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**Boundary conditions for nonlinear hyperbolic systems of conservation laws.**
*(English)*
Zbl 0649.35057

This very interesting paper considers the issue of the admissibility of boundary conditions for systems of hyperbolic conservation laws. Typically conservation laws are treated with respect to the pure Cauchy initial value problem and boundary conditions are not considered. On the other hand applications make the issue of mixed initial-boundary value problems imperative.

In this paper the authors present two approaches. First they suggest an approach based on viscous limits of regularized conservation laws. The approach is shown to actually provide a well posed mixed initial-boundary value problem for both linear hyperbolic systems and scalar nonlinear conservation laws. Second they also suggest an approach based on Riemann problems and show that for linear hyperbolic systems and scalar nonlinear conservation laws the approaches are equivalent and hence the second approach leads to well posed problems as well. The paper is a valuable contribution to the literature on hyperbolic conservation laws.

In this paper the authors present two approaches. First they suggest an approach based on viscous limits of regularized conservation laws. The approach is shown to actually provide a well posed mixed initial-boundary value problem for both linear hyperbolic systems and scalar nonlinear conservation laws. Second they also suggest an approach based on Riemann problems and show that for linear hyperbolic systems and scalar nonlinear conservation laws the approaches are equivalent and hence the second approach leads to well posed problems as well. The paper is a valuable contribution to the literature on hyperbolic conservation laws.

Reviewer: M.Slemrod

### MSC:

35L65 | Hyperbolic conservation laws |

35L50 | Initial-boundary value problems for first-order hyperbolic systems |

35L45 | Initial value problems for first-order hyperbolic systems |

35L60 | First-order nonlinear hyperbolic equations |

### Keywords:

admissibility of boundary conditions; hyperbolic conservation laws; viscous limits; well posed; initial-boundary value problem; Riemann problems
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\textit{F. Dubois} and \textit{P. Le Floch}, J. Differ. Equations 71, No. 1, 93--122 (1988; Zbl 0649.35057)

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### References:

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