The Study Of Hydrodynamic Limits

Hydrodynamics is of a wide range of applications, such as the design and test of plane and aircraft, the calculation of the velocity of the oil in the petroleum transmission pipeline, the prediction of weather, the design of ships, and so on. The hydrodynamic equations include Navier-Stokes equations, Euler equations, Magnetohydrodynamic equations, the equations of the viscoelastic fluids, and so on. The hydrodynamic limits are among the most important problems in the study of the theory of hydrodynamic equations. It mainly studies the relations between the equations in different mechanisms. For example, as the state of the fluids is close to incompressibility, the asymptotic relationship between the compressible Navier-Stokes equations and the incompressible Navier-Stokes equations. This is the so-called incompressible limit.In this dissertation, we will study the incompressible limit of hydrodynamic equations in bounded domains. The main results are as follows:First, we study the incompressible limit of global solutions to the three-dimensional compressible Navier-Stokes equations with “well-prepared” initial data and Navier’s slip boundary condition in a bounded domain as the Mach number tends to zero, which gives the existence and uniqueness of the global strong solution to the corresponding incompressible equations. The main idea is to derive a differential inequality with decay by energy estimates which are uniform in both the Mach number and the time, thus additional difficulties arise. As to the estimates on the boundary, the estimate for the vorticity is difficult to obtain. To overcome this difficulty, we take the advantage of the isothermal coordinates to evaluate the vorticity in local regions near the boundary.Second, we establish the local existence and uniqueness of strong solutions to an Oldroyd-B model for the incompressible viscoelastic fluids with slip boundary condition in a bounded domain ? 2,3?d??R d ? with smooth boundary, via incompressible limits. The main strategy is to derive the uniform energy estimates with respect to the Mach number for the corresponding linearized system, then to obtain the uniform estimates of nonlinear system and the incompressible limit with the aid of successive iterations and the theory of fixed point. The boundary effects here produce more troubles in the high-order derivatives.