Methods and Applications of Analysis

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

Blake Temple and Robin Young

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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.

Article information

Source
Methods Appl. Anal., Volume 17, Number 3 (2010), 225-262.

Dates
First available in Project Euclid: 23 May 2011

Permanent link to this document
https://projecteuclid.org/euclid.maa/1306182405

Mathematical Reviews number (MathSciNet)
MR2785873

Zentralblatt MATH identifier
1244.35090

Subjects
Primary: 35L65: Conservation laws 76N10: Existence, uniqueness, and regularity theory [See also 35L60, 35L65, 35Q30]

Keywords
Compressible Euler periodic solutions conservation laws

Citation

Temple, Blake; Young, Robin. A Liapunov-Schmidt Reduction for Time-Periodic Solutions of the Compressible Euler Equations. Methods Appl. Anal. 17 (2010), no. 3, 225--262. https://projecteuclid.org/euclid.maa/1306182405


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