A Liapunov-Schmidt Reduction for Time-Periodic Solutions of the Compressible Euler Equations
Blake Temple and Robin Young
Abstract: Following the authors' earlier work in [Preprint 2008-033, Preprint 2008-034], we show that the nonlinear eigenvalue problem introduced in [Preprint 2008-034] 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 [Preprint 2008-033]. 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 [Preprint 2008-034], is physically realized in nearby exact solutions of the fully nonlinear compressible Euler equations.