The Low Mach Number Limit Of The Navier-Stokes Equations In Bounded Regions

The study of the low Mach number limit for the hydrodynamic equations not only have important sense to the investigation of mathematical theory,but also provide theoretical support for the numerical simulation and practical application.In fact,the low Mach number limit is a singular limit,thus it is very challenging to verify this limit rigorously,and also a frontier problem in the research of the singular limit of partial differential equations.In this dissertation,we will study the low Mach number limit of all-time strong solutions to the three-dimensional non-isentropic compressible Navier-Stokes equations for perfect gas in bounded domains.The main results are as follows:First,we studies the low Mach number limit and stability of global strong so-lutions to the three-dimensional non-isentropic compressible Navier-Stokes equations with positive thermal conductivity coefficient,where the velocity field and temperature enjoy the Navier slip boundary condition and convective boundary condition,respec-tively.With the uniform estimates with respect to both the Mach number? ?(0,?)and time t ?[0,?),repeatedly applying the local existence theorem,we obtain the global solution to non-isentropic compressible Navier-Stokes equations.Next,with the aid of the uniform estimates and Arzela-Ascoli theorem,we prove that as the Mach number vanishes,the global solution to non-isentropic compressible Navier-Stokes e-quations converges to the one of isentropic incompressible Navier-Stokes equations in t ?[0,+?).The key difficulty is to derive the higher order estimates of the velocity.To overcome this difficulty,the main idea is to estimate the derivatives of vorticity and the divergence of velocity,respectively,based on the analysis of the Navier slip bound-ary condition.Finally,we establish the exponentially asymptotic stability of strong solutions to full compressible Navier-Stokes equations with the help of the uniform estimates and the classical theory on Stokes' problem.Second,we prove the low Mach number limit of all-time strong solutions in another situation that the velocity field satisfies the Dirichlet boundary condition.The strategy here is similar to the one of the above section.However,comparing with the case of slip boundary condition,the boundary effects caused by the Dirichlet boundary condition should create much more difficulties for deriving the estimates of the high-order spatial derivatives.The method of estimating high-order spatial derivatives in the above section does not work here for the Dirichlet boundary condition,since integrations by parts usually fail and the information of the vorticity of velocity on the boundary is not available.In order to overcome this difficulty,we take the advantage of the regularity theory of the Stokes problem,and transform the estimate of ||u||H3 into the one of ||?2divu||L2,then divide the latter into the interior parts and the parts near the boundary.The boundary estimates take the advantage of the isothermal coordinates in local regions near the boundary.In the meantime,we get the uniform estimates of the high-order spatial derivatives of the temperature.In addition,the uniform estimates of these two cases are different,that is,the uniform estimate of ||u||L?(0,T;H2)in the above section is derived,while in this section we only get the uniform estimate of the normal component of uxx near the boundary is lost.