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Preprint 2008-035

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.

Paper:
Available as PDF (321 Kbytes).
Author(s):
Blake Temple,
Robin Young,
Publishing information:
Submitted.
Comments:
Submitted by:
; 2008-10-13.