Existence and Stability Analysis for Nonlinear Free Boundary and Evolution Problems

Free boundary problems arise in many models in physics, engineering, fluid dynamics, and economics. Free boundaries are regions of rapid variations of conditions between two very different states, such as shock waves in gas dynamics. Mathematically, this rapid transition is simplified as occurring infinitely fast along a surface of discontinuity in the partial differential equation governing the physics.

The location of this surface is not known in advance, thus one must solve both for physical states and their boundaries. Significant progress in the study of free boundary problems has been made during the last several decades. However, in the case of nonlinear partial differential equations, and especially equations of mixed type, many important questions are yet to be studied.

The principal investigator (PI) plans to apply the techniques of free boundary problems to study some fundamental multidimensional shock waves in gas dynamics, specifically shock reflection patterns. This involves free boundary problems for nonlinear equations and systems having a complex structure, and thus new methods need to be developed to handle such problems.

Understanding properties of free boundaries, such as regularity, stability and geometric properties, allows for a better analysis and numerical methods in models and applications. Another area of the project is the semigeostrophic system, a model of rotation-dominated atmospheric/ocean flows. It exhibits a rich mathematical structure based on Monge-Kantorovich mass transport theory.

The PI plans to continue the study of the physically realistic case of variable Coriolis parameter in the semigeostrophic model, and also study stability properties of solutions. The project addresses fundamental mathematical models in engineering and atmospheric sciences. Closer interaction with the engineering and meteorological communities is one of the priorities of the project. The project provides research training opportunities for graduate students.

The project consists of two main topics: (1) Free boundary problems in shock analysis. The PI will continue work on self-similar shock reflection for potential flow and for the 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 to the Euler system and in the gradient of the solution for potential flow equation.

The type of the equation may change from hyperbolic to elliptic across the shock. The shock reflection problem can be formulated as a free boundary problem in which the unknowns are the elliptic region and the solution in that region. The PI will continue work on the existence, stability, and regularity of global solutions to the regular reflection, to extend the global existence results to the case of compressible Euler system and three-dimensional reflection by a cone.

Further study includes stability for the regular reflection problem in various classes of solutions. (2) The study of the system of semigeostrophic equations, using methods from Monge-Kantorovich mass transport. The PI will study the semigeostrophic system with variable Coriolis parameter, which is a model that arises from taking into account the curvature of the Earth.

The PI also plans to continue the study of convergence of solutions of the Euler system to solutions of the semigeostrophic system using relative entropy methods.

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.