On the vanishing viscosity approximation to the Cauchy problem for a 2 × 2 system of conservation laws
Annales de l'I.H.P. Analyse non linéaire, Volume 10 (1993) no. 6, p. 627-656
@article{AIHPC_1993__10_6_627_0,
author = {Rubino, Bruno},
title = {On the vanishing viscosity approximation to the Cauchy problem for a 2 $\times$ 2 system of conservation laws},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Gauthier-Villars},
volume = {10},
number = {6},
year = {1993},
pages = {627-656},
zbl = {0806.35117},
mrnumber = {1253605},
language = {en},
url = {http://www.numdam.org/item/AIHPC_1993__10_6_627_0}
}

Rubino, Bruno. On the vanishing viscosity approximation to the Cauchy problem for a 2 × 2 system of conservation laws. Annales de l'I.H.P. Analyse non linéaire, Volume 10 (1993) no. 6, pp. 627-656. http://www.numdam.org/item/AIHPC_1993__10_6_627_0/

[1] G.Q. Chen, X. Ding and P. Luo, Convergence of the Lax-Friedrichs Scheme for Isentropic Gas Dynamics, I, Acta Math. Sci., Vol. 5, 1985, pp. 415-432; II, Acta Math. Sci., Vol. 5, 1985, pp 433-472. | Zbl 0643.76084

[2] G.Q. Chen, Convergence of the Lax-Friedrichs Scheme for Isentropic Gas Dynamics, III, Acta Math. Sci., Vol. 6, 1986, pp. 75-120. | Zbl 0643.76086

[3] K.N. Chueh, C.C. Conley and J.A. Smoller, Positively Invariant Regions for Systems of Non-Linear Diffusion Equations, Indiana Univ. Math. J., Vol. 26, 1977, pp. 372-411. | Zbl 0368.35040

[4] R. Courant and D. Hilbert, Methods of Mathematicals Physics II: Partial Differential Equations, Wiley and Sons, 1962. | Zbl 0099.29504

[5] R.J. Diperna, Compensated Compactness and General Systems of Conservation Laws, Trans. Amer. Math. Soc., Vol. 292, 1985, pp. 383-420. | Zbl 0606.35052

[6] R.J. Diperna, Convergence of Approximate Solutions to Conservation Laws, Arch. Rational Mech. Anal., Vol. 82, 1983, pp. 27-70. | Zbl 0519.35054

[7] R.J. Diperna, Convergence of the Viscosity Method for Isentropic Gas Dynamics, Comm. Math. Phys., Vol. 91, 1983, pp. 1-30. | Zbl 0533.76071

[8] R.J. Diperna, Measure-Valued Solutions to Conservation Laws, Arch. Rational Mech. Anal., Vol. 88, 1985, pp. 223-270. | Zbl 0616.35055

[9] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, 1964. | Zbl 0144.34903

[10] G. Glimm, The Interaction of Non-Linear Hyperbolic Waves, Comm. Pure Appl. Math., Vol. 41, 1988, pp. 569-590. | Zbl 0635.35067

[11] L. Hörmander, Non-Linear Hyperbolic Differential Equations, Lectures notes, 1986- 1987, Lund University, Sweden, 1988, mineographied notes.

[12] E. Isaacson, D. Marchesin, B. Plohr and B. Temple, The Riemann Problem Near a Hyperbolic Singularity: the Classification of Solutions of Quadratic Riemann Problems I, S.I.A.M. J. Appl. Math., Vol. 48, 1988, pp. 1009-1032. | Zbl 0688.35056

[13] E. Isaacson and B. Temple, The Classification of Solutions of Quadratic Riemann Problems II, S.I.A.M. J. Appl. Math., Vol. 48, 1988, pp. 1287-1301; III, S.I.A.M. J. Appl. Math., Vol. 48, 1988, pp. 1302-1318.

[14] P.-T. Kan, On the Cauchy Problem of a 2 × 2 Systems of Non-Strictly Hyperbolic Conservation Laws, Ph.D. Thesis, Courant Institute of Math. Sciences, N.Y. University, 1989.

[15] B.L. Keyfitz and H.C. Kranzer, A System of Non-Strictly Hyperbolic Conservation Laws Arising in Elasticity Theory, Arch. Rational Mech. Anal., Vol. 72, 1980, pp. 219- 241. | Zbl 0434.73019

[16] S. Kružkov, First Order Quasi-Linear Equations with Several Space Variables, Mat. Sb., Vol. 123, 1970, pp. 228-255.

[17] P.D. Lax, Asymptotic Solutions of Oscillatory Initial value Problems, Duke Math. J., Vol. 24, 1957, pp. 627-646. | Zbl 0083.31801

[18] P.D. Lax, Shock Waves and Entropy, in Contributions to Nonlinear Functional Analysis, E. A. ZARANTONELLO Ed., Academic Press, 1971, pp. 603-634. | Zbl 0268.35014

[19] P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, S.I.A.M., Philadelphia, 1973. | Zbl 0268.35062

[20] T.P. Liu, Zero Dissipative Limit for Conservation Laws, in Proceedings of the Conference on Non-linear Variational Problems and Partial Differential Equations, Isola d'Elba, 1990 (to appear).

[21] P. Marcati and A. Milani, The One-Dimensional Darcy's Law as the Limit of a Compressible Euler Flow, J. Differential Equations, 1990, pp. 129-147. | Zbl 0715.35065

[22] F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Vol. 5, 1978, pp. 489-507. | Numdam | Zbl 0399.46022

[23] I.G. Petrovskii, Partial Differential Equations, Iliffe Books, 1967. | Zbl 0163.11706

[24] D.G. Schaeffer and M. Shearer, The Classification of 2 × 2 Systems of Non-Strictly Hyperbolic Conservation Laws, with Application to Oil Recovery, Comm. Pure Appl. Math., Vol. 40, 1987, pp. 141-178. | Zbl 0673.35073

[25] D.G. Schaeffer and M. Shearer, Riemann Problems for Non-Strictly Hyperbolic 2 x 2 Systems of Conservation Laws, Trans. Amer. Math. Soc., Vol. 304, 1987, pp. 267- 306. | Zbl 0656.35081

[26] D. Serre, La compacité par compensation pour les systèmes hyperboliques non linéaires de deux équations a une dimension d'espace, J. Math. Pures Appl., Vol. 65, 1986, pp. 423-468. | Zbl 0601.35070

[27] J.A. Smoller, Shock Waves and Reaction Diffusion Equations, Springer Verlag, 1983. | Zbl 0508.35002

[28] S.L. Sobolev, Partial Differential Equations of Mathematical Physics, Pergamon Press, 1964. | Zbl 0123.06508

[29] L. Tartar, Compensated Compactness and Applications to Partial Differential Equations, in Nonlinear Analysis and Mechanics: Heriott-Watt Symposium, IV, Research Notes in Math., 1979, pp. 136-210. | Zbl 0437.35004

[30] B. Temple, Global Existence of the Cauchy Problem for a Class of 2 x 2 Non-Strictly Hyperbolic Conservation Laws, Adv. in Appl. Math., Vol. 3, 1982, pp. 355-375. | Zbl 0508.76107