Title | Vanishing viscosity solutions of nonlinear hyperbolic systems |

Publication Type | Journal Article |

Year of Publication | 2005 |

Authors | Bianchini, S, Bressan, A |

Journal | Ann. of Math. 161 (2005) 223-342 |

Abstract | We consider the Cauchy problem for a strictly hyperbolic, $n\\\\times n$ system in one space dimension: $u_t+A(u)u_x=0$, assuming that the initial data has small total variation.\\nWe show that the solutions of the viscous approximations $u_t+A(u)u_x=\\\\ve u_{xx}$ are defined globally in time and satisfy uniform BV estimates, independent of $\\\\ve$. Moreover, they depend continuously on the initial data in the $\\\\L^1$ distance, with a Lipschitz constant independent of $t,\\\\ve$. Letting $\\\\ve\\\\to 0$, these viscous solutions converge to a unique limit, depending Lipschitz continuously on the initial data. In the conservative case where $A=Df$ is the Jacobian of some flux function $f:\\\\R^n\\\\mapsto\\\\R^n$, the vanishing viscosity limits are precisely the unique entropy weak solutions to the system of conservation laws $u_t+f(u)_x=0$. |

URL | http://hdl.handle.net/1963/3074 |

## Vanishing viscosity solutions of nonlinear hyperbolic systems

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