Topics in Applied PDE
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.