Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Compensated compactness and general systems of conservation laws


Author: Ronald J. DiPerna
Journal: Trans. Amer. Math. Soc. 292 (1985), 383-420
MSC: Primary 35L65; Secondary 76L05
MathSciNet review: 808729
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Abstract: We outline a general program and present some new results dealing with oscillations in weakly convergent solution sequences to systems of conservation laws. The analysis employs the Young measure and the Tartar-Murat theory of compensated compactness and deals with systems of hyperbolic and elliptic type.


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  • [1] J. M. Ball, On the calculus of variations and sequentially weakly continuous maps, Ordinary and partial differential equations (Proc. Fourth Conf., Univ. Dundee, Dundee, 1976) Springer, Berlin, 1976, pp. 13–25. Lecture Notes in Math., Vol. 564. MR 0637229
  • [2] John M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1976/77), no. 4, 337–403. MR 0475169
  • [3] J. M. Ball, J. C. Currie, and P. J. Olver, Null Lagrangians, weak continuity, and variational problems of arbitrary order, J. Funct. Anal. 41 (1981), no. 2, 135–174. MR 615159, 10.1016/0022-1236(81)90085-9
  • [4] Bernard Dacorogna, Weak continuity and weak lower semicontinuity of nonlinear functionals, Lecture Notes in Mathematics, vol. 922, Springer-Verlag, Berlin-New York, 1982. MR 658130
  • [5] Ronald J. DiPerna, Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J. 28 (1979), no. 1, 137–188. MR 523630, 10.1512/iumj.1979.28.28011
  • [6] R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), no. 1, 27–70. MR 684413, 10.1007/BF00251724
  • [7] Ronald J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91 (1983), no. 1, 1–30. MR 719807
  • [8] Ronald J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal. 88 (1985), no. 3, 223–270. MR 775191, 10.1007/BF00752112
  • [9] James Glimm and Peter D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoirs of the American Mathematical Society, No. 101, American Mathematical Society, Providence, R.I., 1970. MR 0265767
  • [10] Barbara Keyfitz Quinn, Solutions with shocks: An example of an 𝐿₁-contractive semigroup, Comm. Pure Appl. Math. 24 (1971), 125–132. MR 0271545
  • [11] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 0093653
  • [12] Eduardo H. Zarantonello (ed.), Contributions to nonlinear functional analysis, Academic Press, New York-London, 1971. Mathematics Research Center, Publ. No. 27. MR 0366576
  • [13] François Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 3, 489–507 (French). MR 506997
  • [14] François Murat, Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 1, 69–102 (French). MR 616901
  • [15] L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
  • [16] Luc Tartar, The compensated compactness method applied to systems of conservation laws, Systems of nonlinear partial differential equations (Oxford, 1982), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 111, Reidel, Dordrecht, 1983, pp. 263–285. MR 725524
  • [17] -, Solutions oscillantes des equations de Carleman, Séminaire Goulaouic-Meyer-Schwarz, January 1983.
  • [18] A. I. Vol'pert, The spaces $ BV$ and quasilinear equations, Math. USSR-Sb. 2 (1967), 257-267.
  • [19] D. Serre, private communication.
  • [20] N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR-Sb. 10 (1970), 127-243.
  • [21] C. B. Morrey, Jr., Multiple integrals in the calculus of variations, Springer, Berlin, 1966.

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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0808729-4
Article copyright: © Copyright 1985 American Mathematical Society