Abstract

The Euler equations describing the flow of an incompressible, inviscid fluid of uniform density were first published by Euler in 1757. One of the recent results of mathematical fluid dynamics was the discovery that the particle trajectories of such flows are real analytic curves, despite limited regularity of the initial flow (Serfati, Shnirelman, and others). Hence, the flow lines of stationary solutions to the Euler equations are real analytic curves. In this work we consider a two-dimensional stationary flow in a periodic strip. Our goal is to incorporate the analytic structure of the flow lines into the solution of the problem. The equation for the stream function is transformed to new variables, more appropriate for the further analysis. New classes of functions are introduced to take into account the partial analytic structure of solutions. This makes it possible to regard the problem as an analytic operator equation in a complex Banach space. The Implicit Function theorem for complex Banach spaces is applied to establish existence of unique solutions to the problem and the analytic dependence of these solutions on the parameters. Our approach avoids working in the Frechet spaces and using the Nash-Moser-Hamilton Implicit Function Theorem used by the previous authors (Sverak & Choffrut), and provides stronger results.