Topics in Reaction-Diffusion and Fluid Mechanics

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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.