Justification of Limits to the Compressible Euler Equations with Wave Interactions
DescriptionIn the study of gas and fluid dynamics, there are many problems on limiting processes which are believed to be true in physics, but have not been justified rigorously in mathematics so that they remain as challenging mathematical problems.Related to what we are going to discuss in this project, let us first recall the following now well known limiting processes. In the study of rarefied gas, the Boltzmann equation was derived from a system of coupled Newton equations through the Boltzmann-Grad limit; when the Knudsen number tends to zero, the Boltzmann equation converges to the compressible Euler equations; and when the Knudsen number is close to zero, the Boltzmann equation can be approximated by the compressible Navier-Stokes equations. Moreover, when the viscosity and heat conductivity coefficients tend to zero, the Navier-Stokes equations converge to the Euler equations. On the other hand, when the Mach number tends to zero, the compressible systems tend to the corresponding incompressible systems. Furthermore, in some time scale and/or near some boundary, for example, with non-zero temperature gradient, there are other non-classical limiting fluid systems for the Boltzmann equation.In this project, we will focus on the limiting processes to the compressible Euler equations with wave interactions. Precisely, we will first consider the convergence rate of the vanishing viscosity limit of general hyperbolic systems with artificial viscosity. Secondly, we will consider the limits of the Navier-Stokes equations and the Boltzmann equation to the Euler equations. Note that even though there have been extensive studies on these problems and substantial results have been obtained in some settings. The mathematical justification with wave interactions basically remains unsolved.
|Effective start/end date
|1/11/12 → 26/09/16