Large Time Behavior For Two Types Of Hydrodynamic Equations

The hydrodynamics equations,which can reflect the basic mechanical laws of real fluid,have important applications in various fields such as aerospace,oil extraction,marine transportation,etc.In this thesis,we have chosen two types of hydrodynamic equations.This thesis aims to investigate the decay properties of the solutions of the above system as time approaches infinity.It is divided into the following sections:Chapter 1 briefly introduces the research background for the full compressible NavierStokes equations and the two-phase flow model.Based on the previous results,the problem studied in this thesis is clearly presented.Then Chapter 2 gives a brief explanation of the fundamental notations,concepts,and vital lemmas required in this thesis and so on.Chapter 3 investigates the large-time behavior of solutions to the 3D full compressible Navier-Stokes equations with large initial data.There are two main innovations in this chapter:First,by using the high-low frequency decomposition and interpolation techniques,it has been demonstrated that the second-order spatial derivative of the global large solution converges to zero at the L2-rate(1+t)-7/4.This time decay rate coincides with the heat equation,and particularly improves the previous related results.Second,by applying the weighted L2-energy estimates,we establish the space-time decay rate of solution in the weighted Sobolev space Hγ2(γ≥ 0).Chapter 4 mainly studies space-time decay properties of classical solutions to a twophase flow model in R3.There are two primary innovations in this chapter:First,by overcoming the complications arising from the strong coupling between two fluids and utilizing the previous temporal decay results,we proved that the space-time decay rate of the k(∈[0,l])-th order spatial derivative of solution in the weighted Lebesgue space Lγ2(γ≥0)is t-3/4-k/2+γ.Second,by constructing a Lyapunov-type energy inequality,we find that the space-time decay rate of the k(∈[0,l-2])-th order spatial derivatives of the difference between the two velocities of fluid in the weighted Lebesgue space Lγ2(γ≥0)is t-5/4-k/2+γ.The proof is based on the delicate weighted energy estimates and inductive method.Chapter 5 summarizes the primary results of this thesis and presents some suggestions for future research.