The Well-posedness Of Several Fluid Mechanics Equations

The mathematical model of fluid dynamics is derived from the conservation of the mass of the fluid,the conservation of momentum,the conservation of en-ergy and the basic laws of thermodynamics.It plays an important role in theory and scientific calculations in many fields such as hydrodynamics,atmospheric,marine science and petrochemicals.Navier-Stokes system is a basic model for describing fluid dynamics.The study of this model and its coupling model with other equations has always been a hot topic in the research of nonlinear partial differential equations.We mainly focus on the properties of several types of flu-id dynamics equations in this dissertation,and we obtained the well-posedness of the solutions to three types of classical hydrodynamic equations.The main results are stated as follows1.We study the global existence and uniqueness of the strong solution to the coupled incompressible Navier-Stokes and Darcy equations in R2.The a Priori estimates are obtained by using the properties of boundary flattening skill and the Dirichlet-Neumann operator.The local well-posedness of the original problem is obtained by combining the local existence and uniqueness of the approximate solution.Finally,the global existence and uniqueness of strong solutions for the coupling problem are obtained2.We are concerned with the existence of global weak solutions to the com-pressible Navier-Stokes-Poisson equations with the non-flat doping profile when the viscosity coefficients are density-dependent,the data are large and spher-ically symmetric,and we focus on the case where those coefficients vanish on vacuum.We construct suitable approximate system and consider it in annular regions between two balls.The B-D entropy estimation and the strong conver-gence of ?U are obtained.The global solutions are obtained as limits of such approximate solutions.Our proofs are mainly based on the energy and entropy estimates.3.We study the vanishing viscosity of the isentropic compressible Navier-Stokes equations with density dependent viscous coefficient in the presence of the shock wave.Given the shock solution of the corresponding Euler equations,the appropriate approximation problem of the Isentropic Navier-Stokes equation is obtained by reconstructing the original problem,and then we can construct a sequence of smooth solutions which converge to the shock wave as the viscosity tends to zero.