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

John K. Hunter; Yuxi Zheng

doi:10.1007/BF00379260

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

John K. Hunter, Yuxi Zheng

Mathematics

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Abstract

We consider viscosity and dispersion regularizations of the nonlinear hyperbolic partial differential equation (u_t+uu_x)_x=1/2u_x² 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.

Original language	English (US)
Pages (from-to)	355-383
Number of pages	29
Journal	Archive for Rational Mechanics and Analysis
Volume	129
Issue number	4
DOIs	https://doi.org/10.1007/BF00379260
State	Published - Dec 1 1995

All Science Journal Classification (ASJC) codes

Analysis
Mathematics (miscellaneous)
Mechanical Engineering

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10.1007/BF00379260

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

abstract = "We consider viscosity and dispersion regularizations of the nonlinear hyperbolic partial differential equation (ut+uux)x=1/2ux2 with the simplest initial data such that ux 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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AU - Zheng, Yuxi

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N2 - We consider viscosity and dispersion regularizations of the nonlinear hyperbolic partial differential equation (ut+uux)x=1/2ux2 with the simplest initial data such that ux 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.

AB - We consider viscosity and dispersion regularizations of the nonlinear hyperbolic partial differential equation (ut+uux)x=1/2ux2 with the simplest initial data such that ux 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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ER -

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

Abstract

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