Convergence of the viscosity method for isentropic gas dynamics. (English) Zbl 0533.76071
In this paper a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established. The author is concerned with the zero diffusion limit of hyperbolic systems of conservation laws. One natural strategy for proving convergence as the diffusion parameter vanishes is to look for uniform estimates on the amplitude and the derivatives of the approximate solutions and then appeal to a compactness argument in order to extract a strongly convergent subsequence.
Reviewer: I.Teipel
Keywords:
convergence theorem; artificial viscosity; isentropic equations; zero diffusion limit of hyperbolic systems of conservation laws; compactness argumentReferences:
| [1] | Ball, J.M.: On the calculus of variations and sequentially weakly continuous maps. In: Proceedings Dundee Conference on Ordinary and Partial Differential Equations (1976). Lecture Notes in Mathematics, Vol. 564. Berlin, Heidelberg, New York: Springer 1976, pp. 13-25 · Zbl 0348.49004 |
| [2] | Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal.63, 337-403 (1977) · Zbl 0368.73040 · doi:10.1007/BF00279992 |
| [3] | Ball, J.M., Currie, J.C., Olver, P.J.: Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Functional Analysis41, 135-175 (1981) · Zbl 0459.35020 · doi:10.1016/0022-1236(81)90085-9 |
| [4] | Bahkvarov, N.: On the existence of regular solutions in the large for quasilinear hyperbolic systems. Zh. Vychisl. Mat. Fiz.10, 969-980 (1970) |
| [5] | Dacorogna, B.: Weak continuity and weak lower semicontinuity of nonlinear functionals. Lefschetz Center for Dynamical Systems Lecture Notes, Vol. 81-77. Brown University 1981 |
| [6] | Dacorogna, B.: A relaxation theorem and its applications to the equilibrium of gases. Arch. Rat. Mech. Anal. (to appear) · Zbl 0492.49002 |
| [7] | Dacorogna, B.: A generic result for nonconvex problems in the calculus of variations. J. Functional Analysis (to appear) · Zbl 1101.35001 |
| [8] | DiPerna, R.J.: Global solutions to a class of nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math.26, 1-28 (1973) · Zbl 0256.35053 · doi:10.1002/cpa.3160260102 |
| [9] | DiPerna, R.J.: Existence in the large for nonlinear hyperbolic conservation laws. Arch. Rat. Mech. Anal.52, 244-257 (1973) · Zbl 0268.35066 · doi:10.1007/BF00247735 |
| [10] | DiPerna, R.J.: Convergence of approximate solutions to conservation laws. Arch. Rat. Mech. Analysis (1983) (to appear) · Zbl 0519.35054 |
| [11] | Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math.18, 697-715 (1965) · Zbl 0141.28902 · doi:10.1002/cpa.3160180408 |
| [12] | Glimm, J., Lax, P.D.: Decay of solutions of systems of nonlinear hyperbolic conservation laws. Am. Math. Soc.101 (1970) · Zbl 0204.11304 |
| [13] | Lax, P.D.: Hyperbolic systems of conservation laws. II. Commun. Pure Appl. Math.10, 537-566 (1957) · Zbl 0081.08803 · doi:10.1002/cpa.3160100406 |
| [14] | Lax, P.D.: Shock waves and entropy. In: Contributions to nonlinear functional analysis, Zarantonello, E.A. (ed.). New York: Academic Press 1971, pp. 603-634 |
| [15] | Lax, P.D., Wendroff, B.: Systems of conservation laws. Commun. Pure Appl. Math.13, 217-237 (1960) · Zbl 0152.44802 · doi:10.1002/cpa.3160130205 |
| [16] | Murat, F.: Compacité par compensation. Ann. Scula Norm. Sup. Pisa Sci. Fis. Math.5, 489-507 (1978) · Zbl 0399.46022 |
| [17] | Murat, F.: Compacité par compensation: Condition necessaire et suffisante de continuité faible sous une hypotheses de rang constant. Ann. Suola Norm. Sup. Pisa8, 69-102 (1981) · Zbl 0464.46034 |
| [18] | Murat, F.: L’injection du cone positif deH ?1 dansW ?1,q est compacte pour toutq>2 (preprint) |
| [19] | Nishida, T.: Global solutions for an initial boundary value problem of a quasilinear hyperbolic system. Proc. Jpn. Acad.44, 642-646 (1968) · Zbl 0167.10301 · doi:10.3792/pja/1195521083 |
| [20] | Nishida, T., Smoller, J.A.: Solutions in the large for some nonlinear hyperbolic conservation laws. Commun. Pure Appl. Math.26, 183-200 (1973) · Zbl 0267.35058 · doi:10.1002/cpa.3160260205 |
| [21] | Tartar, L.: Une nouvelle méthode de résolution d’équations aux dérivées partielles nonlinéaires. In: Lecture Notes in Mathematics, Vol. 665. Berlin, Heidelberg, New York: Springer 1977, pp. 228-241. |
| [22] | Tartar, L.: Compensated compactness and applications to partial differential equations. In: Research notes in mathematics, nonlinear analysis, and mechanics: Heriot-Watt Symposium, Vol. 4, Knops, R.J. (ed.). New York: Pitman Press 1979 |
| [23] | Tartar, L.: The compensated compactness method applied to systems of conservation laws. In: Systems of nonlinear partial differential equations, Ball, J.M. (ed.). NATO ASI Series, C. Reidel Publishing Co. (1983) · Zbl 0536.35003 |
| [24] | Isaacson, E.: Global solution to a Riemann Problem for a non-strictly hyperbolic system of conservation laws arising in enhanced oil recovery. J. Comp. Phys. (to appear) |
| [25] | Temple, B.: Global solution of the Cauchy problem for a class of 2 {\(\times\)} 2 non-strictly hyperbolic conservation laws. Adv. Appl. Math. (to appear) · Zbl 0508.76107 |
| [26] | Keyfitz, B., Kranzer, H.: A system of non-strictly hyperbolic conservation laws arising in elasticity theory. Arch. Rat. Mech. Anal.72 (1980) · Zbl 0434.73019 |
| [27] | Betsadze, A.V.: Equations of the mixed type. New York: Macmillan Company 1964 |
| [28] | Whitham, G.B.: Linear and nonlinear waves. New York: Wiley 1974 · Zbl 0373.76001 |
| [29] | Rudin, W.: Real and complex analysis. New York: McGraw-Hill 1966 · Zbl 0142.01701 |
| [30] | Chueh, K.N., Conley, C.C., Smoller, J.A.: Positively invariant regions for systems of nonlinear diffusion equations. Ind. Univ. Math. J.26, 372-411 (1977) · Zbl 0368.35040 · doi:10.1512/iumj.1977.26.26029 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
