Theoretical Analysis Of Singular Limits For Several Equations In Fluid Mechanics

The aim of this thesis is to establish the mathematical theory of the small viscos-ity and heat conductivity limit for several equations in fluid mechanics.It consists of two main parts.The first part is devoted to the small viscosity and heat conductivi-ty limit for compressible and incompressible Navier-Stokes equations in energy space L?([0,T],L2(?)),and in the second part,we shall mainly consider the boundary layer behavior of the two-dimensional incompressible heat conducting flows near a physi-cal boundary in small viscosity and heat conductivity limit,and the corresponding mathematical theory.In the first Chapter of this thesis,we shall briefly state the research background of related problems,introduce what we are going to study,and present the main results.In the second Chapter,we present several sufficient conditions for the small vis-cosity(heat conductivity)limit from the incompressible Navier-Stokes equations with Navier slip boundary condition,the compressible Navier-Stokes equations with nonslip condition,and the compressible Navier-Stokes-Fourier equations with nonslip condi-tion to the corresponding inviscid(without heat-conductivity)problems.First,for the vanishing viscosity limit of the incompressible Navier-Stokes equations with Navier slip boundary condition in a bounded domain of R2,when the slip length is smaller than or equal to the order of viscosity,by using an energy method and developing Kato's approach given in[37,71],we obtain several Kato type conditions to guarantee that the weak solution of the Navier-Stokes equations with the Navier condition converges to the strong solution of the associated problem of the Euler equations in the energy space L?([0,T],L2(?)),as the viscosity goes to zero.Meanwhile,for the case that the slip length is greater than the order of viscosity,it is also shown that the convergence holds unconditionally in L?([0,T],L2(?)).Secondly,for the vanishing viscosity limit of the compressible Navier-Stokes equations with non-slip boundary in a smooth bounded domain M3,we deduce a sufficient condition for the convergence to take place in the energy space L?([0,T],L2(?)),by using Kato's idea[37]of constructing an artificial boundary layer.This sufficient condition contains the tangential or the normal compo-nent of velocity only.Finally,we study the vanishing dissipation limit problem for the full compressible Navier-Stokes-Fourier equations with non-slip boundary condition in a smooth bounded domain R3.By using Kato's idea[37],we obtain two sufficient conditions for the convergence of the weak solution of the full Navier-Stokes-Fourier equations to the classical solution of the compressible Euler equations in the ener-gy space L?([0,T],L2(?)),in the small viscosity and heat conductivity limit.These conditions contain the tangential or normal component of velocity,moreover the inte-grability of temperature near the boundary is required due to the appearance of the thermal layer.In the third Chapter,we study the boundary layer behavior,and the well-posedness and blowup of the solution to the boundary layer equation in small viscosity and heat conductivity limit for the two-dimensional incompressible heat conducting flows.In the case that the viscosity and heat conductivity have the same scale,the boundary layer equations are derived by using multi-scale analysis.Then the well-posedness of the boundary layer equations is established by employing the Littlewood-Paley analysis when the datum are analytic with respect to the tangential variable of the boundary.Finally,we obtain a blowup result in a finite time in the Sobolev space for the solution of the boundary layer equation,for a kind of data which are analytic in the tangen-tial variable but do not satisfy the Oleinik monotonicity condition,by constructing a Lyapunov functional.