No contractive metrics for systems of conservation laws
Author:
Blake Temple
Journal:
Trans. Amer. Math. Soc. 288 (1985), 471480
MSC:
Primary 35L65; Secondary 76L05, 76N10
MathSciNet review:
776388
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Abstract: Let be any system of conservation laws satisfying certain generic assumptions on in a neighborhood of space. We prove that for every nondegenerate metric on space there exists states and in such that is a strictly increasing function of in a neighborhood of , where is the admissible solution of with initial data This contrasts with the case of a scalar equation in which is a decreasing function of for all admissible solution pairs and when is taken to be the absolute value norm.
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D. W. Peaceman, Fundamentals of numerical reservoir simulation, Elsevier NorthHolland, New York, 1977.
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H. Rhee, R. Aris and N. R. Amundson, On the theory of multicomponent chromatography, Philos. Trans. Roy. Soc. London Ser. A 267 (1970), 419.
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Blake
Temple, Global solution of the Cauchy problem for a class of
2×2\ nonstrictly hyperbolic conservation laws, Adv. in Appl.
Math. 3 (1982), no. 3, 335–375. MR 673246
(84f:35091), http://dx.doi.org/10.1016/S01968858(82)800109
 [14]
, Systems of conservation laws with invariant submanifolds, Proc. Amer. Math. Soc. (to appear).
 [15]
M. Walsh, S. Bryant, R. Schechter and L. Lake, Precipitation and dissolution of solids attending flow through porous media, University of Texas Preprint, 1982.
 [1]
 R. Aris and N. Amundson, Mathematical methods in chemical engineering, Vol. 2, PrenticeHall, Englewood Cliffs, N. J., 1966. MR 0371183 (51:7404)
 [2]
 R. Courant and K. O. Friedricks, Supersonic flow and shock waves, Wiley, New York, 1948.
 [3]
 J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697715. MR 0194770 (33:2976)
 [4]
 E. Isaacson, Global solution of a Riemann problem for a nonstrictly hyperbolic system of conservation laws arising in enhanced oil recovery, J. Comp. Phys. (to appear).
 [5]
 F. Helfferich and G. Klein, Multicomponent chromatography, Dekker, New York, 1970.
 [6]
 B. Keyfitz, Solutions with shocks: An example of an contractive semigroup, Comm. Pure Appl. Math. 24 (1971), 125132. MR 0271545 (42:6428)
 [7]
 B. Keyfitz and H. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal. 72 (1980), 219241. MR 549642 (80k:35050)
 [8]
 P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 19 (1957), 537566. MR 0093653 (20:176)
 [9]
 , Shock waves and entropy, Contributions to Nonlinear Functional Analysis (E. H. Zarantonello, ed.), Academic Press, New York, 1971, pp. 603634. MR 0393870 (52:14677)
 [10]
 T. P. Liu and C. H. Wang, On a hyperbolic system of conservation laws which is not strictly hyperbolic, MRC Technical Summary Report # 2184, December 29, 1980.
 [11]
 D. W. Peaceman, Fundamentals of numerical reservoir simulation, Elsevier NorthHolland, New York, 1977.
 [12]
 H. Rhee, R. Aris and N. R. Amundson, On the theory of multicomponent chromatography, Philos. Trans. Roy. Soc. London Ser. A 267 (1970), 419.
 [13]
 B. Temple, Global solution of the Cauchy problem for a class of nonstrictly hyperbolic conservation laws, Adv. in Appl. Math. 3 (1982), 335375. MR 673246 (84f:35091)
 [14]
 , Systems of conservation laws with invariant submanifolds, Proc. Amer. Math. Soc. (to appear).
 [15]
 M. Walsh, S. Bryant, R. Schechter and L. Lake, Precipitation and dissolution of solids attending flow through porous media, University of Texas Preprint, 1982.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198507763885
PII:
S 00029947(1985)07763885
Article copyright:
© Copyright 1985
American Mathematical Society
