Topics in Applied PDE

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The proposal focuses on three directions: intense structures in fluid flows, mixing by fluid flows, and effects of chemical attraction in biology, ecology and medicine. Fluid flows exhibit high degree of complexity and can easily develop intense structures. Better fundamental understanding of the mechanisms of creation of intense features in fluid flows is very important for many applications in engineering, weather forecasting and other fields. The first direction of the proposal seeks to study classical equations of fluid mechanics focussing on situations where intense fluid motion can develop spontaneously. Recent novel results obtained by the PI in this direction provide hopes for an essential advance in understanding these complex and important phenomena. In the second direction, a study of most efficient ways of mixing in fluid flow is proposed. Efficient mixing is of critical importance in many applications, ranging from combustion in engines to ecology. In the third direction, the PI will study the role of chemical sensing and chemical attraction for enhancement of reactions in biology. One situation where it is relevant involves healing of the body, where infected or injured tissues releases special compounds which attract immune system cells to fight the infection. Chemical attraction can also be an undesirable effect: some tumors are known to rely on this mechanism for their growth. The PI plans to develop new mathematical tools to analyze these more advanced and better predictive models. The project involves a training component, where junior researchers at all levels will be mentored as scholars and educators and will work on research projects under the guidance of the PI. The first main focus of the proposal is on studying solutions of the classical equations of fluid mechanics, such as incompressible Euler, Navier-Stokes and 2D Boussinesq system. Recent numerical simulations of Hou and Luo suggest a new scenario for singularity formation for solutions of the 3D Euler equation. The scenario is axi-symmetric, and singularity formation happens at the boundary. The scenario also applies for the 2D inviscid Boussinesq system. Inspired by the geometry of the scenario, the PI (jointly with Vladimir Sverak) has constructed examples of solutions of 2D Euler equation with double exponential growth in vorticity gradient. Such growth is known to be sharp. The PI intends to use new insights obtained in the construction to approach more complex and long open questions for 2D Boussinesq system and 3D Euler equation. Methods employed will include functional analysis, Fourier analysis, PDE techniques and novel comparison principles. The second direction concerns efficiency of mixing in fluids given constraints on fluid velocity. Bounds on the mixing rates under various types of constraints and boundary conditions as well as insight into the nature of best mixing flows will be sought. This area lies at the intersection of dynamical systems, PDE and Fourier analysis. The third direction involves effects of chemical attraction between reacting species on the rates of reaction. Chemical attraction will be modeled by the Keller-Segel and related nonlinear terms. The PI proposes to obtain qualitative results about the behavior of solutions in such systems, and to derive precise bounds describing the strength of the effect and its dependence on various key parameters. Sharp bounds on convergence to equlibirum for Fokker-Planck operators with certain natural classes of potentials will also be obtained.