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Transactions of the American Mathematical Society

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Measure-Valued Solutions to Initial-Boundary Value Problems for Certain Systems of Conservation Laws: Existence and Dynamics

Author: Hermano Frid
Journal: Trans. Amer. Math. Soc. 348 (1996), 51-76
MSC (1991): Primary 35L60, 35L50, 35B40; Secondary 76T05
MathSciNet review: 1321574
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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.

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Keywords: Measure-valued solutions, systems of conservation laws, systems of mixed type, initial-boundary value problems
Received by editor(s): March 17, 1994
Additional Notes: Research partially supported by CNPq-Brazil, proc. 302307/86-9.
Article copyright: © Copyright 1996 American Mathematical Society