Dynamics and Stable Structures in Some Nonlinear PDEs

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The goal of this project is to study four stable structures arising in various physical phenomena, modeled by nonlinear partial differential equations:(i) Traveling waves in the Boussinesq model of reactive flows: The Boussinesq system is the simplest system of equations exhibiting behavior of premixed flames in a gravitationally stratified medium.(ii) Attractors to Navier-Stokes equations in thin three-dimensional domains: The working condition of previous research in the area has been that the limiting geometry, as the thickness of the domain vanishes, is flat. This project will investigate the technically more involved case of non-flat limit geometries. This investigation is partially motivated by adapting the model to applications, e.g. in oceanography.(iii) Self-similar, singular solutions to the complex Ginzburg-Landau equation: This equation describes a variety of phenomena, from nonlinear waves to second-order phase transitions. Interest also stems from analogies with the three-dimensional Navier-Stokes equation and the three-dimensional supercritical nonlinear Schrodinger equation.(iv) Rarefaction wave solutions to strictly hyperbolic systems of conservation laws with large data (following on results of NSF grant DMS-0306201).The stability of patterns arising as solutions to equations of mathematical physics, notably related to fluid or gas dynamics, is of central interest to scientists and engineers. The stable patterns are those expected to be observed in experiments. They may be continuous waves, jumps (for example in the density of the studied quantities), or other singularities. Analysis of unstable patterns, solutions of the equations that are non-observable physically, gives important insight into the time evolution of the observed ones. This project analyzes patterns in solutions of several important systems of equations. The applications range from meteorology, blood circulation, lubrication, and combustion in gases, to studies of phase transition phenomena such as super-conductivity, super-fluidity, and liquid crystals.