A Liapunov-Schmidt Reduction for Time-Periodic Solutions of the Compressible Euler Equations

Abstract

Following the authors’ earlier work in "A paradigm for time-periodic sound wave propagation in the compressible Euler equations," and "Time-periodic linearized solutions of the compressible Euler equations and a problem of small divisors," we show that the nonlinear eigenvalue problem introduced in "Time-periodic linearized solutions of the compressible Euler equations and a problem of small divisors," can be recast in the language of bifurcation theory as a perturbation of a linearized eigenvalue problem. Solutions of this nonlinear eigenvalue problem correspond to time periodic solutions of the compressible Euler equations that exhibit the simplest possible periodic structure identified in "A paradigm for time-periodic sound wave propagation in the compressible Euler equations." By a Liapunov-Schmidt reduction we establish and refine the statement of a new infinite dimensional KAM type small divisor problem in bifurcation theory, whose solution will imply the existence of exact time-periodic solutions of the compressible Euler equations. We then show that solutions exist to within an arbitrarily high Fourier mode cutoff. The results introduce a new small divisor problem of quasilinear type, and lend further strong support for the claim that the time-periodic wave pattern described at the linearized level in "Time-periodic linearized solutions of the compressible Euler equations and a problem of small divisors," is physically realized in nearby exact solutions of the fully nonlinear compressible Euler equations.