On the theory of divergence-measure fields and its applications Gui-Qiang Chen Hermano Frid Article Received: 09 December 2001 DOI :
10.1007/BF01233674

Cite this article as: Chen, GQ. & Frid, H. Bol. Soc. Bras. Mat (2001) 32: 401. doi:10.1007/BF01233674 Abstract Divergence-measure fields are extended vector fields, including vector fields inL ^{p} and vector-valued Radon measures, whose divergences are Radon measures. Such fields arise naturally in the study of entropy solutions of nonlinear conservation laws and other areas. In this paper, a theory of divergence-measure fields is presented and analyzed, in which normal traces, a generalized Gauss-Green theorem, and product rules, among others, are established. Some applications of this theory to several nonlinear problems in conservation laws and related areas are discussed. In particular, with the aid of this theory, we prove the stability of Riemann solutions, which may contain rarefaction waves, contact discontinuities, and/or vacuum states, in the class of entropy solutions of the Euler equations for gas dynamics.

Keywords divergence-measure fields normal traces Gauss-Green theorem product rules Radon measures conservation laws Euler equations gas dynamics entropy solutions entropy inequality stability uniqueness vacuum Cauchy problem initial layers boundary layers initial-boundary value problems

Mathematical subject classification Primary: 00-02 26B20 28C05 35L65 35B10 35B35 Secondary: 26B35 26B12 35L67 Dedicated to Constantine Dafermos on his 60^{th} birthday

References [1]

Anzellotti G.,

Pairings between measures and functions and compensated compactness , Ann. Mat. Pura Appl.,

135 (1983), 293–318.

Google Scholar [2]

Baiocchi C. and A. Capelo, Variational and Quasi-Variational Inequalities, Applications to Free-Boundary Problems, Vols.

1,2 (1984), John Wiley: Chichester-New York.

Google Scholar [3]

Bardos C., Le Roux A.Y. and J.C. Nedelec,

First order quasilinear equations with boundary conditions , Comm. Partial Diff. Eqs.,

4 (1979), 1017–1034.

Google Scholar [4]

Bouchitté G. and G. Buttazzo,Characterization of optimal shapes and masses through Monge-Kantorovich equation , Preprint, University of Pisa, February 2000.

[5]

Brezzi F. and M. Fortin,

Mixed and Hydrid Finite Element Methods , Springer-Verlag: New York, 1991.

Google Scholar [6]

Bürger R., Karlsen K.H. and H. Frid,On a free boundary problem for a strongly degenerate quasilinear parabolic equation arising in a model for pressure filtration , submitted for publication in December 2001.

[7]

Chen G.-Q. and H. Frid,

Divergence-measure fields and conservation laws , Arch. Rational Mech. Anal.,

147 (1999), 89–118.

Google Scholar [8]

Chen G.-Q. and H. Frid,

Vanishing viscosity limit for initial-boundary value problems for conservation laws , Contemporary Mathematics,

238 (1999), 35–51.

Google Scholar [9]

Chen G.-Q. and H. Frid,Extended divergence-measure fields and the Euler equations for gas dynamics , submitted for publication in September 2001.

[10]

Chen G.-Q., Frid H. and Y. Li,Uniqueness and stability of Riemann solutions with large oscillation in gas dynamics , Commun. Math. Phys. 2001 (to appear).

[11]

Chen G.-Q., Levermore C.D. and T.-P. Liu,

Hyperbolic conservation laws with stiff relaxation terms and entropy , Comm. Pure Appl. Math.,

47 (1994), 787–830.

Google Scholar [12]

Chen G.-Q. and M. Rascle,

Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws , Arch. Rational Mech. Anal.,

153 (2000), 205–220.

Google Scholar [13]

Dafermos C.M., Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag: Berlin, 1999.

Google Scholar [14]

DiPerna R.,

Convergence of the viscosity method for isentropic gas dynamics , Comm. Math. Phys.,

91 (1983), 1–30.

Google Scholar [15]

DiPerna R.,

Measure-valued solutions to conservation laws , Arch. Rational Mech. Anal.,

88 (1985), 223–270.

Google Scholar [16]

DiPerna R.,

Uniqueness of solutions to hyperbolic conservation laws , Indiana Univ. Math. J.,

28 (1979), 137–188.

Google Scholar [17]

Evans L.C. and R.F. Gariepy, Lecture Notes on Measure Theory and Fine Properties of Functions, CRC Press: Boca Raton, Florida, 1992.

Google Scholar [18]

Federer H., Geometric Measure Theory, Springer-Verlag: Berlin-Heidelberg-New York, 1969.

Google Scholar [19]

Friedrichs K.O. and P.D. Lax,

Systems of conservation equations with a convex extension , Proc. Nat. Acad. Sci. U.S.A.,

68 (1971), 1686–1688.

Google Scholar [20]

Gagliardo E.,

Caratterizioni delle tracce sulla frontiera relativa ad alcune classi di funzioni in n variabli , Rend. Sem. Mat. Univ. Padova,

27 (1957), 284–305.

Google Scholar [21]

Glimm J.,

Solutions in the large for nonlinear hyperbolic systems of equations , Comm. Pure Appl. Math.,

18 (1965), 95–105.

Google Scholar [22]

Glimm J. and P.D. Lax,

Decay of Solutions of Systems of Nonlinear Hyperbolic Conservation Laws , Mem. Amer. Math. Soc.,

101 (1970), AMS: Providence, R.I.

Google Scholar [23]

Harrison J. and A. Norton,

The Gauss-Green theorem for fractal boundaries . Duke Math. J.,

67 (1992), 575–588.

Google Scholar [24]

Harrison J.,

Stokes' theorem for nonsmooth chains , Bull. Amer. Math. Soc. (N.S.),

29 (1993), 235–242.

Google Scholar [25]

Jurkat W.B. and D.J.F. Nonnenmacher,

A generalized n-dimensional Riemann integral and the divergence theorem with singularities , Acta Sci. Math. (Szeged),

59 (1994), 241–256.

Google Scholar [26]

Katsoulakis M. and A. Tzavaras,

Contractive relaxation systems and the scalar multidimensional conservation law , Comm. Partial Diff. Eqs.,

22 (1997), 195–233.

Google Scholar [27]

Kruzkov S.N.,

First order quasilinear equations with several independent variables , Mat. Sb. (N.S.) (Russian),

81 (123) (1970), 228–255.

Google Scholar [28]

Lax P.D.,

Hyperbolic systems of conservation laws , Comm. Pure Appl. Math.,

10 (1957), 537–566.

Google Scholar [29]

Lax P.D.,

Hyperbolic systems of Conservation Laws and the Mathematical Theory of Shock Waves , CBMS.,

11 (1973), SIAM, Philadelphia.

Google Scholar [30]

Lax P.D.,

Shock waves and entropy , In: Contributions to Functional Analysis, ed. E.A. Zarantonello, Academic Press, New York, 1971, pp. 603–634.

Google Scholar [31]

Lions P.L., Perthame B. and E. Tadmor,

A kinetic formulation of multidimensional scalar conservation laws and related equations , J. Amer. Math. Soc.,

7 (1994), 169–191.

Google Scholar [32]

Liu T.-P. and J. Smoller,

On the vacuum state for the isentropic gas dynamics equations , Adv. Appl. Math.,

1 (1980), 345–359.

Google Scholar [33]

Natalini R.,

Convergence to equilibrium for the relaxation approximations of conservation laws , Comm. Pure Appl. Math.,

49 (1996), 795–823.

Google Scholar [34]

Málec J., Nečas J., Rokyta M. and M. Ružička, Weak and Measure-valued Solutions to Evolutionary PDEs. Chapman and Hall: London, 1996.

Google Scholar [35]

Mascia C., Porretta A. and A. Terracina,Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations , Arch. Rational Mech. Anal., (to appear).

[36]

Nonnenmacher D.J.F.,

Sets of finite perimeter and the Gauss-Green theorem with singularities , J. London Math. Soc.,

52 (2) (1995), 335–344.

Google Scholar [37]

Otto F.,First order equations with boundary conditions , Preprint no. 234, SFB 256, Univ. Bonn., 1992.

[38]

Pfeffer W.F.,

Derivation and Integration , Cambridge Tracts in Math.,

140 (2001), Cambridge Univ. Press: Cambridge.

Google Scholar [39]

Rodrigues J.-F.,Obstacle Problems in Mathematical Physics , North-Holland Mathematics Studies,134 (1987), Elsevier Science Publishers B.V.

[40]

Schwartz L., Théorie des Distributions, Actualites Scientifiques et Industrielles,

1091, 1122 , Herman: Paris, 1950–51.

Google Scholar [41]

Szepessy A.,

An existence result for scalar conservation laws using measure valued solutions , Commun. Partial Diff. Eqs.,

14 (1989), 1329–1350.

Google Scholar [42]

Serre D., Systems of Conservation Laws I, II, Cambridge University Press: Cambridge, 2000.

Google Scholar [43]

Shapiro V.,

The divergence theorem for discontinuous vector fields . Ann. Math.,

68 (2) (1958), 604–624.

Google Scholar [44]

Stein E., Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press: Princeton, NJ, 1970.

Google Scholar [45]

Vasseur A.,

Strong traces for solutions to multidimensional scalar conservation laws , Arch. Rational Mech. Anal.

160 (2001), 181–193.

Google Scholar [46]

Volpert A.I.,

The space BV and quasilinear equations , Mat. Sb. (N.S.),

73 (1967), 255–302, Math. USSR Sbornik,

2 (1967), 225–267 (in English).

Google Scholar [47]

Wagner, D.,

Equivalence of the Euler and Lagrange equations of gas dynamics for weak solutions , J. Diff. Eqs.,

68 (1987), 118–136.

Google Scholar [48]

Whitney H., Geometric Integration Theory, Princeton Univ. Press: Princeton, NJ, 1957.

Google Scholar [49]

Whitney H.,

Analytic extensions of differentiable functions defined in closed sets , Trans. Amer. Math. Soc.,

36 (1934), 63–89.

Google Scholar [50]

Ziemer W.P.,

Cauchy flux and sets of finite perimeter , Arch. Rational Mech. Anal.,

84 (1983), 189–201.

Google Scholar [5]

Ziemer W.P., Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Springer-Verlag: New York, 1989.

Google Scholar © Sociedade Brasileira de Matemática 2001

Authors and Affiliations Gui-Qiang Chen Hermano Frid 1. Department of Mathematics Northwestern University Evanston USA 2. Instituto de Matemática Pura e Aplicada-IMPA Rio de Janeiro Brazil