### Abstract

We consider the Cauchy problem for a strictly hyperbolic, n × n system in one-space dimension: u_{t} + A(u)u_{x} = 0, assuming that the initial data have small total variation. We show that the solutions of the viscous approximations u_{t} + A(u)u_{x} = εu _{xx} are defined globally in time and satisfy uniform BV estimates, independent of ε. Moreover, they depend continuously on the initial data in the L^{1} distance, with a Lipschitz constant independent of t, ε. Letting ε → 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} → 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.

Original language | English (US) |
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Pages (from-to) | 223-342 |

Number of pages | 120 |

Journal | Annals of Mathematics |

Volume | 161 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2005 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Annals of Mathematics*,

*161*(1), 223-342. https://doi.org/10.4007/annals.2005.161.223