Singular Limits In Fluid Mechanics

Fluid mechanics is a branch of physics which involves the study of fluids. One of its most important research fields is the hydrodynamics equations, where the Navier-Stokes equations, Euler equations, magneto-hydrodynamic(MHD) equations and viscoelastic hydrodynamic equations are included. The hydrodynamics equations has a wide range of applications, for example, the research on meteorology and water conservancy, the design or test of spacecraft and aircraft, the exploitation and transmission of petroleum, and so on. However, it is very difficult to study the mathematical theory on the existence and behaviors of solutions to the hydrodynamic equations. The singular hydrodynamic limits among the most important problems in this field.The hydrodynamic limit regards the asymptotic relationships between the equations arising from different physical backgrounds. For example, as the state of the fluids is closed to incompressibility, the asymptotic relationship between the compressible MHD equations and incompressible MHD equations, which is the so-called incompressible limit. However, since some large parameters appear in the equations, such as the reciprocal of Mach number, some quantities in energy estimates will tend to infinity as the Mach number goes to zero. Thus the solutions will exhibit singularity in the limiting process.This paper studies the incompressible limit of strong solutions to the full compressible MHD equations in three-dimensional bounded domains. The main results are as follows:Provided the initial data are “well-prepared” and the initial temperature and density are close to constant states, we obtain the singular limit of local solutions to the non-isentropic compressible magneto-hydrodynamic equations for viscous ideal polytropic flows with magnetic diffusions in a three-dimensional bounded domain as the Mach number tends to zero. Thus, we show that the local strong solution converges to the one of incompressible magneto-hydrodynamic equations. The main idea is to derive an energy inequality of exponential type which gives the uniform estimates with respect to the Mach number. Then we obtain the convergence of strong solutions by compactness arguments. Due to the presence of large parameters which are in terms of the reciprocal of Mach number, it is very difficult to get the uniform estimates, in particular, the boundary estimates and vorticity estimates. To overcome these difficulties, we utilize the method of localization near the boundary by setting up the isothermal coordinates.