Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws

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Original languageEnglish
Pages (from-to)1936-1970
Journal / PublicationCommunications in Partial Differential Equations
Issue number11
Online published11 Oct 2013
Publication statusPublished - 2013
Externally publishedYes


We present a new approach to analyze the validation of weakly nonlinear geometric optics for entropy solutions of nonlinear hyperbolic systems of conservation laws whose eigenvalues are allowed to have constant multiplicity and corresponding characteristic fields to be linearly degenerate. The approach is based on our careful construction of more accurate auxiliary approximation to weakly nonlinear geometric optics, the properties of wave front-tracking approximate solutions, the behavior of solutions to the approximate asymptotic equations, and the standard semigroup estimates. To illustrate this approach more clearly, we focus first on the Cauchy problem for the hyperbolic systems with compact support initial data of small bounded variation and establish that the L 1-estimate between the entropy solution and the geometric optics expansion function is bounded by O(ε{lunate}2), independent of the time variable. This implies that the simpler geometric optics expansion functions can be employed to study the behavior of general entropy solutions to hyperbolic systems of conservation laws. Finally, we extend the results to the case with non-compact support initial data of bounded variation. © 2013 Copyright Taylor and Francis Group, LLC.

Research Area(s)

  • Approximate equations, Arbitrary initial data, Asymptotic behavior, Convergence, Entropy solutions, Genuinely nonlinear, Hyperbolic systems of conservation laws, Leading terms, Linearly degenerate, Lipschitz semigroup, Nonstrictly hyperbolic, Riemann solver, Validity, Weakly nonlinear geometric optics

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