Conserving the wrong variables in gas dynamics: A Riemann solution with singular shocks
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
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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
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References
- A. Bressan, Hyperbolic systems of Conservation Laws: The One-Dimensional Cauchy Problem, Oxford University Press, Oxford, 2000. MR 1816648 (2002d:35002)
- C. M. Dafermos, Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method. Arch. Rational Mech. Anal. $\textbf {52}$ (1973), 1–9. MR 0340837 (49:5587)
- C. M. Dafermos and R. J. DiPerna, The Riemann problem for certain classes of hyperbolic systems of conservation laws. J. Differential Equations $\textbf {20}$ (1976), 90–114. MR 0404871 (53:8671)
- B. Deng, Homoclinic bifurcations with nonhyperbolic equilibria. SIAM J. Math. Anal. $\textbf {21}$ (1990), 693–719. MR 1046796 (91g:58200)
- N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations $\textbf {31}$ (1979), 53–98. MR 524817 (80m:58032)
- C. K. R. T. Jones, Geometric singular perturbation theory. Dynamical systems (Montecatini Terme, 1994), Lecture Notes in Mathematics, Vol. 1609, Springer, Berlin, 1995, pp. 44–118. MR 1374108 (97e:34105)
- C. K. R. T. Jones and T. Kaper, A primer on the exchange lemma for fast-slow systems. Multiple-time-scale dynamical systems (Minneapolis, MN, 1997) IMA Vol. Math. Appl. $\textbf {122}$ (2001), 85–132. MR 1846573 (2002g:37022)
- C. K. R. T. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems. J. Differential Equations $\textbf {108}$ (1994), 64–88. MR 1268351 (95c:34085)
- B. L. Keyfitz, Conservation laws, delta shocks and singular shocks. In Nonlinear Theory of Generalized Functions (eds. M. Grosser et al.), Chapman & Hall/CRC Press, Boca Raton, 1999, pp. 99–111. MR 1699874
- B. L. Keyfitz, A new look at singular shocks. Confluentes Matematici, to appear, 2012.
- B. L. Keyfitz and H. C. Kranzer, A viscosity approximation to a system of conservation laws with no classical Riemann solution. In Nonlinear Hyperbolic Problems (Bordeaux, 1998), (eds. C. Carasso et al.), Lecture Notes in Mathematics, Vol. $\textbf {1402},$ Springer, Berlin, 1989, pp. 185–197. MR 1033283 (90k:35168)
- B. L. Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differential Equations, $\textbf {118}$ (1995), 420–451. MR 1330835 (96b:35138)
- H. C. Kranzer and B. L. Keyfitz, A strictly hyperbolic system of conservation laws admitting singular shocks. In Nonlinear Evolution Equations that Change Type (eds. B. L. Keyfitz and M. Shearer), Springer, New York, 1990, pp. 107–125. MR 1074189 (92g:35133)
- M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points – fold and canard points in two dimensions, SIAM J. Math. Anal., $\textbf {33}$ (2001), 286–314. MR 1857972 (2002g:34117)
- 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.
- D. G. Schaeffer, S. Schecter and M. Shearer, Nonstrictly hyperbolic conservation laws with a parabolic line, J. Differential Equations $\textbf {103}$ (1993), 94–126. MR 1218740 (94d:35102)
- S. Schecter, Existence of Dafermos profiles for singular shocks, J. Differential Equations $\textbf {205}$ (2004), 185–210. MR 2094383 (2005k:35269)
- S. Schecter and P. Szmolyan, Composite waves in the Dafermos regularization. J. Dynamics and Differential Equations $\textbf {16}$ (2004), 847–867. MR 2109169 (2005h:35234)
- M. Sever, Distribution solutions of nonlinear systems of conservation laws, Memoirs of the AMS, $\textbf {889}$ (2007), 1–163. MR 2355635 (2008k:35313)
- M. Sever, Large-data solution of a model system for singular shocks, J. of Hyperbolic Differential Equations $\textbf {7}$ (2010), 775–840. MR 2746206 (2012c:35270)
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Additional Information
Barbara Lee Keyfitz
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Email:
bkeyfitz@math.ohio-state.edu
Charis Tsikkou
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Email:
tsikkou@math.ohio-state.edu
Keywords:
Conservation laws,
singular shocks,
Dafermos regularization,
geometric singular perturbation theory,
nonhyperbolicity,
blow-up
Received by editor(s):
February 14, 2012
Published electronically:
May 16, 2012
Additional Notes:
The first author was supported in part by NSF Grant DMS0807569 and by DOE Grant DE-SC0001285
The second author was supported in part by DOE Grant DE-SC0001285
Article copyright:
© Copyright 2012
Brown University
The copyright for this article reverts to public domain 28 years after publication.