Global regularity for 2D Muskat equations with finite slope

Global regularity for 2D Muskat equations with finite slope

We consider the 2D Muskat equation for the interface between two constant density fluids in an incompressible porous medium, with velocity given by Darcy’s law. We establish that as long as the slope of the interface between the two fluids remains bounded and uniformly continuous, the solution remains regular. The proofs exploit the nonlocal nonlinear parabolic nature of the equations through a...

On global regularity of 2D generalized magnetohydrodynamic equations

Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion

Whether or not classical solutions of the 2D incompressible MHD equations without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue. A major result of this paper establishes the global regularity of classical solutions for the MHD equations with mixed partial dissipation and magnetic diffusion. In addition, the global existence, conditional regul...

Global Regularity for the 2D Magneto-Micropolar Equations with Partial Dissipation

Abstract. This paper studies the global existence and regularity of classical solutions to the 2D incompressible magneto-micropolar equations with partial dissipation. The magneto-micropolar equations model the motion of electrically conducting micropolar fluids in the presence of a magnetic field. When there is only partial dissipation, the global regularity problem can be quite difficult. We ...

Global regularity for the 2D Boussinesq equations with partial viscosity terms

In this paper we prove the global in time regularity for the 2D Boussinesq system with either the zero diffusivity or the zero viscosity. We also prove that as diffusivity(viscosity) goes to zero the solutions of the fully viscous equations converges stongly to those of zero diffusion(viscosity) equations. Our result for the zero diffusion system, in particular, solves the Problem no. 3 posed b...