Skip to Main Content

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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Measure-Valued Solutions to Initial-Boundary Value Problems for Certain Systems of Conservation Laws: Existence and Dynamics
HTML articles powered by AMS MathViewer

by Hermano Frid PDF
Trans. Amer. Math. Soc. 348 (1996), 51-76 Request permission

Abstract:

A framework for studying initial-boundary value problems for systems of conservation laws, in what concerns to the existence of measure-valued solutions and their asymptotic behavior, is developed here with the helpful introduction of a class of flux maps which allow a rather complete treatment of these questions including systems of practical importance as those arising in multiphase flow in porous media. The systems of this class may, in general, admit umbilic points, submanifolds where genuine nonlinearity fails, as well as elliptic regions. We prove the existence of measure-valued solutions by using the vanishing viscosity method and, also, finite difference schemes. The main result about the dynamics of the measure@-valued solutions is that for certain special boundary values, given by constant states, the time-averages of these m@-v solutions converge weakly to the Dirac measure concentrated at those states, for a.e. space variable. The rate of convergence of the time-averages of the expected values can be estimated by properties of the flux maps only.
References
    K.N. Chueh, C.C. Conley, J.A. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana University Mathematics Journal 26 (2) (1977), 372–411, MR 55:3541. R.J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rat. Mech. Anal. 8 (1985), 223–270, MR 86g:35121. H. Frid, Existence and asymptotic behavior of measure-valued solutions for three-phase flows in porous media, J. Math. Anal. Appl. (to appear). D. Hoff, J.A. Smoller, Solutions in the large for certain nonlinear parabolic systems, Ann. Inst. Henri Poincaré–Analyse non linéaire 2 (3) (1985), 213–235, MR 87b:35078. E. Isaacson, D. Marchesin, B.V. Plohr, J.B. Temple, Multiphase flow models with singular Riemann problems, Computational and Applied Mathematics (1991), MR 94h:35212. S.N. Kruskov, First-order quasilinear equations with several space variables, Mat. Sb. 123 (1970), 228–255, English translation: Math. USSR Sb. 10 (1970), 217–273, MR 42:2159. P.D. Lax, B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. XIII (1960), 217–237, MR 22:11523. D.W. Peaceman, Fundamentals of Numerical Reservoir Simulation, Elsevier, Amsterdam-New York, 1977. S. Saks, Theory of the Integral, Warsaw, 1937. J.A. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983, MR 84d:35002. L. Tartar, Compensated compactness and applications to partial differential equations, Research Notes in Mathematics, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, ed. R.J. Knops, vol. 4, Pitman Press, New York, 1979, pp. 136–212, MR 81m:35014.
Similar Articles
Additional Information
  • Received by editor(s): March 17, 1994
  • Additional Notes: Research partially supported by CNPq-Brazil, proc. 302307/86-9.
  • © Copyright 1996 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 348 (1996), 51-76
  • MSC (1991): Primary 35L60, 35L50, 35B40; Secondary 76T05
  • DOI: https://doi.org/10.1090/S0002-9947-96-01488-2
  • MathSciNet review: 1321574