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

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 solitons*, Electronic J. of Differential Equations,**2003**(2003), 1-27.**6.**V.G. Danilov, V. M. Shelkovich,*Dynamics of propagation and interaction of -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,*-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)**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC (2000):
35L65,
35L67

Retrieve articles in all journals with MSC (2000): 35L65, 35L67

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

DOI:
https://doi.org/10.1090/S0033-569X-2011-01234-9

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.