Dimension Reduction for Compressible Viscous Fluids

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...

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

Global Smooth Solutions in R3 to Short Wave-Long Wave Interactions Systems for Viscous Compressible Fluids

The short wave-long wave interactions for viscous compressible heat conductive fluids is modeled, following Dias & Frid (2011), by a Benney-type system coupling Navier-Stokes equations with a nonlinear Schrödinger equation along particle paths. We study the global existence of smooth solutions to the Cauchy problem in R3 when the initial data are small smooth perturbations of an equilibrium sta...

Lagrangian Averaging for Compressible Fluids

This paper extends the derivation of the Lagrangian averaged Euler (LAEα) equations to the case of barotropic compressible flows. The aim of Lagrangian averaging is to regularize the compressible Euler equations by adding dispersion instead of artificial viscosity. Along the way, the derivation of the isotropic and anisotropic LAE-α equations is simplified and clarified. The derivation in this ...