Viscous approximation and weak solutions of the 3D axisymmetric Euler equations

Abstract

The three-dimensional axisymmetric Euler equations without swirl can be represented by the conservation of omega(theta)/r along the particle trajectory, where omega(theta) denotes the swirl component of the vorticity. The two-dimensional Euler equation shares a parallel representation. Delort's work has long settled the global existence of weak solutions corresponding to a vortex sheet data of distinguished sign. In contrast, the parallel global existence problem for the axisymmetric Euler equations remains an outstanding open problem. This paper establishes the global existence of weak solutions to the axisymmetric Euler equations without swirl when the initial vorticity omega(theta)(0) obeys omega(theta)(0)/r is an element of L-1 (R-3) boolean AND L-p (R-3) for p is an element of (1,infinity). The approach is the method of viscous approximations. A major step in the proof is to extract a strongly convergent subsequence of solutions to a viscous approximation of the Euler equations. Copyright (C) 2014 John Wiley & Sons, Ltd.