Free Boundary Problems Concerning the Interaction of Elastic Shells with Viscous or Invicid Fluids, the Flows in Porous Media with Non-Uniform Porosity(1/2)

We study a series of theoretical problems in PDEs starting with the problem of finding adiffeomorphism, as well as the estimates of its time derivatives, between two H^k domain for k >n/2 with prescribed Jacobian and boundary data, where the Jacobian is assumed to betime-dependent. The existence of such a diffeomorphism enables us to study the well-posedness ofthe free boundary problems using the ALE formulation without destroying the volume preservingstructure. We will use such a diffeomorphism to study the interaction between fluids (viscous orinvicid) and elastic shells, the incompressible Euler equation without surface tension, and theMuskat problem with variable permeability.