Linear and nonlinear partial differential equations arise in many areas of mathematics, such as differential geometry, stochastic control theory and mathematical finance. This project focuses on the investigations of elliptic and parabolic equations with rough coefficients, systems arising from linear laminates and composite media, and evolutionary equations from fluid mechanics. The first part of the project is to systematically investigate the regularity and the strong solvability in Sobolev spaces for both linear and nonlinear possibly nonlocal equations of elliptic and parabolic type. The novelty of this research is that coefficients are allowed to be merely measurable in some of the independent variables. The second part of the project concerns partial differential systems arising from linear laminates and composite media. The emphases are in local and global smoothness of the gradient of weak solutions to these systems and the higher regularity near boundary points. The third part regards the regularity theory for several models of nonlinear parabolic equations in mathematical fluid mechanics. Several problems about the partial regularity as well as regularity criteria of weak solutions to the Navier-Stokes equations will be addressed.
The proposed research will have significant applications in areas as diverse as biology, physics, economics, and finance. For instance, the equations describing fluid flow have been widely used to model ocean currents, the weather and climate, air flow around a airplane, and motion of stars inside a galaxy, to name a few. They have various important applications such as the optimal design of aerodynamic shapes. To disseminate his work and increase its impact, the P.I. will integrate these research projects into the larger framework of the undergraduate and graduate training at Brown University. Besides recruiting and advising students in science and engineering with a special emphasis on under-represented groups, training students in conducting research in differential equations, the proposed educational activities will also include the organization of a summer workshop on equations in fluid mechanics and mentoring undergraduate mathematical competitions.
