The Riemann problem for general systems of conservation laws

Liu, Tai-Ping

doi:10.1016/0022-0396(75)90091-1

Cited by 299 publications

(239 citation statements)

References 6 publications

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“…In [42], P. Lax constructed solutions to the Riemann problem for a wide class of n × n strictly hyperbolic systems. Further results were provided by T. P. Liu in [45], dealing with systems under generic assumptions. The paper [6] by S. Bianchini provides a fully general construction, valid even for systems not in conservation form.…”

Section: The Riemann Problemmentioning

confidence: 99%

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Hyperbolic Conservation Laws: An Illustrated Tutorial

Bressan

2012

Lecture Notes in Mathematics

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These notes provide an introduction to the theory of hyperbolic systems of conservation laws in one space dimension. The various chapters cover the following topics: 1. Meaning of a conservation equation and definition of weak solutions. 2. Hyperbolic systems. Explicit solutions in the linear, constant coefficients case. Nonlinear effects: loss of regularity and wave interactions. 3. Shock waves: Rankine-Hugoniot equations and admissibility conditions. 4. Genuinely nonlinear and linearly degenerate characteristic fields. Centered rarefaction waves. The general solution of the Riemann problem. Wave interaction estimates. 5. Weak solutions to the Cauchy problem, with initial data having small total variation. Approximations generated by the front-tracking method and by the Glimm scheme. 6. Continuous dependence of solutions w.r.t. the initial data, in the L 1 distance. 7. Characterization of solutions which are limits of front tracking approximations. Uniqueness of entropy-admissible weak solutions. 8. Vanishing viscosity approximations. 9.Extensions and open problems. The survey is concluded with an Appendix, reviewing some basic analytical tools used in the previous sections.Throughout the exposition, technical details are mostly left out. The main goal of these notes is to convey basic ideas, also with the aid of a large number of figures.

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Section: The Riemann Problemmentioning

confidence: 99%

“…This considerably simplifies all further analysis. On the other hand, for strictly hyperbolic systems that do not satisfy the condition (H), basic existence and stability results can still be obtained, but at the price of heavier technicalities [45].…”

Section: A Class Of Hyperbolic Systemsmentioning

confidence: 99%

Hyperbolic Conservation Laws: An Illustrated Tutorial

Bressan

2012

Lecture Notes in Mathematics

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“…As concerns boundary Riemann solvers, a solution of the initial boundary value problem 27) was proposed in [21] in the case of systems in conservation form with only linearly degenerate or genuinely non linear fields: such a boundary Riemann solver is in general different from the one defined by the vanishing viscosity limit (some more precise considerations can be found in Remark 6.1). On the other side, in [25,31,1] and [2] it was considered a quite general boundary condition, which turns out to be compatible with the one defined by the limit of vanishing viscosity approximations: we refer again to Remark 6.1 for a more precise statement.…”

Section: Remark 12mentioning

confidence: 99%

Vanishing viscosity solutions of a 2 \times 2 triangular hyperbolic system with Dirichlet conditions on two boundaries

Spinolo¹

2007

Indiana Univ. Math. J.

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We consider the 2 × 2 parabolic systemswith Dirichlet boundary conditions imposed at x = 0 and at x = l. The matrix A is assumed to be in triangular form and strictly hyperbolic, and the boundary is not characteristic, i.e. the eigenvalues of A are different from 0. We show that, if the initial and boundary data have sufficiently small total variation, then the solution u ε exists for all t ≥ 0 and depends Lipschitz continuously in L 1 on the initial and boundary data.Moreover, as ε → 0 + , the solutions u ε (t) converge in L 1 to a unique limit u(t), which can be seen as the vanishing viscosity solution of the quasilinear hyperbolic systemThis solution u(t) depends Lipschitz continuously in L 1 w.r.t the initial and boundary data. We also characterize precisely in which sense the boundary data are assumed by the solution of the hyperbolic system.2000 Mathematics Subject Classification: 35L65.

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“…The traditional approach for constructing the Riemann problem solutions proceeds by constructing the wave curves associated to each characteristic family, employing admissibility criteria to select the admissible shocks. Effective admissibility criteria for the selection of admissible shocks were proposed by Lax [30] for genuinely nonlinear wave speeds and by Liu [32,33] when the wave speeds lose genuine nonlinearity in finitely many points, and serve to provide a unique solution of (13) for waves of moderate strength.…”

Section: Wave Fan Admissibility Criteriamentioning

confidence: 99%

Remarks on the Contributions of Constantine M. Dafermos to the Subject of Conservation Laws

Tzavaras

2012

Acta Mathematica Scientia

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The Riemann problem for general systems of conservation laws

Cited by 299 publications

References 6 publications

Hyperbolic Conservation Laws: An Illustrated Tutorial

Hyperbolic Conservation Laws: An Illustrated Tutorial

Vanishing viscosity solutions of a 2 \times 2 triangular hyperbolic system with Dirichlet conditions on two boundaries

Remarks on the Contributions of Constantine M. Dafermos to the Subject of Conservation Laws

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