Symmetry Results for Compactly Supported Steady Solutions of the 2D Euler Equations

Abstract

In this paper we present symmetry results regarding compactly supported solutions of the 2D steady Euler equations. Assuming that Omega = {x is an element of R-2 : u( x) not equal 0} is an annular domain, we prove that the streamlines of the flow are circular. We are also able to remove the topological condition on Omega if we impose regularity and nondegeneracy assumptions on u at partial derivative Omega. The proof uses the corresponding stream function solves an elliptic semilinear problem - Delta phi = f (phi) with del phi = 0 at the boundary. One of the main difficulties in our study is that f is not Lipschitz continuous near the boundary values. However, f (phi) vanishes at the boundary values and then we can apply a local symmetry result of F. Brock to conclude. In the case partial derivative(nu)u not equal 0 at partial derivative Omega this argument is not possible. In this case we are able to use the moving plane scheme to show symmetry, despite the possible lack of regularity of f. We think that such result is interesting in its own right and will be stated and proved also for higher dimensions. The proof requires the study of maximum principles, The Hopf lemma and The Serrin corner lemma for elliptic linear operators with singular coefficients.