A Nash–Moser Framework for Finding Periodic Solutions of the Compressible Euler Equations

Abstract

We report on recent progress the authors have made in a long term program to prove the existence of time-periodic shock-free solutions of the compressible Euler equations. We briefly recall our previous results, describe our recent change of direction, and discuss the estimates that must be obtained to get the Nash–Moser method to converge. Assuming these estimates, we present a convergence theorem for the Nash–Moser method. Our approach reduces the problem to that of obtaining small divisor estimates for finite dimensional projections of linearized operators. These operators can be described in detail, and in principle, this reduces the proof of periodic solutions to the calculation of the smallest singular value of an \(N\times N\) matrix.