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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Compensated compactness and general systems of conservation laws
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by Ronald J. DiPerna PDF
Trans. Amer. Math. Soc. 292 (1985), 383-420 Request permission


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.
  • 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) Lecture Notes in Math., Vol. 564, Springer, Berlin, 1976, pp. 13–25. MR 0637229
  • John M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1976/77), no. 4, 337–403. MR 475169, DOI 10.1007/BF00279992
  • J. M. Ball, J. C. Currie, and P. J. Olver, Null Lagrangians, weak continuity, and variational problems of arbitrary order, J. Functional Analysis 41 (1981), no. 2, 135–174. MR 615159, DOI 10.1016/0022-1236(81)90085-9
  • 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, DOI 10.1007/BFb0096144
  • Ronald J. DiPerna, Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J. 28 (1979), no. 1, 137–188. MR 523630, DOI 10.1512/iumj.1979.28.28011
  • R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), no. 1, 27–70. MR 684413, DOI 10.1007/BF00251724
  • Ronald J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91 (1983), no. 1, 1–30. MR 719807, DOI 10.1007/BF01206047
  • Ronald J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal. 88 (1985), no. 3, 223–270. MR 775191, DOI 10.1007/BF00752112
  • 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
  • Barbara Keyfitz Quinn, Solutions with shocks: An example of an $L_{1}$-contractive semigroup, Comm. Pure Appl. Math. 24 (1971), 125–132. MR 271545, DOI 10.1002/cpa.3160240203
  • P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
  • Eduardo H. Zarantonello (ed.), Contributions to nonlinear functional analysis, Academic Press, New York-London, 1971. Mathematics Research Center, Publ. No. 27. MR 0366576
  • François Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 3, 489–507 (French). MR 506997
  • 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
  • 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
  • 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
  • —, Solutions oscillantes des equations de Carleman, Séminaire Goulaouic-Meyer-Schwarz, January 1983. A. I. Vol’pert, The spaces $BV$ and quasilinear equations, Math. USSR-Sb. 2 (1967), 257-267. D. Serre, private communication. N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR-Sb. 10 (1970), 127-243. C. B. Morrey, Jr., Multiple integrals in the calculus of variations, Springer, Berlin, 1966.
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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 292 (1985), 383-420
  • MSC: Primary 35L65; Secondary 76L05
  • DOI:
  • MathSciNet review: 808729