## Colloquium - Xin Liu

**Date: ** January 25, 2022

**Time: ** 4:00PM - 5:00PM

**Location: ** ZOOM

**Speaker: **Xin Liu, WIAS

**Description: **

**Title:** Second law of thermodynamics and asymptotic limits in compressible flows

**Abstract:** In this talk, I will first discuss the second law of thermodynamics in the compressible Navier-Stokes equations for viscous, compressible flows. In particular, the singularity/degeneracy caused by the vacuum, i.e., density connecting to zero, might create large/singular growth of entropy next the vacuum region. Previous study has shown that the Cauchy/initial value problem for CNS fails to describe the boundedness of entropy. However, as a quantity measuring chaoticness, entropy has no reason to be unbounded. To investigate the entropy-bounded solutions in CNS, we investigate 1) equilibrium of radiation gaseous star, 2) self-similar solutions to CNS and its stability, as examples of flows with or without bounded entropy. On the second part of the talk, depending on time permitted, I will discuss a two scales problem in the compressible Navier-Stokes equations, involving the Mach number and the aspect ratio. By taking different orders of the asymptotic limits, one can formally derive the incompressible Navier-Stokes equations, the primitive equations, and the compressible primitive equations. The relations between these equations can be presented as the PE diagram. To rigorously justify these asymptotic limits have been one of the major topics in the community. Together with previous study, we will fully justify all asymptotic limits for the isentropic flows, i.e., flows with constant entropy.