Mathematical Sciences: Aspects of Compressible and Incompressible Flows

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The principal investigator will study certain qualitative properties of incompressible viscous fluids, compressible fluids and mixtures of fluids by using techniques of modern functional analysis and numerical analysis. In particular, she will study the long-time asymptotic behavior of solutions of the Navier-Stokes equations in two and three space dimensions, the equations of incompressible magnetohydrodynamics in three dimensions, and some integro-differential equations that model compressible flows. She will also investigate the global existence in time of solutions of a set of multiphase equations. Very often one is interested in the behavior of a physical system only after a long time. For example, once a process is started one might expect that a single type of stable behavior is the dominant observable output. Deviations from this stable behavior might be very much unwanted. The principal investigator will devote most of her efforts to study how various real-world fluid systems behave for a very long time by using modern techniques from applied mathematics and numerical analysis.