Title | On the convergence rate of vanishing viscosity approximations |

Publication Type | Journal Article |

Year of Publication | 2004 |

Authors | Bressan, A, Yang, T |

Journal | Comm. Pure Appl. Math. 57 (2004) 1075-1109 |

Abstract | Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound $\\\\big\\\\|u(t,\\\\cdot)-u^\\\\ve(t,\\\\cdot)\\\\big\\\\|_{\\\\L^1}= \\\\O(1)(1+t)\\\\cdot \\\\sqrt\\\\ve|\\\\ln\\\\ve|$ on the distance between an exact BV solution $u$ and a viscous approximation $u^\\\\ve$, letting the viscosity coefficient $\\\\ve\\\\to 0$. In the proof, starting from $u$ we construct an approximation of the viscous solution $u^\\\\ve$ by taking a mollification $u*\\\\phi_{\\\\strut \\\\sqrt\\\\ve}$ and inserting viscous shock profiles at the locations of finitely many large shocks, for each fixed $\\\\ve$. Error estimates are then obtained by introducing new Lyapunov functionals which control shock interactions, interactions between waves of different families and by using sharp decay estimates for positive nonlinear waves. |

URL | http://hdl.handle.net/1963/2915 |

DOI | 10.1002/cpa.20030 |

## On the convergence rate of vanishing viscosity approximations

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