Cavity singularity and maximal regularity of the compressible Stokes equations: some numerical examples

Abstract

In this thesis we study two problems. First we study a linearized compressible Stokes system on the T-shaped cavity domain, with a suitable boundary condition. The T-shaped domain is a well-known model problem in aerodynamics. In the T-shaped cavity domain the fluid flows consist of three essential phenomena that are the flows in the top region with inflow and outflow, the ones in the bottom(cavity) region and the interface fluctuation between two flows. We establish a piecewise maximal regularity by constructing a vector function cor- responding to the pressure jump on the interior curve, without subtracting the corner singularities. Secondly, a complete regularity theory is an open question for compressible Stokes equations on polygonal domains with zero Dirichlet conditions. As a partial answer we study a nonlinear compressible Stokes system on the infinite sector, with inflow and outflow conditions vanishing at the vertex. We construct the velocity vector u and the density function ρ of the following forms u(x,y)=r^λ (ϕ(θ) e + ψ(θ) e'),θ∈(0,ω), ρ(x,y)=r^(λ-1) σ(θ). where ω is the opening angle of the sector; φ, ψ and σ are the solutions for a nonlinear boundary value problem with any positive number λ≠nπ/ω for integer n; r=(x^2+y^2 )^(1/2), θ are the polar coordinates at the origin, e = (cosθ,sinθ) and e' = (−sinθ,cosθ). We also give some numerical examples.