Mathematical Problems in Nonlinear Conservation Laws and Related Applications

研究人员研究非线性守恒律中涉及激波的数学问题和相关应用，以及有效的新的非线性技术的分析和发展。激波常出现在双曲型、椭圆-双曲型和抛物型-双曲型非线性守恒律的解中。我们的工作分为三个主题，分别对应于三类非线性守恒律：(1)多维跨音速激波、自由边界问题及相关的非线性问题；(2)双曲型守恒律中含激波的熵解的散度场；(3)各向异性退化扩散-对流方程及相关非线性问题的激波解。这三个主题的统一数学主题是激波问题和相关的非线性技术。该计划的目标有两个：(1)研究涉及激波的重要的非线性问题，以获得新的物理见解，指导有效的非线性技术的制定，并找到提出非线性守恒定律的正确的数学框架，并开发稳定且快速收敛的数值方法；(2)分析和发展非线性技术，包括自由边界方法、动力学方法、补偿正则性方法和相关势能技术，以形成新的、更有效的非线性技术，并解决守恒定律及其相关应用中的各种更重要的非线性问题。该研究项目涉及的数学问题包括：气体动力学、水力学、弹性力学、多相流、燃烧、磁流体动力学、半导体、相变、动力学、生物物理、沉积-固结过程、材料科学和图像处理。该奖项将支持关于非线性守恒律中涉及激波的数学问题的可解性及其相关应用的研究，以及应用分析和数值分析中非线性方法的分析和发展。这项研究将使人们对非线性现象，特别是激波，有更深入的了解。鉴于冲击波在自然界中的丰富性和应用范围，了解冲击波的行为并为其进一步研究发展方法论和一套非线性技术将产生广泛的影响。

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.