Study on the global regularity for two-dimensional magnetohydrodynamic and Boussinesq equations

Abstract

Whether the classical solutions of two-dimensional incompressible ideal MHD equations or inviscid Boussinesq equations can develop a finite time singularity or globally regular for all time from smooth initial data with finite energy is an outstanding open problem in fluid dynamics. We study these equations to explore how far one can go beyond these two cases and still can prove the global regularity.

First, the global regularity for the 2D MHD equations with horizontal dissipation and horizontal diffusion is studied. We prove that the horizontal components of any solution admit a global bound in any Lebesgue space L^{2r}, 1\leq r <\infty and the bound grows no faster than the order of \sqrt{r\log r} as r increases. Furthermore, we prove that any possible blow-up can be controlled by the L^\infty-norm of the horizontal components. We establish the global regularity of slightly regularized 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion. The global regularity issue of the MHD equation with horizontal dissipation and horizontal magnetic diffusion is extremely hard. The classical energy method does not work. By using the techniques from the Littlewood- Paley decomposition and logarithmic bound for the horizontal components, we are able to resolve the global regularity issue of the 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion.

Second, the global well-posedness for the 2D Euler-Bousinesq equations with a singular velocity is investigated. We prove the global existence and uniqueness of the solutions to the initial value problem of 2D Euler-Boussinesq equations when the velocity field is double logarithmically more singular than the standard velocity field given by the Biot-Savart law.

Third, the global existence, and uniqueness for the 2D Navier-Stokes-Boussinesq equation with more general dissipation is studied. We prove that the solution is globally regular even the critical dissipation is logarithmically weaker.