Mathematical Structure in Fluid Mechanics

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VasseurDms-0607953 The investigator carries out a qualitative study of partialdifferential equations arising as models in fluid mechanics andchemistry. Understanding the properties of these models is afundamental challenge both in mathematics and science. Theregularity and stability of solutions are issues strongly relatedto nontrivial physical phenomena like turbulence and phasetransitions that are complex multiscale problems. Analyzingmathematical models of these phenomena often requires thedevelopment of novel tools encompassing techniques from analysis,applied mathematics, and stochastics. This project, whichaddresses both theoretical and applied issues, spans fromconservation laws to Navier-Stokes equations, kinetic equations,and motion of particles in fluids. It requires tools from thetheories of nonlinear partial differential equations, asymptoticanalysis, probability, and stochastics. The discovery of mathematical structure in physical modelsis of great importance for a deeper understanding of the physicalphenomena themselves. It gives also some hints of the range ofvalidity of the models. In this project, the investigatordevelops and studies new models involving couplings between fluidmechanics and physics far from equilibrium. This field hasimportant applications in science and engineering. For example,it models pollutants in a river or combustion of gasoline in anengine. The applied and mathematical challenges stem from thecomplexity of the phenomena.