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On a nonlinear hyperbolic variational equation: II. The zero-viscosity and dispersion limits

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Abstract

We consider viscosity and dispersion regularizations of the nonlinear hyperbolic partial differential equation (u t+uux)x=1/2u 2x with the simplest initial data such that u x blows up in finite time. We prove that the zero-viscosity limit selects a unique global weak solution of the partial differential equation without viscosity. We also present numerical experiments which indicate that the zero-dispersion limit selects a different global weak solution of the same initial-value problem.

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Authors and Affiliations

  1. Department of Mathematics and Institute for Theoretical Dynamics, University of California at Davis, USA

    John K. Hunter & Yuxi Zheng

  2. Department of Mathematics, Indiana University at Bloomington, USA

    John K. Hunter & Yuxi Zheng

Authors
  1. John K. Hunter
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  2. Yuxi Zheng
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Communicated by C. Dafermos

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Hunter, J.K., Zheng, Y. On a nonlinear hyperbolic variational equation: II. The zero-viscosity and dispersion limits. Arch. Rational Mech. Anal. 129, 355–383 (1995). https://doi.org/10.1007/BF00379260

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  • DOI: https://doi.org/10.1007/BF00379260

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