Abstract
We establish some basic lower semicontinuity properties for a class of weighted metrics in BV. These Riemann-type metrics, uniformly equivalent to the L 1 distance, are defined in terms of the Glimm interaction potential. They are relevant in the study of nonlinear hyperbolic systems of conservation laws, being contractive w.r.t. the corresponding flow of solutions.
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References
A. Bressan, A locally contractive metric for systems of conservation laws, Ann. Scuola Norm. Sup. Pisa IV-22 (1995), 109–135.
A. Bressan, The semigroup approach to systems of conservation laws, Mathematica Contemporanea 10 (1996), 21–74.
A. Bressan and R. M. Colombo, The semigroup generated by 2 x 2 conservation laws, Arch. Rational Mech. Anal. 133 (1995), 1–75.
A. Bressan and R. M. Colombo, Unique solutions of 2 x 2 conservation laws with large data, Indiana Univ. Math. J. 44 (1995), 677–725.
A. Bressan, G. Crasta and B. Piccoli, Well-posedness of the Cauchy problem for n x n systems of conservation laws, preprint S.I.S.S.A., Trieste, 1996.
A. Bressan and A. Marson, A variational calculus for discontinuous solutions of systems of conservation laws, Comm. Part. Diff. Equat. 20 (1995), 1491–1552.
M. Crandall, The semigroup approach to first-order quasilinear equations in several space variables, Israel J. Math. 12 (1972), 108–132.
K. Deimling, “Nonlinear Functional Analysis”, Springer-Verlag, Berlin, 1985.
L. C. Evans and R. F. Gariepy, “Measure Theory and Fine Properties of Functions”, CRC Press, 1992.
J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715.
S. Kruzkov, First order quasilinear equations with several space variables, Math. USSR Sbornik 10 (1970), 217–243.
P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957), 537–566.
J. Smoller, “Shock Waves and Reaction-Diffusion equations”, Springer-Verlag, New York, 1983.
B. Temple, No L 1-contractive metrics for systems of conservation laws, Trans. Amer. Math. Soc. 288 (1985), 471–480.
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Baiti, P., Bressan, A. (1997). Lower semicontinuity of weighted path length in BV. In: Colombini, F., Lerner, N. (eds) Geometrical Optics and Related Topics. Progress in Nonlinear Differential Equations and Their Applications, vol 32. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-2014-5_3
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DOI: https://doi.org/10.1007/978-1-4612-2014-5_3
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7381-3
Online ISBN: 978-1-4612-2014-5
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