Boundary conditions for nonlinear hyperbolic systems of conservation laws.

*(English)*Zbl 0649.35057This 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:

[1] | Ballou, D, Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions, Trans. amer. math. soc., 152, 441-460, (1970) · Zbl 0207.40401 |

[2] | Bardos, C, Introduction aux problèmes hyperboliques non linéaires, () · Zbl 0549.35078 |

[3] | Bardos, C; Leroux, A.Y; Nedelec, J.C, First order quasilinear equations with boundary conditions, Comm. partial differential equations, 4, No. 9, 1017-1034, (1979) · Zbl 0418.35024 |

[4] | Benabdallah, A, The “-system” on an interval, C. R. acad. sci. Paris Sér. 1 math, 303, No. 4, 123-126, (1986) · Zbl 0607.35061 |

[5] | Dubois, F; Le Floch, P, Boundary condition for systems of hyperbolic conservation laws, C. R. acad. sci. Paris Sér. 1 math., 304, No. 3, 75-78, (1987) · Zbl 0634.35046 |

[6] | Godunov, S.K, Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. sb, 47, No. 3, 271, (1959) · Zbl 0171.46204 |

[7] | Goodmann, J, Initial boundary value problems for hyperbolic systems of conservation laws, () |

[8] | Gustafsson, B; Kreiss, H.O; Sundström, A, Stability theory of difference approximations for mixed initial boundary value problems, II, Math. comp., 26, 649-686, (1972) · Zbl 0293.65076 |

[9] | Howes, F.A, Multidimensional initial-boundary value problems with strong linearities, Arch. rational mech. anal., 91, No. 2, 153-168, (1986) · Zbl 0593.35010 |

[10] | Kreiss, H.O, Initial boundary value problems for hyperbolic systems, Comm. pure appl. math., 23, 277-298, (1970) · Zbl 0193.06902 |

[11] | Kruzkov, S.N, First order quasi-linear systems in several independant variables, Math. USSR sb., 10, No. 2, 217-243, (1970) |

[12] | Ladyzenskaya, O.A; Uralceva, N.N, Boundary problems for linear and quasilinear parabolic equations, Amer. math. soc. transl. (2), 47, 217-299, (1965) |

[13] | Lax, P.D, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. pure appl. math., 7, 159-193, (1954) · Zbl 0055.19404 |

[14] | Lax, P.D, Hyperbolic systems of conservation laws, II, Comm. pure. appl. math., 10, 537-566, (1957) · Zbl 0081.08803 |

[15] | Lax, P.D, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, (1973), SIAM Philadelphia · Zbl 0268.35062 |

[16] | \scP. Le Floch, Boundary conditions for scalar nonlinear conservation laws, Math. Methods Appl. Sci. in press. · Zbl 0679.35065 |

[17] | Le Floch, P; Nedelec, J.C, Weighted scalar conservation laws, C.R. acad. sci. Paris Sér 1 math., 301, No. 17, 1301-1304, (1985) |

[18] | Internal Report No. 144, Centre de Mathématiques Appliquées de l’École Polytechnique, Trans. Amer. Math. Soc., in press. |

[19] | Leroux, A.Y, Approximation de quelques problèmes hyperboliques non-linéaires, Thèse d’état, (1979), Rennes |

[20] | Liu, T.P, Initial-boundary value problems for gas dynamics, Arch. rational mech. anal., 64, 137-168, (1977) · Zbl 0357.35016 |

[21] | Nishida, T; Smoller, J, Mixed problems for nonlinear conservation laws, J. differential equations, 23, No. 2, 244-269, (1977) · Zbl 0303.35052 |

[22] | Rockafellar, T, Convex analysis, (1972), Princeton Univ. Princeton NJ, Press · Zbl 0224.49003 |

[23] | Serre, D, Oscillating data in one nonlinear hyperbolic system, C.R. acad. sci. Paris Sér. 1 math., No. 3, 115-118, (1986) · Zbl 0606.35050 |

[24] | Smoller, J, Shock waves and reaction-diffusion equations, (1983), Springer-Verlag New York · Zbl 0508.35002 |

[25] | Viviand, H; Veuillot, J.P, Méthodes pseudoinstationnaires pour le calcul d’écoulements transsoniques, Publication ONERA 1978-4, (1978) |

[26] | Volpert, A.I, The space BV and quasilinear equations, Mat. sb., 73, No. 115, 255-302, (1967) |

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