Mathematical Problems in Nonlinear Conservation Laws and Related Applications

研究涉及激波非线性守恒定律的数学问题及其应用，同时分析和开发高效的新非线性技术。激波常出现在双曲型、混合椭圆-双曲型和混合抛物-双曲型非线性守恒律的解中。提出的工作是在三个相互关联的主题对应的非线性守恒定律的三种类型：(1)多维跨音速冲击，自由边界问题，和相关的非线性问题；(2)具有双曲守恒律激波的熵解的散度测量场；(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。在每一个，都提出了涉及激波的非线性问题和新的数学技术。这三个主题的统一的数学主题是冲击波问题和相关的非线性技术。该计划的目标是双重的：(1)研究涉及冲击波的重要非线性问题，以获得新的物理见解，指导有效非线性技术的制定，并找到正确的数学框架，在其中提出非线性守恒定律和发展稳定和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的势能技术，以制定新的、更有效的非线性技术，解决守恒定律和相关应用中的各种更重要的非线性问题。本研究项目涉及的数学问题涉及气体动力学、水力学、弹性、多相流、燃烧、磁流体力学、半导体、相变、动力学理论、生物物理学、沉积-固结过程、材料科学和图像处理等领域。该奖项将支持研究涉及激波非线性守恒定律及其相关应用的数学问题的可解性，其解的定性行为，以及应用分析和数值分析中非线性方法的分析和发展。这一研究将导致对非线性现象，特别是激波的更深入的理解，并将为应用提供更有效的非线性方法和理论。鉴于激波在自然界中所涉及的应用的丰富性和范围，本项目将通过理解激波的行为、开发方法和一套非线性技术来进一步研究激波，从而产生广泛的影响。

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.