Magnetohydrodynamic turbulent flows for viscous incompressible fluids through the lenses of harmonic analysis

Abstract

This thesis is divided into three main parts devoted to the study of magnetohydrodynamics (MHD) turbulence flows. Part I consists of introduction and background (or preliminary) materials which were crucially important in the process. The main body of the thesis is included in parts II and III. In Part II, new regularity results for stochastic heat equations in probabilistic evolution spaces of Besov type are established, which in turn were used to establish global and local in time existence and uniqueness results for stochastic MHD equations. The existence result holds with positive probability which can be made arbitrarily close to one. The work is carried out by blending harmonic analysis tools, such as Littlewood-Paley decomposition, Jean-Micheal Bony paradifferential calculus and stochastic calculus. Our global existence result is new in three-dimensional spaces and is published in [148](Sango and Tegegn, Harmonic analysis tools for stochastic magnetohydrodynamics equations in Besov spaces, International Journal of Modern Physics B, World Scientific, 2016, 30). Our results in this part are novel; they introduced Littlewood-Paley theory and paradifferential calculus for stochastic partial differential equation. In Part III, we studied Kolmogorov's spectral theory for MHD equations with reasonably smooth external forces applied to both velocity and magnetic fields.