Regularity Problems in Mathematical Fluid Mechanics

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I propose to study the regularity of solutions for several representative partial differential equations in mathematical fluid mechanics.**Equations from fluid mechanics have always played an important role in the development of the theory of partial differential equations. Among the many open problems these equations present, one of the most important is their "well-posedness" regarding the existence and uniqueness of the solutions. The key to settling the well-posedness problem is to understand how the solutions can stay regular or form finite-time singularities. The significance of this problem is two-fold. From the mathematical point of view, regularity (or lack thereof) of solutions is the very first and most fundamental issue to be settled in any theory of partial differential equations; from the physical point of view, regularity of solutions directly relates to the validity of the equations as mathematical models for physical phenomena. **There are three major obstacles to successful mathematical analysis of equations from mathematical fluid mechanics: nonlinear terms, nonlocal operators, and coupling between unknown quantities. These difficulties have inspired the invention of several new methods and techniques for partial differential equations in recent years. However the progress is still far from satisfactory. **I plan to contribute to the theory of partial differential equations through detailed study of four representative systems: the two-dimensional generalized magnetohydrodynamical (GMHD) equations, the Euler-Poincare equations, a one-dimensional nonlinear nonlocal system, and the Onsager model for liquid crystals. These equations are chosen to achieve a balance of difficulty/impact and accessibility. On one hand, all four exhibit most, if not all, of the three difficulties in mathematical fluid mechanics: nonlinearity, nonlocality, and coupling. As a consequence, progress in the study of these equations would shed light on the study of other fluid mechanical equations. Furthermore, as many mathematical models in chemistry, biology, and engineering are derived using ideas from fluid mechanics, the proposed research will also have impact on those fields. On the other hand, there is evidence that the well-posedness problem of these systems, though still open, are among the more tractable ones in the many open problems in mathematical fluid mechanics. Therefore, these equations are ideal for the training of HQP. **The outcome of the proposed research will significantly improve our understanding of nonlinearity, nonlocality, and coupling in partial differential equations and will shed light on the study of a wide variety of equations from not only fluid mechanics but also other fields such as mathematical biology. It will also contribute to our understanding of turbulence. Progress in the proposed research will be of interest to both the partial differential equations community and the fluid mechanics community. Part of the proposed research will also draw attention from the community of nonlinear functional analysis. The proposed research will benefit from existing and potential national and international collaborations. Through working on the proposed projects, HQP will receive comprehensive training in partial differential equations, harmonic analysis, nonlinear functional analysis, fluid mechanics, and scientific computing.