Vanishing viscosity limits for axisymmetric flows with boundary

2018/06/13 by Abe, Ken
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1806.04811

Abstract

We construct global weak solutions of the Euler equations in an infinite cylinder Π=\x∈ ℝ³ | x_h=(x₁,x₂), r=|x_h|<1\ for axisymmetric initial data without swirl when initial vorticity ω₀=ω^θ₀e_θ satisfies ω^θ₀/r∈ L^q for q∈ [3/2,3). The solutions constructed are Hölder continuous for spatial variables in Π if in addition that ω^θ₀/r∈ L^s for s∈ (3,∞) and unique if s=∞. The proof is by a vanishing viscosity method. We show that the Navier-Stokes equations subject to the Neumann boundary condition is globally well-posed for axisymmetric data without swirl in L^p for all p∈ [3,∞). It is also shown that the energy dissipation tends to zero if ω^θ₀/r∈ L^q for q∈ [3/2,2], and Navier-Stokes flows converge to Euler flow in L² locally uniformly for t∈ [0,∞) if additionally ω^θ₀/r∈ L^∞. The L²-convergence in particular implies the energy equality for weak solutions.

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