This project is concerned with the study of interface problems arising in fluid mechanics and nonlinear wave equations with emphasis given to the study of singular limits of solutions to these problems which include 1) free boundary problems arising in inviscid fluid flows with and without surface tension; 2) free boundary problems arising in magneto-hydro-dynamics; and 3) existence and regularity of minimal surfaces in Minkowski spaces viewed as sharp interfaces of singular limits of solutions to nonlinear wave equation with singular potentials. A common thread of these problems is that they have a variational formulation which has a geometric interpretation. By focusing on the relationship between the analytical aspects and geometric aspects of these nonlinear hyperbolic PDEs, the PI will develop new methods and techniques to systematically obtain energy estimates, free boundary regularity bounds, and well-posedness. The research will be based on the geometric methods that arise either from variational principles or from the geometric nature of the problems and analytical methods that arise in studying existence, uniqueness, regularity, and long-time behavior of solutions to nonlinear PDEs.
These problems are prototypical for the study of free boundary regularity and rapidly-oscillating solutions in hyperbolic partial differential equations. Their resolution should be an important contribution to our understanding of this area. They are a cohesive set of problems that will require new methods and techniques that may well be useful in other area of PDEs. Fluid interface problems arise in many physical, medical, and engineering models. Problems involving fluid-vacuum interfaces arise in the study of water waves and astrophysics including the shape of stars. Problems involving fluid-fluid interfaces arise in multi-phase fluid flows while problems involving fluid-deformable structure interfaces arise in biomedical modeling such as cell deformation. Interface problems in magneto-hydro-dynamics are central to the theoretical and practical study of producing energy by fusion. The free boundary problems to be studied in this project may shed light on the instabilities present in the magnetic fields used for confining plasmas. We will also study the interface between hot and cool plasmas, i.e., the transport barrier. A systematic mathematical analysis of these problems should help us understand how various types of instabilities originate. The identification of ill-posed problems should also be very beneficial for modeling considerations and applications.
