Conserving the wrong variables in gas dynamics: A Riemann solution with singular shocks

Keyfitz, Barbara; Tsikkou, Charis

doi:10.1090/S0033-569X-2012-01317-1

Authors: Barbara Lee Keyfitz and Charis Tsikkou
Journal: Quart. Appl. Math. 70 (2012), 407-436
MSC (2010): Primary 35L65, 35L67; Secondary 34E15, 34C37
DOI: https://doi.org/10.1090/S0033-569X-2012-01317-1
Published electronically: May 16, 2012
MathSciNet review: 2986129
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a system of two equations derived from isentropic gas dynamics with no classical Riemann solutions. We show existence of unbounded self-similar solutions (singular shocks) of the Dafermos regularization of the system. Our approach is based on the blowing-up approach of geometric singular perturbation theory.

References

Alberto Bressan, Hyperbolic systems of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000. The one-dimensional Cauchy problem. MR 1816648
Constantine M. Dafermos, Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method, Arch. Rational Mech. Anal. 52 (1973), 1–9. MR 340837, DOI https://doi.org/10.1007/BF00249087
C. M. Dafermos and R. J. DiPerna, The Riemann problem for certain classes of hyperbolic systems of conservation laws, J. Differential Equations 20 (1976), no. 1, 90–114. MR 404871, DOI https://doi.org/10.1016/0022-0396%2876%2990098-X
Bo Deng, Homoclinic bifurcations with nonhyperbolic equilibria, SIAM J. Math. Anal. 21 (1990), no. 3, 693–720. MR 1046796, DOI https://doi.org/10.1137/0521037
Neil Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations 31 (1979), no. 1, 53–98. MR 524817, DOI https://doi.org/10.1016/0022-0396%2879%2990152-9
Christopher K. R. T. Jones, Geometric singular perturbation theory, Dynamical systems (Montecatini Terme, 1994) Lecture Notes in Math., vol. 1609, Springer, Berlin, 1995, pp. 44–118. MR 1374108, DOI https://doi.org/10.1007/BFb0095239
Tasso J. Kaper and Christopher K. R. T. Jones, A primer on the exchange lemma for fast-slow systems, Multiple-time-scale dynamical systems (Minneapolis, MN, 1997) IMA Vol. Math. Appl., vol. 122, Springer, New York, 2001, pp. 65–87. MR 1846573, DOI https://doi.org/10.1007/978-1-4613-0117-2_3
C. K. R. T. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Differential Equations 108 (1994), no. 1, 64–88. MR 1268351, DOI https://doi.org/10.1006/jdeq.1994.1025
Barbara Lee Keyfitz, Conservation laws, delta-shocks and singular shocks, Nonlinear theory of generalized functions (Vienna, 1997) Chapman & Hall/CRC Res. Notes Math., vol. 401, Chapman & Hall/CRC, Boca Raton, FL, 1999, pp. 99–111. MR 1699874

B. L. Keyfitz, A new look at singular shocks. Confluentes Matematici, to appear, 2012.

Barbara Lee Keyfitz and Herbert C. Kranzer, A viscosity approximation to a system of conservation laws with no classical Riemann solution, Nonlinear hyperbolic problems (Bordeaux, 1988) Lecture Notes in Math., vol. 1402, Springer, Berlin, 1989, pp. 185–197. MR 1033283, DOI https://doi.org/10.1007/BFb0083875

Barbara Lee Keyfitz and Herbert C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differential Equations 118 (1995), no. 2, 420–451. MR 1330835, DOI https://doi.org/10.1006/jdeq.1995.1080

Herbert C. Kranzer and Barbara Lee Keyfitz, A strictly hyperbolic system of conservation laws admitting singular shocks, Nonlinear evolution equations that change type, IMA Vol. Math. Appl., vol. 27, Springer, New York, 1990, pp. 107–125. MR 1074189, DOI https://doi.org/10.1007/978-1-4613-9049-7_9

M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions, SIAM J. Math. Anal. 33 (2001), no. 2, 286–314. MR 1857972, DOI https://doi.org/10.1137/S0036141099360919

M. Mazzotti, Non-classical composition fronts in nonlinear chromatography - Delta-shock, Indust. & Eng. Chem. Res., $\textbf {48}$ (2009), 7733–7752.

M. Mazzotti, A. Tarafder, J. Cornel, F. Gritti and G. Guiochon, Experimental evidence of a delta-shock in nonlinear chromatography, J. Chromatography. A, $\textbf {1217(13)}$ (2010), 2002–2012.

David G. Schaeffer, Stephen Schecter, and Michael Shearer, Non-strictly hyperbolic conservation laws with a parabolic line, J. Differential Equations 103 (1993), no. 1, 94–126. MR 1218740, DOI https://doi.org/10.1006/jdeq.1993.1043

Stephen Schecter, Existence of Dafermos profiles for singular shocks, J. Differential Equations 205 (2004), no. 1, 185–210. MR 2094383, DOI https://doi.org/10.1016/j.jde.2004.06.013

Stephen Schecter and Peter Szmolyan, Composite waves in the Dafermos regularization, J. Dynam. Differential Equations 16 (2004), no. 3, 847–867. MR 2109169, DOI https://doi.org/10.1007/s10884-004-6698-2

Michael Sever, Distribution solutions of nonlinear systems of conservation laws, Mem. Amer. Math. Soc. 190 (2007), no. 889, viii+163. MR 2355635, DOI https://doi.org/10.1090/memo/0889

Michael Sever, Large-data solution of a model system for singular shocks, J. Hyperbolic Differ. Equ. 7 (2010), no. 4, 775–840. MR 2746206, DOI https://doi.org/10.1142/S0219891610002281