This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The research focuses on the mathematical analysis of the evolutionary solutions to the equations from gas dynamics and fluid flow. In gas dynamics the goal is to gain a deeper understanding of relativistic kinetic equations coupled with their internally generated fully Lorentz invariant forces. Attention will be focused on questions of existence, uniqueness, regularity and decay properties of strong solutions to these equations. The second program of research studies incompressible fluid dynamics in both the axisymmetric and three dimensional cases. The goal of this research is to obtain lower bounds on putative blow-up rates and to determine their stability under perturbations. This project can be considered relevant for physical phenomena such as hurricanes, tornadoes, and whirlpools that are fairly close to axisymmetric. This proposal aims to moreover establish connections between these subject areas, and further explore other broad questions closely related to both projects.
The proposed research contains an interdisciplinary approach to questions arising in non-linear partial differential equations in areas ranging from incompressible fluid flow to gas dynamics. The physical phenomena that these equations model are encountered all across the natural world. Methods the investigator plans to use involve energy based techniques which have proven to be flexible and robust for studying regularity of a variety of nonlinear partial differential equations. The PI's research vision includes developing new tools to study fundamental analytical problems about these basic models; these tools are expected to be generalizable to a wide array of important problems in applied non-linear PDE. The PI plans to disseminate knowledge obtained through the proposed activity among both pure and applied mathematicians. The PI will continue to explore, develop and implement diverse methods to integrate mathematical research at levels ranging from undergraduate projects to the vanguard of current research.