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 Free Access

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. C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin-New York, 2000. MR 1763936 (2001m:35212)
3. V. G. Danilov, Generalized Solution Describing Singularity Interaction, International J. of Mathematics and Mathematical Sciences 29 (2002), 481-494. MR 1896252 (2003m:35149)
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. of Applied Mathematics 18 (2007), 537-570. MR 2360622 (2009a:35272)
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, V. M. Shelkovich, Dynamics of propagation and interaction of $\delta$ -shock waves in conservation law systems, J. Differential Equations 211 (2005), 333-381. MR 2125546 (2006f:35173)
7. V. G. Danilov, D. Mitrović, Weak asymptotics of shock wave formation process, Nonlinear Analysis: Theory, Methods and Applications, 61 (2005), 613-635. MR 2126617 (2006a:35193)
8. V. G. Danilov, D. Mitrović, Delta shock wave formation in the case of triangular hyperbolic system of conservation laws , J. Differential Equations 245 (2008), 3704-3734. MR 2462701 (2009h:35264)
9. V. G. Danilov, D. Mitrović, Smooth approximations of global in time solutions to scalar conservation laws, Abstract and Applied Analysis, 2009 (2009), 1-26. MR 2485639 (2010d:35224)
10. V. G. Danilov, G. A. Omelyanov, and V. M. Shelkovich, Weak asymptotics method and interaction of nonlinear waves. In: Asymptotic Methods for Wave and Quantum Problems, M. V. Karasev, ed., AMS Transl., Ser. 2, Vol. 208, 33-164. MR 1995392 (2004f:35021)
11. V. G. Danilov, V. M. Shelkovich, Delta-shock wave type solution of hyperbolic systems of conservation laws, Quart. Appl. Math. 63 (2005), 401-427. MR 2169026 (2006j:35158)
12. R. Flores-Espinosa, G. A. Omelyanov, Weak asymptotics for the problem of interaction of two shock waves, Nonlinear Phenomena in Complex Systems, 8 (2005), 331-341. MR 2240498 (2007c:35108)
13. R. Flores-Espinosa, G. A. Omelyanov, Asymptotic behavior for the centered-rarefaction appearance problem. Electronic J. Differential Equations, 2005 (2005), 1-25. MR 2195544 (2006j:35160)
14. M. G. Garcia, G. A. Omelyanov, Kink-antikink interaction for semilinear wave equation with a small parameter, Electronic J. of Differential Equations, 2009 (2009), 1-26. MR 2495850 (2010d:35233)
15. M. G. Garcia, R. Flores-Espinosa, G. A. Omelyanov, Interaction of shock waves in gas dynamics. Uniform in time asymptotics, International J. Mathematics and Mathematical Sciences, 19 (2005), 3111-3126. MR 2206088 (2006i:35235)
16. J. Glimm, D. Marchesin, O. McBryan, Unstable fingers in two phase flows, Commun. Pure. Appl. Math., 34 (1981), 53-75. MR 600572 (82e:76069)
17. A. M. Il'in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problem, Nauka, Moscow, 1989, English transl., American Mathematical Society, Providence, RI, 1992. MR 1182791 (93g:35016)
18. A. M. Il'in, S. V. Zakharov, From a weak discontinuity to the gradient catastrophe, Mat. Sb. 193 (2001), 3-18. MR 1867014 (2002k:35198)
19. G. Kossioris, I. Shyuichi, Geometric singularities for solutions of single conservation law, Arch. Rational Mech. Anal., 139 (1997), 255-290. MR 1480242 (98j:35116)
20. S. N. Kruzhkov, First order quasilinear equations in several independent variables, Math. USSR Sb. 10 (1970), 217-243.
21. D. A. Kulagin, G. A. Omelyanov, Interaction of kinks for semilinear wave equations with a small parameter, Nonlinear Analysis: Theory, Methods and Applications, 65 (2006), 347-378. MR 2228433 (2007f:35197)
22. A. M. Il'in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Nauka, Moscow, 1989, English transl., American Mathematical Society, Providence, RI, 1992. MR 1182791 (93g:35016)
23. A. M. Il'in, S. V. Zakharov, From a weak discontinuity to the gradient catastrophe, Mat. Sb. 193 (2001), 3-18. MR 1867014 (2002k:35198)
24. D. Mitrovic, V. Bojkovic, V. Danilov, Linearization of the Riemann problem for a triangular system of conservation laws and delta shock wave formation process, Mathematical Methods in the Applied Sciences, 33 (2010), 904-921. MR 2662315 (2011d:35310)
25. D. Mitrovic, J. Susic, Global in time solution to Hopf equation and application to a non-strictly hyperbolic system of conservation laws, Electronic J. of Differential Equations, 2007 (2007), 1-22. MR 2349942 (2008f:35241)
26. S. Nakane, Formation of shocks for a single conservation law, SIAM Journal of Math. Anal., 19 (1988), 1391-1408. MR 965259 (89k:35142)
27. E. Yu. Panov, V. M. Shelkovich, $\delta'$ -Shock waves as a new type of solutions to systems of conservation laws, J. of Differential Equations 228 (2006), 49-86. MR 2254184 (2007f:35188)
28. D. Wagner, The Riemann problem in two space dimensions for a single conservation law, SIAM J. Math. Anal., 14 (1983), 534-559. MR 697528 (84f:35092)