Remarks on the theory of the divergence-measure fields

Frid, Hermano

doi:10.1090/S0033-569X-2012-01311-5

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
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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
Ronald J. DiPerna, Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J. 28 (1979), no. 1, 137–188. MR 523630, DOI https://doi.org/10.1512/iumj.1979.28.28011
Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
Lawrence C. Evans, Weak convergence methods for nonlinear partial differential equations, CBMS Regional Conference Series in Mathematics, vol. 74, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1034481
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