Remarks on the theory of the divergence-measure fields
Author:
Hermano Frid
Journal:
Quart. Appl. Math. 70 (2012), 579-596
MSC (2010):
Primary 26B20, 28C05, 35L65, 35B35; Secondary 26B35, 26B12, 35L67
DOI:
https://doi.org/10.1090/S0033-569X-2012-01311-5
Published electronically:
May 9, 2012
MathSciNet review:
2986135
Full-text PDF Free Access
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Abstract: We review the theory of the (extended) divergence-measure fields providing an up-to-date account of its basic results established by Chen and Frid (1999, 2002), as well as the more recent important contributions by Silhavý (2008, 2009). We include a discussion on some pairings that are important in connection with the definition of normal trace for divergence-measure fields. We also review its application to the uniqueness of Riemann solutions to the Euler equations in gas dynamics, as given by Chen and Frid (2002). While reviewing the theory, we simplify a number of proofs allowing an almost self-contained exposition.
References
- Gui-Qiang Chen and Hermano Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal. 147 (1999), no. 2, 89–118. MR 1702637, DOI https://doi.org/10.1007/s002050050146
- Gui-Qiang Chen and Hermano Frid, On the theory of divergence-measure fields and its applications, Bol. Soc. Brasil. Mat. (N.S.) 32 (2001), no. 3, 401–433. Dedicated to Constantine Dafermos on his 60th birthday. MR 1894566, DOI https://doi.org/10.1007/BF01233674
- Gui-Qiang Chen and Hermano Frid, Extended divergence-measure fields and the Euler equations for gas dynamics, Comm. Math. Phys. 236 (2003), no. 2, 251–280. MR 1981992, DOI https://doi.org/10.1007/s00220-003-0823-7
- Gui-Qiang Chen and Hermano Frid, Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations, Trans. Amer. Math. Soc. 353 (2001), no. 3, 1103–1117. MR 1804414, DOI https://doi.org/10.1090/S0002-9947-00-02660-X
- Gui-Qiang Chen, Hermano Frid, and Yachun Li, Uniqueness and stability of Riemann solutions with large oscillation in gas dynamics, Comm. Math. Phys. 228 (2002), no. 2, 201–217. MR 1911734, DOI https://doi.org/10.1007/s002200200615
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2010. MR 2574377
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- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
- M. Šilhavý, The divergence theorem for divergence measure vectorfields on sets with fractal boundaries, Math. Mech. Solids 14 (2009), no. 5, 445–455. MR 2532602, DOI https://doi.org/10.1177/1081286507081960
- M. Šilhavý, Normal currents: structure, duality pairings and div-curl lemmas, Milan J. Math. 76 (2008), 275–306. MR 2465994, DOI https://doi.org/10.1007/s00032-007-0081-9
- David H. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Differential Equations 68 (1987), no. 1, 118–136. MR 885816, DOI https://doi.org/10.1016/0022-0396%2887%2990188-4
References
- Chen, G.-Q. and Frid, H.. Divergence-measure fields and hyperbolic conservation laws. Arch. Ration. Mech. Anal. 147 (1999), no. 2, 89 –118. MR 1702637 (2000d:35136)
- Chen, G-Q. and Frid, H.. On the theory of divergence-measure fields and its applications. Bol. Soc. Brasil. Mat. (N.S.) 32 (2001), no. 3, 401–433. MR 1894566 (2003c:28011)
- Chen, G.-Q. and Frid, H.. Extended divergence-measure fields and the Euler equations for gas dynamics. Comm. Math. Phys. 236 (2003), no. 2, 251–280. MR 1981992 (2004f:35113)
- Chen, G.-Q. and Frid, H., Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations, Trans. Amer. Math. Soc. 353 (2001), 1103–1117. MR 1804414 (2001m:35211)
- Chen, G.-Q., Frid, H., and Y. Li, Uniqueness and stability of Riemann solutions with large oscillation in gas dynamics, Commun. Math. Phys. 228 (2002), no.2, 201–217. MR 1911734 (2003c:76108)
- Dafermos, C.M., “Hyperbolic Conservation Laws in Continuum Physics” (Third Edition). Springer-Verlag, Berlin, Heidelberg, 1999, 2005, 2010. MR 2574377 (2011i:35150)
- DiPerna, R., Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J. 28 (1979), 137–188. MR 523630 (80i:35119)
- Evans, L. C. and Gariepy, R. F., “Lecture Notes on Measure Theory and Fine Properties of Functions”. CRC Press: Boca Raton, Florida, 1992. MR 1158660 (93f:28001)
- Evans, L. C., “Weak Convergence Methods for Nonlinear Partial Differential Equations”. CBMS No. 174, American Mathematical Society, Providence, Rhode Island, 1988. MR 1034481 (91a:35009)
- Federer, H., “Geometric Measure Theory”. Springer-Verlag: Berlin-Heidelberg-New York, 1969. MR 0257325 (41:1976)
- Silhavý, M., The divergence theorem for divergence measure vectorfields on sets with fractal boundaries. Math. Mech. Solids 14 (2009), no. 5, 445–455. MR 2532602 (2010i:26023)
- Silhavý, M., Normal currents: structure, duality pairings and div-curl lemmas. Milan J. Math. 76 (2008), 275–306. MR 2465994 (2010c:49081)
- Wagner, D., Equivalence of the Euler and Lagrange equations of gas dynamics for weak solutions. J. Diff. Eqs. 68 (1987), 118–136. MR 885816 (88i:35100)
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Additional Information
Hermano Frid
Affiliation:
Instituto de Matemática Pura e Aplicada - IMPA, Estrada Dona Castorina, 110, Rio de Janeiro, RJ 22460-320, Brazil
Email:
hermano@impa.br
Keywords:
Divergence-measure fields,
normal traces,
Gauss-Green theorem,
product rule
Received by editor(s):
January 9, 2012
Published electronically:
May 9, 2012
Dedicated:
Dedicated to Costas Dafermos on his 70th birthday
Article copyright:
© Copyright 2012
Brown University