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.

**1**K. N. Chueh, C. C. Conley, and J. A. Smoller,*Positively invariant regions for systems of nonlinear diffusion equations*, Indiana Univ. Math. J.**26**(1977), no. 2, 373–392. MR**0430536****2**Ronald J. DiPerna,*Measure-valued solutions to conservation laws*, Arch. Rational Mech. Anal.**88**(1985), no. 3, 223–270. MR**775191**, 10.1007/BF00752112**3**H. Frid,*Existence and asymptotic behavior of measure-valued solutions for three-phase flows in porous media*, J. Math. Anal. Appl. (to appear).**4**David Hoff and Joel Smoller,*Solutions in the large for certain nonlinear parabolic systems*, Ann. Inst. H. Poincaré Anal. Non Linéaire**2**(1985), no. 3, 213–235 (English, with French summary). MR**797271****5**Eli L. Isaacson, Dan Marchesin, Bradley J. Plohr, and J. Blake Temple,*Multiphase flow models with singular Riemann problems*, Mat. Apl. Comput.**11**(1992), no. 2, 147–166 (English, with English and Portuguese summaries). MR**1235737****6**S. N. Kružkov,*First order quasilinear equations with several independent variables.*, Mat. Sb. (N.S.)**81 (123)**(1970), 228–255 (Russian). MR**0267257****7**Peter Lax and Burton Wendroff,*Systems of conservation laws*, Comm. Pure Appl. Math.**13**(1960), 217–237. MR**0120774****8**D.W. Peaceman,*Fundamentals of Numerical Reservoir Simulation*, Elsevier, Amsterdam-New York, 1977.**9**S. Saks,*Theory of the Integral*, Warsaw, 1937.**10**Joel Smoller,*Shock waves and reaction-diffusion equations*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR**688146****11**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**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-96-01488-2

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