Regularity And Singular Set Of Weak Solutions To Partial Differential Equations

This thesis is devoted to study the regularity of incompressible Navier-Stokes equations in weak spaces andΓ-convergence of the energy functional for Landau-Lifshitz ferromagnetic model with Dirichlet boundary condition, then we wil develop a variational inequality method to the Ginzburg-Landau functional and obtain the existence of local minimizers with vortics.First of all, we will talk about some conceptions and problems of the partial differential equations above, including the study of regularity for weak solutions to Navier-Stokes equations, Landau-Lifshitz equation and functional, the original definition ofΓ-convergence, also about the superconducting materials with vortex pinning to Ginzburg-Landau functional in 3-dimensional case.Secondly, we will show some regularity criteria of weak solution and existence of minimizers:1. the Cauchy problem for the 3-dimensional Navier-Stokes equations, we will establish some Serrin type regularity criterion in weak spaces involving the summa-bility of the pressure or the gradient of the pressure;2.Γ-convergence of Landau-Lifshitz ferromagnetic model in the presence of Bloch wall in the disk of (?)~2 with Dirichlet boundary condition;3. the existence of local minimizers with vortices locating in the pinning regions.