Mathematical Sciences: Nonlinear Evolution Equations

该项目致力于研究非线性演化方程的各个方面。它们是偏微分方程、经典调和分析和数学物理的交叉点。主要研究领域包括在波传播问题中作为近似模型出现的几种色散系统。重点是论点和技术的发展，主要来自傅立叶分析，它提供了更好的描述这些模型的解的行为。该项目的第二部分致力于研究与粘性流动有关的问题。它们包括二维密度相关的Navier-Stokes方程解的长时间行为。研究了数据最小规则下的可压缩理想流体系统。在具体问题中，这些模型是否能很好地近似物理现象，取决于其解的具体性质，主要是那些描述其长期行为的性质。因此，通过本研究获得的解的定性特征为数值近似方案的测试提供了一个基准。这种非线性偏微分方程的应用分析与他们的数值研究和物理现象的建模是平等的伙伴，所有这些相互作用，给出了一个完整的数学图像。

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.