The investigator studies mathematical problems involving shock waves in nonlinear conservation laws and related applications, along with the analysis and development of efficient new nonlinear techniques. Shock waves often occur in solutions of nonlinear conservation laws of hyperbolic type, mixed elliptic-hyperbolic type, and mixed parabolic-hyperbolic type. The proposed work is in three interrelated topics corresponding to the nonlinear conservation laws of three types: (1) multidimensional transonic shocks, free boundary problems, and related nonlinear problems; (2) divergence-measure fields for entropy solutions with shocks to hyperbolic conservation laws; (3) solutions with shocks to anisotropic degenerate diffusion-convection equations and related nonlinear problems. In each, both nonlinear problems involving shock waves and new mathematical techniques are proposed. The unifying mathematical theme of the three topics is shock wave problems and related nonlinear techniques. The objective of this proposed program is twofold: (1) investigate important nonlinear problems involving shock waves to gain new physical insights, to guide the formulation of efficient nonlinear techniques, and to find the correct mathematical frameworks in which to pose the nonlinear conservation laws and develop the numerical methods that converge stably and rapidly; (2) analyze and develop nonlinear techniques including free boundary methods, kinetic methods, compensated regularity methods, and related potential techniques to formulate new, more efficient nonlinear techniques and to solve various more important nonlinear problems in conservation laws and related applications.
The mathematical problems in this research program arise in such areas as gas dynamics, hydraulics, elasticity, multiphase flow, combustion, magnetohydrodynamics, semiconductor, phase transitions, kinetic theory, biophysics, sedimentation-consolidation processes, material science, and image processing. The award will support research on the solvability of these mathematical problems involving shock waves in nonlinear conservation laws and related applications, the qualitative behavior of their solutions, as well as the analysis and development of nonlinear methods in applied analysis and numerical analysis. This research will lead to a deeper understanding of nonlinear phenomena, especially for shock waves, and will provide more efficient nonlinear methods and theories for applications. Given the richness and the range of applications for which shock waves are involved in nature, this project will have a broad impact by understanding the behavior of shock waves and developing methodology and a set of nonlinear techniques for their further study.
