Topics in Reaction-Diffusion and Fluid Mechanics

主题在反应扩散和流体力学摘要建议researchAlexander Kiselev研究人员打算研究各种数学问题，出现在流体流动的建模和流动和扩散过程的相互作用。它们涉及动力系统，谱理论和某些偏微分方程解的存在性和正则性问题。一个问题是通过研究某些对流扩散偏微分方程来理解流体流动对扩散的增强。PI将研究流动的几何和动力学特性与其增强扩散的效率之间的联系。另一个主题是耗散准地转和Navier-Stokes方程的存在性和性质。特别感兴趣的是这些方程的解的一些新的不等式的推导和非线性项在流体力学的偏微分方程中的影响的调查，要研究的方程模型重要的物理过程，但其数学性质目前还没有得到很好的理解。它们包括燃烧，大气和海洋建模，空气动力学和天体物理学以及许多生物模型，包括形态发生的问题。希望这些结果将有助于更好地理解科学和工程中的基本问题，从前沿形成和传播到湍流。

Topics in Reaction-Diffusion and Fluid MechanicsAbstract of Proposed ResearchAlexander Kiselev The investigator intends to study a variety of mathematical questions that arise in the modeling of fluid flows and the interaction of flows and diffusion processes. They involve issues in dynamical systems, spectral theory and the existence and regularity of solutions of certain partial differential equations. One problem is to understand the enhancement of diffusion by fluid flow by studying certain advection-diffusion partial differential equations. The PI will investigate the link between the geometric and dynamical properties of flows and their efficiency in enhancing diffusion. Another topic is the existence and properties of the dissipative quasi-geostrophic and Navier-Stokes equations. A particular interest is the derivation of some new inequalities for the solutions of these equations and the investigation of the effects of nonlinear terms in the partial differential equations of fluid mechanics.The equations to be studied model important physical processes yet their mathematical properties are not well understood at present. They include issues in combustion, atmospheric and ocean modeling, aerodynamics and astrophysics as well as many biological models including morphogenesis. It is hoped the results will lead to better understanding of fundamental issues in science and engineering ranging from front formation and propagation to turbulence.