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.
Similar content being viewed by others
References
Bressan, A.: The unique limit of the Glimm scheme. Arch. Ration. Mech. Anal. 130, 205–230 (1995)
Bressan, A., Liu, T.P., Yang, T.: \({L}_1\) stability estimates for \(n\times n\) conservation laws. Arch. Ration. Mech. Anal. 149, 1–22 (1999)
Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)
Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18, 697–715 (1965)
Glimm, J., Lax, P.D.: Decay of solutions of systems of nonlinear hyperbolic conservation laws. Memoirs Am. Math. Soc. 101 (1970)
John, F.: Formation of singularities in one-dimensional nonlinear wave propagation. Commun. Pure Appl. Math. 27, 377–405 (1974)
Liu, T.-P.: Decay to \(N\)-waves of solutions of general systems of nonlinear hyperbolic conservation laws. Commun. Pure Appl. Math. 30, 585–610 (1977)
Majda, A.: Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences, no. 53. Springer, Berlin (1984)
Moser, J.: A new technique for the construction of solutions of nonlinear differential equations. Proc. Nat. Acad. Sci. 47, 1824–1831 (1961)
Riemann, B.: The propagation of planar air waves of finite amplitude, classic papers in shock compression science. In: Johnson J.N., Cheret R. (eds.) Springer, Berlin (1998)
Tadmor, E.: Convergence of spectral methods for nonlinear conservation laws. SIAM J. Numer. Anal. 26, 30–44 (1989)
Temple, B., Young, R.: The large time stability of sound waves. Commun. Math. Phys. 179, 417–466 (1996)
Temple, B., Young, R.: A paradigm for time-periodic sound wave propagation in the compressible Euler equations. Methods Appl. Anal. 16(3), 341–364 (2009)
Temple, B., Young, R.: A Liapunov–Schmidt reduction for time-periodic solutions of the compressible Euler equations. Methods Appl. Anal. 17(3), 225–262 (2010)
Temple, B., Young, R.: Time-periodic linearized solutions of the compressible Euler equations and a problem of small divisors. SIAM J. Math. Anal. 43(1), 1–49 (2011)
Young, R.: Sup-norm stability for Glimm’s scheme. Commun. Pure Appl. Math. 46, 903–948 (1993)
Acknowledgments
Temple supported in part by NSF Applied Mathematics Grant Number DMS-040-6096. Young supported in part by NSF Applied Mathematics Grant Number DMS-010-4485
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Temple, B., Young, R. A Nash–Moser Framework for Finding Periodic Solutions of the Compressible Euler Equations. J Sci Comput 64, 761–772 (2015). https://doi.org/10.1007/s10915-014-9851-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-014-9851-z