On the Regularity of Solutions of PDEs via Harmonic Analysis Methods

Abstract

Thegist of this thesis is that, by employing the pure mathematical machinery of functional analytical methods, harmonic analysis, measure theory, distribution theory, and energy methods, we analyze the regularity of generalized solutions of weakly formulated PDEs and present improved regularity criteria that will ensure the smoothness of their weak solutions. Different kinds of regularity criteria involving pressure, vorticity, velocity, logarithmic, component reduction, one-directional derivatives, etc. are proved in Besov, critical Besov, and anisotropic Lorentz spaces for various fluid dynamical systems. One of Clay’s millennium problems, the smoothness of the Navier-Stokes equation, is intimately related to all the systems that we are going to analyze in this thesis and is a fundamental open problem of well-posedness and regularity that arises from the turbulent behavior of f lows over a period of time. Our goal is to obtain the regularity in more general critical function spaces that will ensure the regularity of the systems in that particular time interval. In this research work, we deal with the unsteady fluid problems on the entire three-dimensional spatial domain and in the finite-time interval