Research On Solutions Of Incompressible Navier-Stokes Equations In Exterior Domains

In this paper,we first state some well-known results on the existence and uniqueness of global solution,spatial asymptotics,large-time behaviors of strong solutions or weak solutions for incompressible Navier-Stokes(N-S)equations in exterior domains recently.As we all know,many mathematicians have studied fluids with constant density,and a lot of famous results have been obtained.Leray[46]pointed out the existence of global weak solutions with finite energy for the incompressible N-S equations with constant density in the whole space.The asymptotic behavior of Leray's solution(uL,pL)at infinity was solved by Gilbarg and Weinberger[22]in the 1970s.They proved that the pressure has a limit at infinity and if uL is bounded,there exists a constant vector u? such that(?)Han[23,4,25,26]first improved and expanded the decay rate of three-dimensional incompressible N-S flow with time based on previous studies,and later proved the decay of some high-order norms of three-dimensional incompressible N-S flow.But the uniqueness of smooth solution is still an open problem for N?3 dimensions.On the other hand,uniqueness may be represented in smaller function classes,in which global existence has not been proved.Although the results from the mathematical point of view have greatly promoted the study,it is difficult for real fluid to be homogeneous.Therefore,We prove the unique existence of solutions of the three-dimensional incompressible density-dependent N-S equations in an exterior domain with small non-decaying boundary data.