Stability and Instability Analysis of Compressible Fluid with non-slip Boundary Condition

Description

In recent years, there has been tremendous progress on the mathematical theoriesof the fundamental equation in the boundary layer theory, that is, the Prandtl systemderived by Prandtl in 1904 from the incompressible Navier-Stokes equations with nonslipboundary condition. The results include the well-posedness theories in variousfunction spaces, stability and instability analysis, inviscid limits. On the other hand,there are also studies on the corresponding problems about the compressible Navier-Stokes equations that involves not only viscous layer in velocity but also thermal layerin temperature.For well-posedness theories, in addition to the classical approach based on theCrocco transformation introduced by Oleinik in 1960s, approaches based on energymethods were introduced recently by several authors in addition to the abstract Cauchy-Kowaleskaya theory in the analytic framework. For the compressible fluid, the studyon the boundary layer for the isentropic flow is similar to the incompressible fluid.However, if one considers the full Navier-Stokes equations, the phenomena are verydifferent, for example, different thickness of layers of velocity and temperature appearif the vanishing rates of the viscosity and heat conductivity coefficients are different.New systems of boundary layers for these physical measurements have been derivedand some preliminary study in the two space dimension (denoted by 2D) was given inour recent work.Besides the importance of the well-posedness theories, the instability of the boundarylayer is one of the key issues because it is related to the separation of the boundarylayer that leads to rich phenomena in fluid dynamics, such as turbulence. For theclassical Prandtl equations, the study on the instability of boundary layer is extensiveboth in theory and numerics. In particular, recently there are some elegant works onthe instability of 2D shear flow by Grenier, G’erad-Varet and Dormy, and others.The purpose of this project is to investigate the stability and instability mechanismof boundary layer for compressible fluid. The study aims to solve the following problems:whether the non-degenerate critical point in the shear flow leads to both linearand nonlinear stability in 2D; whether the temperature layer has different stability andinstability structures; how these criteria play a role in three space dimension (denotedby 3D); how about the flow other than the shear flow.We believe that the progress on these problems will further enrich of the mathematicaltheories of the Prandtl system, in particular for the compressible fluid.