Study on the global well-posedness for the two-dimensional Boussinesq and Lans-alpha magnetohydrodynamics equations

Abstract

We investigate the global (in time) regularity problem for two different models; generalized two-dimensional Boussinesq and Lans-alpha magnetohydrodynamics system. First, the global regularity of 2D incompressible generalized Euler-Boussinesq equations has been studied. We establish the global existence and uniqueness of solutions to the initial-value problem when the velocity field is "double logarithmically" more singular than the one given by the Biot-Savart law. This global regularity result goes beyond the critical case. Secondly, we consider the two-dimensional Navier-Stokes-Boussinesq equations with logarithmically super-critical dissipation. By implementing Besov space technique, the global well-posedness of initial value problem is established. These results improve the existing results of super-critical Boussinesq system of equations. Finally, we study the two-dimensional generalized Lans-alpha magnetohydrodynamics system. We mainly focus on Lans-alpha magnetohydrodynamics system of equations with logarithmically weaker dissipation than full dissipation together with zero diffusion or zero dissipation and logarithmically weaker diffusion than full diffusion. In both cases, we are successful to resolve global regularity issues.