Conditional regularity for evolution equations in fluid dynamics

Fluid dynamics is a vast and fundamental field influencing many aspects of science and engineering. The rigorous mathematical analysis of its governing equations such as Navier-Stokes is crucial for understanding the behavior and underpinning mechanisms of many physical processes. It is a well-established fact that weak solutions to fluid dynamics equations exist. However, the question of the global existence and uniqueness of smooth solutions (the regularity problem) remains beyond the scope of present mathematical techniques. Researchers try to find ways how to narrow the gap between the known existence of weak solutions and the proof of full regularity. One of the ways to address this problem is through the study of conditional regularity of weak solutions: the solutions are supposed to satisfy some reasonable additional properties under which the full smoothness of solutions follows. This area has seen a huge amount of progress in the last 25 years.  The conditions have been imposed on various physical variables of the flows, such as velocity and its components, presure, vorticity, or their various combinations. Furthermore, researches utilize a range of functional spaces, including Sobolev and Besov spaces, among others. A comparison of the state of the art twenty years ago with our present understanding demonstrates the substantial progress achieved in recent years. Nevetheless, there remains a number open questions requiring rigorous investigation. This dissertation will select and focus on some key challenges among these problems.