Mathematical Theory Of Serval Inviscid Limit Problems In Fluid Mechanics

In this doctoral thesis,we study the inviscid limit problems in fluid mechanics,de-velop the boundary layer theory for serval typical models and discuss the hydrodynamic stability mechanism.Mainly,we consider the well-posedness of boundary layer prob-lems for the two dimensional compressible isentropic flows,the validity of boundary layer theory in the three dimensional incompressible flows rotating with high frequency.Moreover,we analyze the nonlinear stability of the three dimensional incompressible flows around a Poiseuille flow in pipes.In the introduction,we briefly introduce some physical backgrounds of the asymp-totic analysis of fluid dynamics for inviscid limit problems,and recall some known works on boundary layer theory and hydrodynamic stability.Besides,we describe the problems to be studied,present the main results and the structure of this thesis.In Chapter 2,we investigate the asymptotic behavior of the two dimensional com-pressible isentropic flows with no-slip boundary condition as the viscosity tends to zero.By using the method of multi-scale analysis,we derive the boundary layer prob-lems via the asymptotic expansions of the density and velocity field.Away from the physical boundary,the flow is found to be approached by the corresponding inviscid flow,and boundary layers appear near the physical boundary,they are described by the compressible Prandtl equations.The steady equations can be reformed as a s-calar degenerate parabolic equation by using von Mises transformation,provided the tangential velocity is strictly positive along the flow.As a result,this problem with inflow admits a unique classical solution at least locally in the flow direction,and it exists globally if the outflow density decreases in the flow direction.For either steady boundary layer problems without inflow or the unsteady case,we rewrite the system as a scalar degenerated parabolic equation through Crocco's transformation.Under Oleinik's monotonic assumption on the tangential velocity with respect to the normal variable of the boundary,we prove the local well-posedness of the boundary layer prob-lems by adapting the method Oleinik used.Moreover,we obtain the global existence of a weak solution to the problem derived from unsteady boundary layer problems by the Crocco transformation,provided the outflow density decreases along the flow direction.In Chapter 3,we consider the local well-posedness of unsteady Prandtl equations,when Oleinik's monotonicity condition fails on a non-degenerate curve and the veloc-ity polynomially decays in the normal variable.We construct two energy functionals with the same order of vorticity in a small neighborhood of the critical curve and in the domains of monotonic velocity,respectively.Once the prior boundedness of the total energy is established,we obtain the local well-posedness in the Gevrey class by constructing approximation solutions.In Chapter 4,we study the asymptotic limit of the incompressible rotating flows in three dimensions as the viscosity and the period of rotation tend to zero simulta-neously.To avoid the high frequency oscillation of velocity in time,the initial data and force are assumed to be well-prepared.We derive the limit equations and the Ekman boundary layer by the formal asymptotic expansion of the velocity in viscosity and period.Furthermore,by constructing the second order approximation solution,and obtaining the high order estimates of the remainder,we obtain the validity of the boundary layer expansion,rigorously.Finally,the stability of three dimensional incompressible flows about pipeline Poiseuille flow is considered in Chapter 5.By defining cylindrical Helmholtz-Leray projector and applying the Fourier-Laplace transform,we obtain the fourth order or-dinary problem for the stream function of the linearized problem.From the dispersion relationship deduced by boundary conditions,the spectral stability for large Reynolds Poiseuille flow is proved.As a result of utilizing the classical semigroup theory,we deduce that the original nonlinear problem admits a unique global solution,and its perturbation around pipeline Poiseuille flow is nonlinearly stable.