Mathematical Sciences: Theory and Applications of Wave Front Propagation in Inhomogeneous Media

该奖项支持对非均匀介质中的非线性波的数学研究，重点关注在非线性偏微分方程式中出现的波前现象的性质。特别是，将讨论反应扩散方程和多孔介质类型的方程。这些方程来源于燃烧理论中的预混火焰传播问题和地下水流中的溶质运移问题。在小扩散快反应极限下，研究周期系数存在时的一般反应扩散前沿传播问题。从数值角度研究了二维反应扩散前沿猝灭问题，研究了准周期随机介质中反应扩散前沿传播的存在性。还将讨论均匀和非均匀介质中溶质运移的浓集锋面传播问题。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。

This award supports mathematical research on nonlinear waves inhomogeneous media, focusing on properties of wave front phenomena arising in nonlinear partial differential equations. In particular reaction-diffusion equations and porous medium type equations will be treated. These equations come from premixed flame propagation problems in combustion theory and solute transport problems in underground water flow. Work will be done investigating general reaction-diffusion front propagation in the presence of periodic coefficients under the small diffusion fast reaction limit. Also to be studied is the reaction-diffusion front quenching problem in two dimensions from a numerical point of view and investigations into the existence of reaction-diffusion front propagation in quasiperiodic and random media. The concentrate front propagation for solute transport in homogeneous and inhomogeneous media will also be taken up. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.