Mathematical Sciences: Nonlinear Evolution Equations

数学科学：非线性演化方程

• 批准号: 9531834
• 负责人: Gustavo Ponce
• 金额: --
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9531834&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Nonlinear; Evolution; Equations

Abstract Ponce The project is devoted to research into various aspect of nonlinear evolution equations. They are at the intersection of partial differential equations, classical harmonic analysis and mathematical physics. The principal areas include the study of several dispersive systems arising as approximate models in wave propagation problems. The emphasis is on the development of arguments and techniques, mainly from Fourier analysis, which provide a better description of the behavior of the solutions of such models.The second part of the project is devoted to the study of problems related to viscous flows. They include the long time behavior of solutions to the 2-D density dependent Navier-Stokes equations. Also the system for compressible ideal fluids under minimal regularity of the data is studied. In concrete problems the validity of these models as a good approximation of the physical phenomenon depends on specific properties of their solutions, mainly those describing their long time behavior. Thus, the qualitative features of solutions obtained through this research provide a benchmark against which numerical approximation schemes can be tested. This applied analysis of nonlinear partial differential equations is an equal partner with their numerical study and the modelling of the physical phenomena, all of which interact to give a full mathematical picture.

摘要 Ponce 该项目致力于研究非线性演化方程的各个方面。它们处于偏微分方程、经典调和分析和数学物理的交叉点。主要领域包括研究作为波传播问题的近似模型出现的几种色散系统。重点是论据和技术的发展，主要来自傅立叶分析，它们可以更好地描述此类模型的解的行为。该项目的第二部分致力于研究与粘性流相关的问题。它们包括二维密度相关纳维-斯托克斯方程解的长期行为。还研究了数据最小规律性下的可压缩理想流体系统。在具体问题中，这些模型作为物理现象的良好近似的有效性取决于其解决方案的特定属性，主要是那些描述其长期行为的属性。因此，通过本研究获得的解的定性特征提供了测试数值逼近方案的基准。非线性偏微分方程的应用分析与数值研究和物理现象建模是同等的伙伴，所有这些相互作用共同给出完整的数学图景。