Shock wave formation process for a multidimensional scalar conservation law
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
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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.
References
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References
- 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.
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- V. G. Danilov, Generalized Solution Describing Singularity Interaction, International J. of Mathematics and Mathematical Sciences 29 (2002), 481–494. MR 1896252 (2003m:35149)
- 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)
- 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.
- 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)
- 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)
- 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)
- 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)
- V. G. Danilov, G. A. Omel$’$yanov, 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)
- 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)
- R. Flores-Espinosa, G. A. Omel$’$yanov, Weak asymptotics for the problem of interaction of two shock waves, Nonlinear Phenomena in Complex Systems, 8 (2005), 331–341. MR 2240498 (2007c:35108)
- R. Flores-Espinosa, G. A. Omel$’$yanov, Asymptotic behavior for the centered-rarefaction appearance problem. Electronic J. Differential Equations, 2005 (2005), 1–25. MR 2195544 (2006j:35160)
- M. G. Garcia, G. A. Omel$’$yanov, Kink-antikink interaction for semilinear wave equation with a small parameter, Electronic J. of Differential Equations, 2009 (2009), 1–26. MR 2495850 (2010d:35233)
- M. G. Garcia, R. Flores-Espinosa, G. A. Omel$’$yanov, 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)
- J. Glimm, D. Marchesin, O. McBryan, Unstable fingers in two phase flows, Commun. Pure. Appl. Math., 34 (1981), 53–75. MR 600572 (82e:76069)
- 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)
- 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)
- G. Kossioris, I. Shyuichi, Geometric singularities for solutions of single conservation law, Arch. Rational Mech. Anal., 139 (1997), 255–290. MR 1480242 (98j:35116)
- S. N. Kruzhkov, First order quasilinear equations in several independent variables, Math. USSR Sb. 10 (1970), 217–243.
- D. A. Kulagin, G. A. Omel$’$yanov, 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)
- 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)
- 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)
- 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)
- 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)
- S. Nakane, Formation of shocks for a single conservation law, SIAM Journal of Math. Anal., 19 (1988), 1391–1408. MR 965259 (89k:35142)
- 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)
- 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)
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Additional Information
V. G. Danilov
Affiliation:
Moscow Technical University of Communication and Informatics, Aviamotornaya 8a, 111024 Moscow, Russia
Email:
danilov@miem.edu.ru
D. Mitrovic
Affiliation:
Faculty of Mathematics, University of Montenegro, Cetinjski put bb, 81000 Podgorica, Montenegro
Address at time of publication:
Faculty of Mathematics, University of Bergen, Johannes Bruns gate 12, 5007 Bergen
Email:
matematika@t-com.me
Keywords:
global approximate solution,
weak asymptotic method
Received by editor(s):
April 24, 2009
Published electronically:
June 28, 2011
Additional Notes:
The work of V. G. Danilov is supported by RFFI grant 05-01-00912, DFG Project 436 RUS 113/895/0-1.
Article copyright:
© Copyright 2011
Brown University
The copyright for this article reverts to public domain 28 years after publication.