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Singular solutions of a fully nonlinear 2 × 2 system of conservation laws

Published online by Cambridge University Press:  30 August 2012

Henrik Kalisch
Affiliation:
Department of Mathematics, University of Bergen, Postbox 7800, 5020 Bergen, Norway (henrik.kalisch@math.uib.no)
Darko Mitrović
Affiliation:
Department of Mathematics, University of Montenegro, Cetinjski put bb, 81000 Podgorica, Montenegro (matematika@t-com.me)
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Abstract

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Existence and admissibility of δ-shock solutions is discussed for the non-convex strictly hyperbolic system of equations

The system is fully nonlinear, i.e. it is nonlinear with respect to both unknowns, and it does not admit the classical Lax-admissible solution for certain Riemann problems. By introducing complex-valued corrections in the framework of the weak asymptotic method, we show that a compressive δ-shock solution resolves such Riemann problems. By letting the approximation parameter tend to zero, the corrections become real valued, and the solutions can be seen to fit into the framework of weak singular solutions defined by Danilov and Shelkovich. Indeed, in this context, we can show that every 2 × 2 system of conservation laws admits δ-shock solutions.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2012

References

1.Albeverio, S. and Danilov, V. G., Global in time solutions to Kolmogorov–Feller pseudodifferential equations with small parameter, Russ. J. Math. Phys. 18 (2011), 1025.Google Scholar
2.Brio, M., Admissibility conditions for weak solutions of nonstrictly hyperbolic systems, in Proc. Int. Conf. on Hyperbolic Problems, Springer Lecture Notes in Mathematics, pp. 4650 (Springer, 1988).Google Scholar
3.Chen, G.-Q. and Liu, H., Formation of δ-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. Analysis 34 (2003), 925938.Google Scholar
4.Colombeau, J.-F., New generalized functions and multiplication of distributions (North-Holland, Amsterdam, 1984).Google Scholar
5.Colombeau, J.-F. and Oberguggenberger, M., On a hyperbolic system with a compatible quadratic term: generalized solutions, delta waves, and multiplication of distributions, Commun. PDEs 15 (1990), 905938.CrossRefGoogle Scholar
6.Maso, G. Dal, LeFloch, P. and Murat, F., Definition and weak stability of non-conservative products, J. Math. Pures Appl. 74 (1995), 483548.Google Scholar
7.Danilov, V. G., Remarks on vacuum state and uniqueness of concentration process, Electron. J. Diff. Eqns 2008 (2008), 110.Google Scholar
8.Danilov, V. G. and Mitrović, D., Weak asymptotic of shock wave formation process, Nonlin. Analysis 61 (2005), 613635.Google Scholar
9.Danilov, V. G. and Mitrović, D., Delta shock wave formation in the case of triangular system of conservation laws, J. Diff. Eqns 245 (2008), 37043734.CrossRefGoogle Scholar
10.Danilov, V. G. and Shelkovich, V. M., Dynamics of propagation and interaction of δ-shock waves in conservation law system, J. Diff. Eqns 211 (2005), 333381.Google Scholar
11.Danilov, V. G., Omel′yanov, G. A. and Shelkovich, V. M., Weak asymptotic method and interaction of nonlinear waves, in Asymptotic methods for wave and quantum problems (ed. Karasev, M.), American Mathematical Society Translations Series, Volume 208, pp. 33165 (American Mathematical Society, Providence, RI, 2003).Google Scholar
12.Ercole, G., Delta shock waves as self similar viscosity limits, Q. Appl. Math. 58 (2000), 177199.CrossRefGoogle Scholar
13.Grosser, M., Kunzinger, M., Oberguggenberger, M. and Steinbauer, R., Geometric theory of generalized functions with applications to general relativity, Mathematics and Its Applications, Volume 537 (Kluwer Academic, Dordrecht, 2001).Google Scholar
14.Hayes, B. and LeFloch, P. G., Measure-solutions to a strictly hyperbolic system of conservation laws, Nonlinearity 9 (1996), 15471563.Google Scholar
15.Huang, F., Existence and uniqueness of discontinuus solutions for a hyperbolic system, Proc. R. Soc. Edinb. A 127 (1997), 11931205.Google Scholar
16.Huang, F., Well posdeness for pressureless flow, Commun. Math. Phys. 222 (2001), 117146.Google Scholar
17.Huang, F., Weak solution to pressureless type system, Commun. PDEs 30 (2005), 283304.CrossRefGoogle Scholar
18.Joseph, K. T., A Riemann problem whose viscosity solution contains δ-measures, Asymp. Analysis 7 (1993), 105120.CrossRefGoogle Scholar
19.Keyfitz, B. and Kranzer, H. C., A viscosity approximation to a system of conservation laws with no classical Riemann solution, in Proc. Int. Conf. on Hyperbolic Problems, Lecture Notes in Mathematics, Volume 1402, pp. 185197 (Springer, 1989).Google Scholar
20.Keyfitz, B. and Kranzer, H. C., A strictly hyperbolic system of conservation laws admitting singular shocks, in Nonlinear evolution equations that change type (ed. Keyfitz, B. and Shearer, M.), IMA Volumes in Mathematics and Its Applications, Volume 27, pp. 107125 (Springer, 1990).Google Scholar
21.Korchinski, C., Solution of a Riemann problem for a 2 × 2 system of conservation laws possessing no classical weak solution, PhD Thesis, Adelphi University (1977).Google Scholar
22.LeFloch, P. G., An existence and uniqueness result for two nonstrictly hyperbolic systems, in Nonlinear evolution equations that change type (ed. Keyfitz, B. and Shearer, M.), IMA Volumes in Mathematics and Its Applications, Volume 27, pp. 126138 (Springer, 1990).Google Scholar
23.LeVeque, R. J., The dynamics of pressureless dust clouds and delta waves, J. Hyperbol. Diff. Eqns 1 (2004), 315327.CrossRefGoogle Scholar
24.Liu, Y.-P. and Xin, Z., Overcompressive shock waves, in Nonlinear evolution equations that change type (ed. Keyfitz, B. and Shearer, M.), IMA Volumes in Mathematics and Its Applications, Volume 27, pp. 139145 (Springer, 1990).Google Scholar
25.Mitrović, D. and Nedeljkov, M., Delta shock waves as a limit of shock waves, J. Hyperbol. Diff. Eqns 4 (2007), 629653.Google Scholar
26.Nedeljkov, M., Delta and singular delta locus for one-dimensional systems of conservation laws, Math. Meth. Appl. Sci. 27 (2004), 931955.Google Scholar
27.Nedeljkov, M., Singular shock waves in interactions, Q. Appl. Math. 66 (2008), 281302.Google Scholar
28.Nedeljkov, M., Shadow waves: entropies and interactions for delta and singular shocks, Arch. Ration. Mech. Analysis 197 (2010), 489537.Google Scholar
29.Nedeljkov, M., Pilipović, S. and Scarpalézos, D., The linear theory of Colombeau generalized functions, Pitman Research Notes in Mathematics Series, Volume 385 (Longman, Harlow, 1998).Google Scholar
30.Omel′yanov, G. A. and Segundo-Caballero, I., Asymptotic and numerical description of the kink/antikink interaction, Electron. J. Diff. Eqns 2010 (2010), 150.Google Scholar
31.Panov, E. Y. and Shelkovich, V. M., δ′-shock waves as a new type of solutions to systems of conservation laws, J. Diff. Eqns 228 (2006), 4986.CrossRefGoogle Scholar
32.Sheng, W. and Zhang, T., The Riemann problem for transportation equations in gas dynamics, Mem. Am. Math. Soc. 137 (1999), 177.Google Scholar
33.Sun, M., Delta shock waves for the chromatography equations as self-similar viscosity limits, Q. Appl. Math. 69 (2011), 425443.CrossRefGoogle Scholar
34.Tan, D., Zhang, T. and Zheng, Y., Delta shock waves as a limits of vanishing viscosity for a system of conservation laws, J. Diff. Eqns 112 (1994), 132.CrossRefGoogle Scholar
35.Volpert, A. I., The space BV and quasilinear equations, Math. USSR Sb. 2 (1967), 225267.CrossRefGoogle Scholar
36.Yang, H., Riemann problems for class of coupled hyperbolic system of conservation laws, J. Diff. Eqns 159 (1999), 447484.Google Scholar