Decay with a rate for noncompactly supported solutions of conservation laws
Author:
Blake Temple
Journal:
Trans. Amer. Math. Soc. 298 (1986), 4382
MSC:
Primary 35L65
MathSciNet review:
857433
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Abstract: We show that solutions of the Cauchy problem for systems of two conservation laws decay in the supnorm at a rate that depends only on the norm of the initial data. This implies that the dissipation due to increasing entropy dominates the nonlinearities in the problem at a rate depending only on the norm of the initial data. Our results apply to any BV initial data satisfying and . The problem of decay with a rate independent of the support of the initial data is central to the issue of continuous dependence in systems of conservation laws because of the scale invariance of the equations. Indeed, our result implies that the constant state is stable with respect to perturbations in . This is the first stability result in an norm for systems of conservation laws. It is crucial that we estimate decay in the supnorm since the total variation does not decay at a rate independent of the support of the initial data. The main estimate requires an analysis of approximate characteristics for its proof. A general framework is developed for the study of approximate characteristics, and the main estimate is obtained for an arbitrary number of equations.
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, On supnorm bounds in Glimm's method, Mathematics Research Center Technical Summary Reports, #2855, University of Wisconsin, Madison.
 [1]
 ILiang Chern, On the decay of a strong wave for hyperbolic conservation laws in one space dimension, Thesis, Courant Institute, New York Univ., 1982.
 [2]
 C. Courant and K. O. Friedrichs, Supersonic flow and shock waves, SpringerVerlag, Berlin and New York, 1948.
 [3]
 C. Dafermos, Application of the invariance principle for compact processes. II. Asymptotic behavior of solutions of a hyperbolic conservation law, J. Differential Equations 11 (1972), 416424. MR 0296476 (45:5536)
 [4]
 R. DiPerna, Decay and asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws, Indiana Univ. Math. J. 24 (1975), 10471071. MR 0410110 (53:13860)
 [5]
 , Decay of solutions of hyperbolic systems of conservation laws with a convex extension, Arch. Rational Mech. Anal. 69 (1977), 146. MR 0454375 (56:12626)
 [6]
 J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 695715. MR 0194770 (33:2976)
 [7]
 J. Glimm and P. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Mem. Amer. Math. Soc. No. 101 (1970). MR 0265767 (42:676)
 [8]
 B. Keyfitz, Solutions with shocks: an example of an contractive semigroup, Comm. Pure Appl. Math. 24 (1971), 125132. MR 0271545 (42:6428)
 [9]
 P. Lax, Hyperbolic systems of conservation laws, II. Comm. Pure Appl. Math. 10 (1957), 537566. MR 0093653 (20:176)
 [10]
 T. P. Liu, Asymptotic behavior of solutions of general system of nonlinear hyperbolic conservation laws, Indiana Univ. J. 27 (1978), 211253. MR 481315 (80b:35103)
 [11]
 , Admissible solutions of hyperbolic conservation laws, Mem. Amer. Math. Soc. No. 240 (1981). MR 603391 (82i:35116)
 [12]
 , Decay to waves of solutions of general system of nonlinear hyperbolic conservation laws, Comm. Pure Appl. Math. 30 (1977), 585610. MR 0450802 (56:9095)
 [13]
 , Largetime behavior of solutions of initial and initialboundary value problem of general system of hyperbolic conservation laws, Comm. Math. Phys. 55 (1977), 163177. MR 0447825 (56:6135)
 [14]
 , Linear and nonlinear largetime behaviors of solutions of hyperbolic conservation laws, Comm. Pure Appl. Math. 30 (1977), 767796. MR 0499781 (58:17556)
 [15]
 , The deterministic version of the Glimm scheme, Comm. Math. Phys. 57 (1977), 135148. MR 0470508 (57:10259)
 [16]
 L. Luskin and J. B. Temple, The existence of a global weak solution to the nonlinear Waterhammer problem, Comm. Pure Appl. Math. 35 (1982), 697735. MR 668411 (84h:35103)
 [17]
 J. A. Smoller, Shock waves and reactiondiffusion equations, SpringerVerlag, New York, 1980.
 [18]
 T. 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)
 [19]
 , Solutions in the large for the nonlinear hyperbolic conservation laws of gas dynamics, J. Differential Equations 41 (1981), 96161. MR 626623 (82i:35117)
 [20]
 , Systems of conservaton laws with coinciding shock and rarefaction curves, Contemp. Math. 17 (1983), 143151.
 [21]
 , Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc. 280 (1983), 781795. MR 716850 (84m:35080)
 [22]
 , No contractive metrics for systems of conservation laws, Trans. Amer. Math. Soc. 288 (1985),
 [23]
 , On supnorm bounds in Glimm's method, Mathematics Research Center Technical Summary Reports, #2855, University of Wisconsin, Madison.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198608574336
PII:
S 00029947(1986)08574336
Keywords:
Riemann problem,
random choice method,
decay,
stability,
continuous dependence,
conservation laws,
Cauchy problem
Article copyright:
© Copyright 1986
American Mathematical Society
