Study On The Well-Posedness Of Complex Fluids Related To Navier-Stokes Equation

The incompressible Navier-Stokes equation describes the motion of an incompress-ible fluid with viscosity.For a long time,an important problem concerned by physicists and mathematicians is whether there exists a global smooth solution to the 3D incom-pressible Navier-Stokes equations.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 fluid motion and magnetic induction equations.The theoreti-cal study not only has the diffculty of Navier-Stokes equations,but also has new diffculties caused by strong coupling.This paper focuses on the well-posedness of three-dimensional incompressible Hall-MHD equation with variable density and three-dimensional incom-pressible rotating fluid with variable density,as follows:In the frst part,we focus on the well-posedness problem of the three dimensional incompressible resistive and viscous Hall-magnetohydrodynamics system(Hall-MHD)with variable density.We mainly prove existence and uniqueness issues of the density-dependent incompressible Hall-magnetohydrodynamic system in the whole space R~3.In the second part,we consider well-posedness result of the Cauchy Problem for the incompressible rotating fluids with variable density in R~3.Under the Lagrangian coordinate transformation,by using the contraction mapping principle theorem,one can obtain the local existence and uniqueness of the solution for the inhomogeneous rotating Navier-Stokes equation provided that the initial density is bounded.