Taiwanese Journal of Mathematics

Multiplication of Distributions and Travelling Wave Solutions for the Keyfitz-Kranzer System

Carlos Orlando R. Sarrico

Full-text: Open access

PDF File (327 KB)

Abstract

The present paper concerns the study of distributional travelling waves for the model problem $u_{t} + (u^{2}-v)_{x} = 0$, $v_{t} + (u^{3}/3-u)_{x} = 0$, also called the Keyfitz-Kranzer system. In the setting of a product of distributions, which is not defined by approximation processes, we are able to define a rigourous concept of a solution which extends the classical solution concept. As a consequence, we will establish necessary and sufficient conditions for the propagation of distributional profiles and explicit examples are given. A survey of the main ideas and formulas for multiplying distributions is also provided.

Article information

Source
Taiwanese J. Math., Volume 22, Number 3 (2018), 677-693.

Dates
Received: 21 February 2017
Revised: 15 November 2017
Accepted: 29 November 2017
First available in Project Euclid: 16 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1513393252

Digital Object Identifier
doi:10.11650/tjm/171104

Mathematical Reviews number (MathSciNet)
MR3807332

Subjects
Primary: 46F10: Operations with distributions 35D99: None of the above, but in this section 35L67: Shocks and singularities [See also 58Kxx, 76L05]

Keywords
products of distributions conservation laws travelling shock waves travelling delta waves travelling waves which are not measures

Citation

Sarrico, Carlos Orlando R. Multiplication of Distributions and Travelling Wave Solutions for the Keyfitz-Kranzer System. Taiwanese J. Math. 22 (2018), no. 3, 677--693. doi:10.11650/tjm/171104. https://projecteuclid.org/euclid.twjm/1513393252


Export citation

References

  • A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Bull. Un. Mat. Ital. B (7) 2 (1988), no. 3, 641–656.
    Zentralblatt MATH: 0653.49002
  • J.-J. Cauret, J.-F. Colombeau, A. Y. LeRoux, Discontinuous generalized solutions of nonlinear nonconservative hyperbolic equations, J. Math. Anal. Appl. 139 (1989), no. 2, 552–573.
    Zentralblatt MATH: 0691.35057
    Digital Object Identifier: doi:10.1016/0022-247X(89)90129-7
  • J.-F. Colombeau and A. LeRoux, Multiplications of distributions in elasticity and hydrodynamics, J. Math. Phys. 29 (1988), no. 2, 315–319.
    Zentralblatt MATH: 0646.76007
    Digital Object Identifier: doi:10.1063/1.528069
  • G. Dal Maso, P. G. Lefloch and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9) 74 (1995), no. 6, 483–548.
    Zentralblatt MATH: 0853.35068
    Mathematical Reviews (MathSciNet): MR1365258
  • V. G. Danilov, V. P. Maslov and V. M. Shelkovich, Algebras of the singularities of singular solutions to first-order quasi-linear strictly hyperbolic systems, Theoret. Mat. Phys. 114 (1998), no. 1, 1–42.
    Mathematical Reviews (MathSciNet): MR1756560
    Zentralblatt MATH: 0946.35049
    Digital Object Identifier: doi:10.4213/tmf827
  • V. G. Danilov and D. Mitrovic, Delta shock wave formation in the case of triangular hyperbolic system of conservation laws, J. Differerential Equations 245 (2008), no. 12, 3704–3734.
    Mathematical Reviews (MathSciNet): MR2462701
    Zentralblatt MATH: 1192.35120
    Digital Object Identifier: doi:10.1016/j.jde.2008.03.006
  • Yu. V. Egorov, On the theory of generalized functions, Uspekhi Mat. Nauk 45 (1990), no. 5, 3–40.
    Mathematical Reviews (MathSciNet): MR1084986
    Zentralblatt MATH: 0754.46034
  • B. T. Hayes and P. G. LeFloch, Measure solutions to a strictly hyperbolic system of conservation laws, Nonlinearity 9 (1996), no. 6, 1547–1563.
    Mathematical Reviews (MathSciNet): MR1419460
    Zentralblatt MATH: 0908.35075
    Digital Object Identifier: doi:10.1088/0951-7715/9/6/009
  • H. Kalisch and D. Mitrović, Singular solutions of a fully nonlinear $2 \times 2$ system of conservation laws, Proc. Edinb. Math. Soc. (2) 55 (2012), no. 3, 711–729.
    Mathematical Reviews (MathSciNet): MR2975250
    Digital Object Identifier: doi:10.1017/S0013091512000065
    Zentralblatt MATH: 1286.35162
  • B. L. Keyfitz, Conservation laws, delta-shocks and singular shocks, in: Nonlinear Theory of Generalized Functions (Vienna, 1997), 99–111, Chapman & Hall/CRC Res. Notes Math 401, Chapman & Hall/CRC, Boca Raton, FL, 1999.
  • ––––, Singular shocks: retrospective and prospective, Confluentes Math. 3 (2011), no. 3, 445–470.
    Zentralblatt MATH: 1237.35113
    Digital Object Identifier: doi:10.1142/S1793744211000424
  • B. L. Keyfitz and H. C. Kranzer, The Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy, J. Differential Equations 47 (1983), no. 1, 35–65.
    Zentralblatt MATH: 0521.35035
    Digital Object Identifier: doi:10.1016/0022-0396(83)90027-X
  • ––––, Spaces of weighed measures for conservation laws with singular shock solutions, J. Differential Equations 118 (1995), no. 2, 420–451.
    Zentralblatt MATH: 0821.35096
    Digital Object Identifier: doi:10.1006/jdeq.1995.1080
  • V. P. Maslov, Non-standard characteristics in asymptotic problems, Russian Math. Surveys 38 (1983), no. 6, 1–42.
    Zentralblatt MATH: 0562.35007
  • V. P. Maslov and G. A. Omel'yanov, Asymptotic soliton-form solutions of equations with small dispersion, Russian Math. Surveys 36 (1981), no. 3, 73–149.
    Zentralblatt MATH: 0494.35080
    Digital Object Identifier: doi:10.1070/RM1981v036n03ABEH004248
  • 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.
    Zentralblatt MATH: 1189.35178
  • C. O. R. Sarrico, About a family of distributional products important in the applications, Portugal. Math. 45 (1988), no. 3, 295–316.
    Mathematical Reviews (MathSciNet): MR973164
    Zentralblatt MATH: 0664.46042
  • ––––, Distributional products and global solutions for nonconservative inviscid Burgers equation, J. Math. Anal. Appl. 281 (2003), no. 2, 641–656.
    Zentralblatt MATH: 1026.35078
    Digital Object Identifier: doi:10.1016/S0022-247X(03)00187-2
  • ––––, New solutions for the one-dimensional nonconservative inviscid Burgers equation, J. Math. Anal. Appl. 317 (2006), no. 2, 496–509.
    Zentralblatt MATH: 1099.35121
    Digital Object Identifier: doi:10.1016/j.jmaa.2005.06.037
  • ––––, Collision of delta-waves in a turbulent model studied via a distribution product, Nonlinear Anal. 73 (2010), no. 9, 2868–2875.
    Mathematical Reviews (MathSciNet): MR2678648
    Zentralblatt MATH: 1198.35051
    Digital Object Identifier: doi:10.1016/j.na.2010.06.036
  • ––––, Products of distributions and singular travelling waves as solutions of advection-reaction equations, Russ. J. Math. Phys. 19 (2012), no. 2, 244–255.
    Mathematical Reviews (MathSciNet): MR2926328
    Zentralblatt MATH: 1284.35112
    Digital Object Identifier: doi:10.1134/S1061920812020100
  • ––––, Products of distributions, conservation laws and the propagation of $\delta'$-shock waves, Chin. Ann. Math. Ser. B 33 (2012), no. 3, 367–384.
    Zentralblatt MATH: 1288.46030
    Digital Object Identifier: doi:10.1007/s11401-012-0713-4
  • ––––, The multiplication of distributions and the Tsodyks model of synapses dynamics, Int. J. Math. Anal. (Ruse) 6 (2012), no. 21-24, 999–1014.
  • ––––, A distributional product approach to $\delta$-shock wave solutions for a generalized pressureless gas dynamics system, Int. J. Math. 25 (2014), no. 1, 1450007, 12 pp.
    Zentralblatt MATH: 1298.46038
    Digital Object Identifier: doi:10.1142/S0129167X14500074
  • ––––, The Brio system with initial conditions involving Dirac masses: a result afforded by a distributional product, Chin. Ann. Math. Ser. B 35 (2014), no. 6, 941–954.
    Mathematical Reviews (MathSciNet): MR3271424
    Zentralblatt MATH: 1315.46044
    Digital Object Identifier: doi:10.1007/s11401-014-0862-8
  • ––––, The Riemann problem for the Brio system: a solution containing a Dirac mass obtained via a distributional product, Russ. J. Math. Phys. 22 (2015), no. 4, 518–527.
    Mathematical Reviews (MathSciNet): MR3431175
    Zentralblatt MATH: 1344.46034
    Digital Object Identifier: doi:10.1134/S1061920815040111
  • C. O. R. Sarrico and A. Paiva, Products of distributions and collision of a $\delta$-wave with a $\delta'$-wave in a turbulent model, J. Nonlinear Math. Phys. 22 (2015), no. 3, 381–394.
  • D. G. Schaeffer, S. Schecter and M. Shearer, Non-strictly hyperbolic conservation laws with a parabolic line, J. Differential Equations 103 (1993), no. 1, 94–126.
    Zentralblatt MATH: 0809.35056
    Digital Object Identifier: doi:10.1006/jdeq.1993.1043
  • S. Schecter, Existence of Dafermos profiles for singular shocks, J. Differential Equations 205 (2004), no. 1, 185–210.
    Zentralblatt MATH: 1077.35095
    Digital Object Identifier: doi:10.1016/j.jde.2004.06.013
  • L. Schwartz, Théorie des distributions, Hermann, Paris, 1966.
  • M. Willem, Analyse harmonique réele, Hermann, Paris, 1995.
    Zentralblatt MATH: 0839.43001

Mathematical Society of the Republic of China

Taiwanese Journal of Mathematics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more