Abstract
We consider the problem of self-similar zero-viscosity limits for systems ofN conservation laws. First, we give general conditions so that the resulting boundary-value problem admits solutions. The obtained existence theory covers a large class of systems, in particular the class of symmetric hyperbolic systems. Second, we show that if the system is strictly hyperbolic and the Riemann data are sufficiently close, then the resulting family of solutions is of uniformly bounded variation and oscillation. Third, we construct solutions of the Riemann problem via self-similar zero-viscosity limits and study the structure of the emerging solution and the relation of self-similar zero-viscosity limits and shock profiles. The emerging solution consists ofN wave fans separated by constant states. Each wave fan is associated with one of the characteristic fields and consists of a rarefaction, a shock, or an alternating sequence of shocks and rarefactions so that each shock adjacent to a rarefaction on one side is a contact discontinuity on that side. At shocks, the solutions of the self-similar zero-viscosity problem have the internal structure of a traveling wave.
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References
C. M. Dafermos, Solution of the Riemann problem for a class of hyperbolic conservation laws by the viscosity method,Arch. Rational Mech. Anal. 52 (1973), 1–9.
C. M. Dafermos, Structure of solutions of the Riemann problem for hyperbolic systems of conservations laws,Arch Rational Mech. Anal. 53 (1974), 203–217.
C. M. Dafermos &R. J. Diperna, The Riemann problem for certain classes of hyperbolic systems of conservation laws,J. Diff. Eqns. 20 (1976), 90–114.
C. M. Dafermos, Admissible wave fans in nonlinear hyperbolic systems,Arch. Rational Mech. Anal. 106 (1989), 243–260.
R. J. Diperna, Singularities of solutions of nonlinear hyperbolic systems of conservation laws,Arch. Rational Mech. Anal. 60 (1975), 75–100.
H. T. Fan, A limiting “viscosity” approach to the Riemann problem for materials exhibiting change of phase (II),Arch. Rational Mech. Anal. 116 (1992), 317–338.
H. T. Fan, One-phase Riemann problem and wave interactions in systems of conservation laws of mixed type,SIAM J. Math. Anal. 24 (1993), 840–865.
G. B. Folland,Real Analysis. Modern Techniques and their Applications, Wiley Interscience, New York, 1984.
J. Glimm, Solutions in the large for nonlinear hyperbolic systems of conservation laws,Comm. Pure Appl. Math. 18 (1965), 697–715.
A. S. Kalašnikov, Construction of generalized solutions of quasi-linear equations of first order without convexity conditions as limits of solutions of parabolic equations with a small parameter,Dokl. Akad. Nauk SSSR 127 (1959), 27–30 (in Russian).
B. Keyfitz &H. Kranzer, A viscosity approximation to a system of conservation laws with no classical Riemann solution, inProceedings of International Conference on Hyperbolic Problems, Bordeaux, 1988.
P. D. Lax, Hyperbolic Systems of Conservation Laws II,Comm. Pure Appl. Math. 10 (1957), 537–566.
P. D. Lax, Shock waves and entropy, inContributions to Nonlinear Functional Analysis.E. H. Zarantonello, ed., Academic Press, New York, 1971, pp. 603–634.
T.-P. Liu, The Riemann problem for general 2 × 2 conservation laws,Trans. Amer. Math. Soc. 199 (1974), 89–112.
T.-P. Liu, The Riemann problem for general systems of conservation laws,J. Diff. Eqns. 18 (1975), 218–234.
T.-P. Liu, The entropy condition and the admissibility of shocks,J. Math. Anal. Appl. 53 (1976), 78–88.
T.-P. Liu, Admissible solutions of hyperbolic conservation laws,Memoirs Amer. Math. Soc. 240 (1981), 1–78.
A. Majda &R. L. Pego, Stable viscosity matrices for systems of conservation laws,J. Diff. Eqns. 56 (1985), 229–262.
P. H. Rabinowitz,Théorie du Degré Topologique et Applications à des Problèmes aux Limites non Linéaires, rédigé parH. Berestycki, Laboratoire d'Analyse Numérique, Université Paris VI, 1975.
M. Slemrod, A limiting “viscosity” approach to the Riemann problem for materials exhibiting change of phase,Arch. Rational Mech. Anal. 105 (1989), 327–365.
M. Slemrod, A comparison of two viscous regularizations of the Riemann problem for Burgers's equation,SIAM J. Math. Anal. 26 (1995), 1415–1424.
M. Slemrod &A. E. Tzavaras, A limiting viscosity approach for the Riemann problem in isentropic gas dynamics,Indiana Univ. Math. J. 38 (1989), 1047–1074.
M. Slemrod &A. E. Tzavaras, Self-similar fluid-dynamic limits for the Broadwell system,Arch. Rational Mech. Anal. 122 (1993), 353–392.
A. E. Tzavaras, Wave structure induced by fluid dynamic limits in the Broadwell model,Arch. Rational Mech. Anal. 127 (1994), 361–387.
A. E. Tzavaras, Elastic as limit of viscoelastic response, in a context of self-similar viscous limits,J. Diff. Eqns. 123 (1995), 305–341.
V. A. Tupciev, Collapse of an arbitrary discontinuity for a system of two quasilinear first order equations,Z. Vycisl. Mat. i Mat. Fiz. 4 (1964), 817–825. English translation:USSR Comput. Math. Math. Phys. 4 (1964), 36–48.
V. A. Tupciev, On the method of introducing viscosity in the study of problems involving the decay of a discontinuity,Dokl. Akad. Nauk SSSR 211 (1973), 55–58. English translation:Soviet Math. Dokl. 14 (1973), 978–982.
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Tzavaras, A.E. Wave interactions and variation estimates for self-similar zero-viscosity limits in systems of conservation laws. Arch. Rational Mech. Anal. 135, 1–60 (1996). https://doi.org/10.1007/BF02198434
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DOI: https://doi.org/10.1007/BF02198434