### Abstract

A natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension, u_{1} + f(u)_{x} = 0, uε{lunate}R^{m}, is studied. These matrices are admissible in the sense that small-amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system u_{t} + f(u)_{x} = v(Du_{x})_{x} as ifv → 0 if D is in this class. The class is determined by a linearized stability requirement: The Cauchy problem for the equation u_{1} + f′(u_{0}) u_{x} = vDu_{xx} should be well posed in L^{2} uniformly in v as v → 0. Previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion.

Original language | English (US) |
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Pages (from-to) | 229-262 |

Number of pages | 34 |

Journal | Journal of Differential Equations |

Volume | 56 |

Issue number | 2 |

DOIs | |

State | Published - 1985 |

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### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Differential Equations*,

*56*(2), 229-262. https://doi.org/10.1016/0022-0396(85)90107-X

**Stable viscosity matrices for systems of conservation laws.** / Majda, Andrew; Pego, Robert L.

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 56, no. 2, pp. 229-262. https://doi.org/10.1016/0022-0396(85)90107-X

}

TY - JOUR

T1 - Stable viscosity matrices for systems of conservation laws

AU - Majda, Andrew

AU - Pego, Robert L.

PY - 1985

Y1 - 1985

N2 - A natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension, u1 + f(u)x = 0, uε{lunate}Rm, is studied. These matrices are admissible in the sense that small-amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system ut + f(u)x = v(Dux)x as ifv → 0 if D is in this class. The class is determined by a linearized stability requirement: The Cauchy problem for the equation u1 + f′(u0) ux = vDuxx should be well posed in L2 uniformly in v as v → 0. Previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion.

AB - A natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension, u1 + f(u)x = 0, uε{lunate}Rm, is studied. These matrices are admissible in the sense that small-amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system ut + f(u)x = v(Dux)x as ifv → 0 if D is in this class. The class is determined by a linearized stability requirement: The Cauchy problem for the equation u1 + f′(u0) ux = vDuxx should be well posed in L2 uniformly in v as v → 0. Previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion.

UR - http://www.scopus.com/inward/record.url?scp=0002124586&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0002124586&partnerID=8YFLogxK

U2 - 10.1016/0022-0396(85)90107-X

DO - 10.1016/0022-0396(85)90107-X

M3 - Article

VL - 56

SP - 229

EP - 262

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 2

ER -