Some Studies On The Incompressible Navier-Stokes Equations And Related Problems

This thesis is composed of four parts.In the first part,the existence and uniqueness of the incompressible Navier-Stokes equations with damping are studied.By the Galerkin method,we show that the Cauchy problem of the Navier-Stokes equations with dampingα｜u|^β-1u (α＞0) has global weak solutions for anyβ≥1,global strong solutions for anyβ≥7/2 and that the strong solution is unique for any 7/2≤β≤5.In the second part,we investigate the large time behavior of weak solutions for the Cauchy problem of the Navier-Stokes equations with dampingα｜u｜^β-1u(α＞0,β≥3).By the Fourier splitting method,we show that the decay rate of the solutions is -3/2,which is the same as the classical incompressible Navier-Stokes equations.In the third part,we mainly study the global existence and stability of the solutions for the nonhomogeneous incompressible Navier-Stokes equations in three-dimensional bounded or unbounded domains.By delicate energy estimates,under some more regular conditions and some compatibility condition on the initial data,we obtain that if the initial data are small perturbation on those of a known strong solution,then there exists a global solution,which defined on(0,∞) for the nonhomogeneous incompressible Navier-Stokes equations and is a perturbation of the known one.The strong solution of the nonhomogeneous incompressible Navier-Stokes equations is stable.In the fourth part,we give some remarks on the two dimensional Boussinesq equations.By the Schauder fixed points,we get that when the initial vorticity is smooth,there exists a unique classical solutions for the Cauchy problem of the two dimensional Boussinesq equations.When the initial vorticity belongs to L¹(R²)(or the Radon measure space),whether there exists global existence of the solutions is still open.It will be investigated in future.