Nonlinear Waves

New York University, New York NY

Investigators

Abstract

Nonlinear Waves Abstract of Proposed Research Jalal Shatah This project will develop new geometric and analytical methods for systematically obtaining energy estimates and proving regularity of solutions for nonlinear hyperbolic equations. Geometric methods are ones that arise either from variational principles or from the geometric nature of the problems. Analytical methods are ones that arise in studying existence, uniqueness, regularity, and long-time behavior of solutions to nonlinear hyperbolic equations. The project will also involve the study of singular limits of solutions to parameter dependent equations. We shall concentrate on the analysis of free boundary value problems arising in fluid flows, with and without surface tension, on problems in magneto-hydro-dynamics and on problems arising from the study of minimal surfaces in Minkowski space. The three major goals of this project are: 1. to identify geometric and physical quantities whose control is sufficient to prevent the waves from breaking down and ensure global regularity of solutions; 2. to derive geometric and analytical criteria for stability of fluid interfaces; 3. to investigate the transition from Kelvin-Helmholtz instability to Rayleigh-Taylor instability; 4. to study the transition from compressible to incompressible interface problems; 5. to study the regularity of the free boundary for plasmas surrounded by a vacuum and the interface problem between hot and cool plasmas; and 6. to develop new techniques of deriving asymptotic equations for small-amplitude, high-frequency perturbations for singularly perturbed wave equations. These problems are prototypical for the study of free boundary regularity and rapidly-oscillating solutions in hyperbolic 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.