Research On Regularity Of Complex Fluids Related To Navier-Stokes Equation

The three-dimensional incompressible Navier-Stokes equations are widely used in astrophysics,aerospace,materials science and other felds.The existence of the global smooth solution of the three-dimensional incompressible Navier-Stokes equations is one of the famous millennium problems.The study of this kind of related problems not only has profound theoretical signifcance in mathematics,but also has application background in real life.The basic equations of magnetohydrodynamics(MHD)are coupled by the Navier-Stokes equations of fuid motion and magnetic induction equations.The study of mathematical problems not only has the diffculty of Navier-Stokes equations,but also has new diffculties caused by strong coupling.This paper focuses on the regularity of weak solutions of MHD equations and the existence of low regularity solutions of Hall-MHD equations,as follows :In the frst part,we prove the smoothness of the weak solution of the MHD equations in Lorentz space.By using the incompressibility of the fuid and the magnetic feld divergence constraints and the nonlinear term structure conditions of the equation,we propose a Prodi-Serrin type criterion for the weak solution of the MHD equations based on a component of the velocity feld and the critical norm of the magnetic feld.The global smoothness of the weak solution of the equations is obtained.In the second part,we solve the global existence problem of solutions in the critical space for the initial conditions of the three-dimensional incompressible generalized Hall-MHD equations.