-Monge-Ampere Equations

The PI proposes to study a free boundary problem of compressible Euler-Monge-Ampere (CEMA) equations in this project, which is a fully nonlinear counterpart of compressible Euler-Poisson equations of hydrodynamical model for plasma. The advantage to use this fully nonlinear model is that a nite electric led for point charges is allowed as a fully nonlinear model of electro-static interaction. In this project, the PI focuses on the local-in-time theory and proposes to study the following problems: 1) For the free boundary problem of CEMA system, identify suitable stability condition analogue to the Taylor sign condition for that of Euler equations, and derive the local-in-time a priori estimates of solutions under this condition, 2) For the free boundary problem of CEMA system, establish the local-in-time well-posedness in suitable Sobolev spaces under the identi ed stability condition. It is challenging to study uids and related free boundary problems arisingfrom uid dynamics and plasma sciences. The fully nonlinear nature of Monge-Ampere equation coupled with the uid type free boundary makes the study extremely challenging and interesting. The success of this project will set a basis to long time (global or almost global) well-posedness theory of the free boundary problem of compressible Euler-Monge-Ampere (CEMA) equations and also contributes to the develop-ment of analytic and geometric tools for the study of uids and related free boundary problems.