Mathematical Problems in Nonlinear Conservation Laws and Related Applications

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The investigator studies mathematical problems involvingshock waves in nonlinear conservation laws and related applications,along with the analysis and development of efficient new nonlineartechniques. Shock waves often occur in solutions of nonlinearconservation laws of hyperbolic type, mixed elliptic-hyperbolic type,and mixed parabolic-hyperbolic type. The proposed work is in threeinterrelated topics corresponding to the nonlinear conservation lawsof three types:(1) multidimensional transonic shocks, free boundary problems, andrelated nonlinear problems;(2) divergence-measure fields for entropy solutions with shocks tohyperbolic conservation laws;(3) solutions with shocks to anisotropic degeneratediffusion-convection equations and related nonlinear problems.In each, both nonlinear problems involving shock waves and newmathematical techniques are proposed. The unifying mathematical themeof the three topics is shock wave problems and related nonlineartechniques. The objective of this proposed program is twofold:(1) investigate important nonlinear problems involving shockwaves to gain new physical insights, to guide the formulation ofefficient nonlinear techniques, and to find the correct mathematicalframeworks in which to pose the nonlinear conservation laws anddevelop the numerical methods that converge stably and rapidly;(2) analyze and develop nonlinear techniques including free boundarymethods, kinetic methods, compensated regularity methods, and relatedpotential techniques to formulate new, more efficient nonlineartechniques and to solve various more important nonlinear problemsin conservation laws and related applications. The mathematical problems in this research program arise in suchareas 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 thesemathematical problems involving shock waves in nonlinear conservationlaws and related applications, the qualitative behavior of theirsolutions, as well as the analysis and development of nonlinearmethods in applied analysis and numerical analysis.This research will lead to a deeper understanding of nonlinearphenomena, especially for shock waves, and will provide moreefficient nonlinear methods and theories for applications.Given the richness and the range of applications for whichshock waves are involved in nature, this project will havea broad impact by understanding the behavior of shock waves anddeveloping methodology and a set of nonlinear techniquesfor their further study.