Shock wave formation process for a multidimensional scalar conservation law

Danilov, V.; Mitrovic, D.

doi:10.1090/S0033-569X-2011-01234-9

Authors: V. G. Danilov and D. Mitrovic
Journal: Quart. Appl. Math. 69 (2011), 613-634
MSC (2000): Primary 35L65, 35L67
DOI: https://doi.org/10.1090/S0033-569X-2011-01234-9
Published electronically: June 28, 2011
MathSciNet review: 2893992
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We construct a global smooth approximate solution to a multidimensional scalar conservation law describing the shock wave formation process for initial data with small variation. In order to solve the problem, we modify the method of characteristics by introducing ``new characteristics'', nonintersecting curves along which the (approximate) solution to the problem under study is constant. The procedure is based on the weak asymptotic method, a technique which appeared to be rather powerful for investigating nonlinear waves interactions.

1. I. Berre, H. K. Dahle, K. H. Karlsen, K.-A. Lie, J. R. Natvig, Time-of-Flight + Fast Marching + Transport Collapse: An Alternative to Streamlines for Two-Phase Porous Media Flow with Capillary Forces?, In Computational Methods in Water Resources (Delft, The Netherlands, 2002), 995-1002, Elsevier, 2002.
2. Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2000. MR 1763936
3. V. G. Danilov, Generalized solutions describing singularity interaction, Int. J. Math. Math. Sci. 29 (2002), no. 8, 481–494. MR 1896252, https://doi.org/10.1155/S0161171202012206
4. V. G. Danilov, Weak asymptotic solution of phase-field system in the case of confluence of free boundaries in the Stefan problem with underheating, European J. Appl. Math. 18 (2007), no. 5, 537–569. MR 2360622, https://doi.org/10.1017/S0956792507007061
5. V. G. Danilov, G. A. Omelianov, Weak asymptotic method for the study of propagation and interaction of infinitely narrow $\delta$ solitons, Electronic J. of Differential Equations, 2003 (2003), 1-27.
6. V. G. Danilov and V. M. Shelkovich, Dynamics of propagation and interaction of 𝛿-shock waves in conservation law systems, J. Differential Equations 211 (2005), no. 2, 333–381. MR 2125546, https://doi.org/10.1016/j.jde.2004.12.011
7. V. Danilov and D. Mitrovic, Weak asymptotics of shock wave formation process, Nonlinear Anal. 61 (2005), no. 4, 613–635. MR 2126617, https://doi.org/10.1016/j.na.2005.01.034
8. V. G. Danilov and D. Mitrovic, Delta shock wave formation in the case of triangular hyperbolic system of conservation laws, J. Differential Equations 245 (2008), no. 12, 3704–3734. MR 2462701, https://doi.org/10.1016/j.jde.2008.03.006
9. V. G. Danilov and D. Mitrovic, Smooth approximations of global in time solutions to scalar conservation laws, Abstr. Appl. Anal. , posted on (2009), Art. ID 350762, 26. MR 2485639, https://doi.org/10.1155/2009/350762
10. V. G. Danilov, G. A. Omel′yanov, and V. M. Shelkovich, Weak asymptotics method and interaction of nonlinear waves, Asymptotic methods for wave and quantum problems, Amer. Math. Soc. Transl. Ser. 2, vol. 208, Amer. Math. Soc., Providence, RI, 2003, pp. 33–163. MR 1995392, https://doi.org/10.1090/trans2/208/02
11. V. G. Danilov and V. M. Shelkovich, Delta-shock wave type solution of hyperbolic systems of conservation laws, Quart. Appl. Math. 63 (2005), no. 3, 401–427. MR 2169026, https://doi.org/10.1090/S0033-569X-05-00961-8
12. R. Flores Espinoza and G. A. Omel′yanov, Weak asymptotics for the problem of interaction of two shock waves, Nonlinear Phenom. Complex Syst. 8 (2005), no. 4, 331–341. MR 2240498
13. Ruben Flores-Espinoza and Georgii A. Omel′yanov, Asymptotic behavior for the centered-rarefaction appearance problem, Electron. J. Differential Equations (2005), No. 148, 25. MR 2195544
14. Martin G. Garcia and Georgii A. Omel’yanov, Kink-antikink interaction for semilinear wave equations with a small parameter, Electron. J. Differential Equations (2009), No. 45, 26. MR 2495850
15. M. G. García-Alvarado, R. Flores-Espinoza, and G. A. Omel′yanov, Interaction of shock waves in gas dynamics: uniform in time asymptotics, Int. J. Math. Math. Sci. 19 (2005), 3111–3126. MR 2206088, https://doi.org/10.1155/IJMMS.2005.3111
16. J. Glimm, D. Marchesin, and O. McBryan, Unstable fingers in two-phase flow, Comm. Pure Appl. Math. 34 (1981), no. 1, 53–75. MR 600572, https://doi.org/10.1002/cpa.3160340104
17. A. M. Il′in, Matching of asymptotic expansions of solutions of boundary value problems, Translations of Mathematical Monographs, vol. 102, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by V. Minachin [V. V. Minakhin]. MR 1182791
18. S. V. Zakharov and A. M. Il′in, From a weak discontinuity to a gradient catastrophe, Mat. Sb. 192 (2001), no. 10, 3–18 (Russian, with Russian summary); English transl., Sb. Math. 192 (2001), no. 9-10, 1417–1433. MR 1867014, https://doi.org/10.1070/SM2001v192n10ABEH000599
19. Shyuichi Izumiya and Georgios T. Kossioris, Geometric singularities for solutions of single conservation laws, Arch. Rational Mech. Anal. 139 (1997), no. 3, 255–290. MR 1480242, https://doi.org/10.1007/s002050050053
20. S. N. Kruzhkov, First order quasilinear equations in several independent variables, Math. USSR Sb. 10 (1970), 217-243.
21. D. A. Kulagin and G. A. Omel′yanov, Interaction of kinks for semilinear wave equations with a small parameter, Nonlinear Anal. 65 (2006), no. 2, 347–378. MR 2228433, https://doi.org/10.1016/j.na.2005.06.015
22. A. M. Il′in, Matching of asymptotic expansions of solutions of boundary value problems, Translations of Mathematical Monographs, vol. 102, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by V. Minachin [V. V. Minakhin]. MR 1182791
23. S. V. Zakharov and A. M. Il′in, From a weak discontinuity to a gradient catastrophe, Mat. Sb. 192 (2001), no. 10, 3–18 (Russian, with Russian summary); English transl., Sb. Math. 192 (2001), no. 9-10, 1417–1433. MR 1867014, https://doi.org/10.1070/SM2001v192n10ABEH000599
24. D. Mitrovic, V. Bojkovic, and V. G. Danilov, Linearization of the Riemann problem for a triangular system of conservation laws and delta shock wave formation process, Math. Methods Appl. Sci. 33 (2010), no. 7, 904–921. MR 2662315, https://doi.org/10.1002/mma.1226
25. Darko Mitrovic and Jela Susic, Global solution to a Hopf equation and its application to non-strictly hyperbolic systems of conservation laws, Electron. J. Differential Equations (2007), No. 114, 22. MR 2349942
26. Shizuo Nakane, Formation of shocks for a single conservation law, SIAM J. Math. Anal. 19 (1988), no. 6, 1391–1408. MR 965259, https://doi.org/10.1137/0519102
27. E. Yu. Panov and V. M. Shelkovich, 𝛿’-shock waves as a new type of solutions to systems of conservation laws, J. Differential Equations 228 (2006), no. 1, 49–86. MR 2254184, https://doi.org/10.1016/j.jde.2006.04.004
28. David H. Wagner, The Riemann problem in two space dimensions for a single conservation law, SIAM J. Math. Anal. 14 (1983), no. 3, 534–559. MR 697528, https://doi.org/10.1137/0514045