Multi-scale Analysis of Compressible Viscous and Rotating Fluids

Transport in Viscous Rotating Fluids

We consider a uniformly rotating viscous incompressible fluid and estimate particle transport in the vertical direction (parallel to the rotation axis). We prove that for short time and regular initial data, strong rotation suppresses the vertical gradient of flow maps. The proof uses a diffusive Lagrangian formalism, and the suppression of the vertical gradient is a natural and direct byproduc...

Heat Convection of Compressible Viscous Fluids. I

The stationary problem for the heat convection of compressible fluid is considered around the equilibrium solution with the external forces in the horizontal strip domain z0 < z < z0 + 1 and it is proved that the solution exists uniformly with respect to z0 ≥ Z0 . The limit system as z0 → +∞ is the Oberbeck-Boussinesq equations. Mathematics Subject Classification (2010). Primary 35Q35; Secondar...

A Singular Limit for Compressible Rotating Fluids

We consider a singular limit problem for the NavierStokes system of a rotating compressible fluid, where the Rossby and Mach numbers tend simultaneously to zero. The limit problem is identified as the 2-D Navier-Stokes system in the “horizontal” variables containing an extra term that accounts for compressibility in the original system.

Mathematical theory of compressible, viscous, and heat conducting fluids

We review some recent results on the Navier–Stokes–Fourier system governing the evolution of a general compressible, viscous, and heat conducting fluid. We discuss several concepts of weak solutions, in particular, using the implications of the Second law of thermodynamics. We introduce the concept of relative entropy and dissipative solution and show the principle of weak-strong uniqueness. We...

Problems in the Theory of Viscous Compressible Fluids