Nonlinear Free Boundary and Evolution Problems

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

Free boundary problems arise in many models in physics, engineering, fluid dynamics, and economics. Free boundaries correspond to sharp changes in the variables describing the problem. While significant progress has been made in the study of free boundary problems, many important questions have yet to be studied in the case of nonlinear partial differential equations and especially equations of mixed type. For these cases, a better understanding of properties of free boundaries, such as stability, regularity and geometric structure, would make it possible to study complex phenomena in real-world applications. The primary focus of this project is the study of shock reflection problems in gas dynamics, one of the most fundamental multidimensional shock wave problems. The project will also have a broad impact through close interactions with engineering and meteorological communities and through training of graduate students. The project consists of two main topics: (1) Free boundary problems in shock analysis. The PI will continue his work on self-similar shock reflection for potential flow and for full and isentropic Euler system. Shock reflection problems arise in many physical situations. Moreover, such problems are important in the mathematical theory of multidimensional conservation laws since their solutions are building blocks and asymptotic attractors of general solutions to the multidimensional Euler equations for compressible fluids. Self-similar equations of compressible fluid dynamics are of mixed elliptic-hyperbolic type. Shocks correspond to discontinuities in the solution for Euler system, and in the gradient of the solution for potential flow equation. Type of equation may change from hyperbolic to elliptic across the shock. Shock reflection problem can be formulated as a free boundary problem in which unknown are the elliptic (subsonic) region and solution in the elliptic region. The PI will continue his work on existence, stability and regularity of global solutions of the regular reflection, to extend the global existence results to the case of compressible Euler system, which is a fundamental model of gas dynamics. Further study includes uniqueness and stability for regular reflection problem in various classes of solutions. (2) Another area of the proposed research is semi-geostrophic system. The PI will study semi-geostrophic system with variable Coriolis parameter. Such model arises from taking into account the curvature of the Earth. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.