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The validity of nonlinear geometric optics for weak solutions of conservation laws

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Abstract

The method of weakly nonlinear geometric optics is one of the main formal perturbation techniques used in analyzing nonlinear wave motion for hyperbolic systems. The tacit assumption in using such perturbation methods is that the corresponding solutions of the hyperbolic system remain smooth; since shock waves typically form in such solutions, these assumptions are rarely satisfied in practice. Nevertheless, in a variety of applied contexts, these methods give qualitatively reliable answers for discontinuous weak solutions. Here we give a rigorous proof for the validity of nonlinear geometric optics for general weak solutions of systems of hyperbolic conservation laws in a single space variable. The methods of proof do not mimic the formal construction of weakly nonlinear asymptotics but instead rely on structural symmetries of the approximating equations, stability estimates for intermediate asymptotic times, and the rapid decay in variation of weak solutions for large asymptotic times.

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Author notes

  1. Andrew Majda

    Present address: Department of Mathematics, Princeton University, 08544, Princeton, N.J., USA

Authors and Affiliations

  1. Department of Mathematics, Duke University, 27706, Durham, North Carolina, USA

    Ronald J. DiPerna & Andrew Majda

  2. Department of Mathematics, University of California, 94720, Berkeley, California, USA

    Ronald J. DiPerna & Andrew Majda

Authors
  1. Ronald J. DiPerna
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  2. Andrew Majda
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Additional information

Communicated by A. Jaffe

Partially supported by NSF Grant No. DMS-8301135

Partially supported by NSF Grant No. MCS-81-02360 and ARO Grant No. 483964-25530

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DiPerna, R.J., Majda, A. The validity of nonlinear geometric optics for weak solutions of conservation laws. Commun.Math. Phys. 98, 313–347 (1985). https://doi.org/10.1007/BF01205786

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  • DOI: https://doi.org/10.1007/BF01205786

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