Mathematical Sciences: Microlocal Analysis and Nonlinear Waves

数学科学：微局域分析和非线性波

• 批准号: 9401248
• 负责人: Mark Williams
• 金额: $ 4.5万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-15 至 1996-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401248&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Microlocal; Analysis; Nonlinear; Mathematical; Sciences; Microlocal; Analysis; Nonlinear

9401248 Williams This award supports mathematical research on problems arising in the theory and application of nonlinear partialdifferential equations. Particular emphasis is placed on the use of microlocal methods in the study of hyperbolic equations, developing methods for understanding diffraction of singularities by smooth obstacles in several types of nonlinear wave problems, resonant reflection and transmission of continuous multidimensional oscillations. Work will also be done on nonlinear optics for multidimensional shocks. A major goal of the project is to introduce the consideration of boundaries into rigorous nonlinear geometrical optics, which has until now dealt mostly with problems in free space. The importance of boundaries is indicated, for example, by the fact that multidimensional shock front and vortex sheet problems can be formulated as nonlinear hyperbolic free boundary problems. Also, boundaries can produce striking effects (such as resonant reflection) even in continuous solutions. 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. ***

9401248 WILLIAMS该奖项支持有关非线性partialDifferentifentifentic方程的理论和应用中引起的问题的数学研究。 特别强调在双曲方程研究中使用微局部方法，开发了通过几种非线性波问题的平滑障碍，共同反射和连续多维振荡的传播来理解奇异性的衍射方法。 对于多维冲击的非线性光学，还将完成工作。 该项目的一个主要目标是将考虑边界的考虑到严格的非线性几何光学元件中，该光学元素必须迄今主要处理自由空间中的问题。 例如，通过以下事实表明了边界的重要性，即可以将多维冲击锋和涡流板问题作为非线性双曲线自由边界问题提出。 同样，即使在连续的溶液中，边界也会产生惊人的效果（例如谐振反射）。 部分微分方程构成了物理世界数学建模的基础。 数学分析的作用并不是创建方程，而是提供有关解决方案的定性和定量信息。 这可能包括有关独特性，光滑和成长的问题的答案。 此外，分析通常会开发出解决这些近似值准确性的解决方案和估计值的方法。 ***