Topics in Regularity Theory of Partial Differential Equations

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This project focuses on mathematical research in the regularity theory of certain partial differential equations that arise naturally in physics and engineering. A central topic in the research of linear and nonlinear partial differential equations is the smoothness of solutions under various regularity assumptions on coefficients and domains. Many equations and systems from physics and engineering involve discontinuous coefficients or domains with sharp corners or cusps, or even fractal structures. For instance, fiber-reinforced materials may have fiber inclusions that are very closely spaced and may even touch. In such cases, much of the classical theory of partial differential equations is no longer applicable. Another research topic of major importance is hydrodynamic turbulence in fluid and gas dynamics. The study of turbulence leads to new ideas in scientific computing, differential equations, and statistics, and it has been characterized as one of the most fascinating problems in fluid mechanics.The principal investigator will carry out research closely related to the aforementioned topics and will attempt to address some of the open problems in these areas. He will focus his attention on several projects that can be gathered into three main topical areas. First, the project will develop new methods to study certain elliptic equations associated with composite materials (e.g., elasticity problems, conductivity problems). The principal investigator is particularly interested in the fine regularity of solutions to these equations, including higher-order regularity and more quantitative derivative estimates. Second, the project will explore nonstandard Sobolev estimates for local (or nonlocal) nonlinear elliptic and parabolic equations with irregular coefficients or kernels, in possibly irregular domains. Finally, the project will investigate well-posedness and regularity issues for certain deterministic and stochastic fluid equations such as the magnetohydrodynamics equations.