Global Regularity Aspects of Equations in Hydrodynamics

The question of the global regularity of the two-dimensional magnetohydrodynamics system without viscous dissipation is currently unknown. A challenging problem concerning the global regularity of the two-and-a-half and three-dimensional Hall-magnetohydrodynamics system is still open. The global regularity of two-dimensional and three-dimensional Kuramoto–Sivashinsky equations are not solved as well. Chapter 2 explores some cancellations and bounds within the Hall term for both two-and-a-half dimensional and three-dimensional cases, as well as various regularity criteria. The two-a-half-dimensional Hall equation and Hall-magnetohydrodynamics system are also proved to be globally well-posed when magnetic dissipation is considered at the below-critical level in the horizontal direction and at the supercritical level in the vertical direction. The purpose of Chapter 3 is to introduce and prove some global regularity criteria on the Sobolev and Besov spaces in dimensions two and three. In Chapter 4, a two-and-a-half-dimensional magnetohydrodynamics system is presented, and its global well-posedness is demonstrated. Furthermore, a magnetohydrodynamics system without viscous dissipation is introduced, and a global regularity criterion is derived.