Taiwanese Journal of Mathematics

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

Carlos Orlando R. Sarrico

Full-text: Open access

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.
  • 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.
  • J.-F. Colombeau and A. LeRoux, Multiplications of distributions in elasticity and hydrodynamics, J. Math. Phys. 29 (1988), no. 2, 315–319.
  • 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.
  • 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.
  • 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.
  • Yu. V. Egorov, On the theory of generalized functions, Uspekhi Mat. Nauk 45 (1990), no. 5, 3–40.
  • B. T. Hayes and P. G. LeFloch, Measure solutions to a strictly hyperbolic system of conservation laws, Nonlinearity 9 (1996), no. 6, 1547–1563.
  • 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.
  • 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.
  • 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.
  • ––––, Spaces of weighed measures for conservation laws with singular shock solutions, J. Differential Equations 118 (1995), no. 2, 420–451.
  • V. P. Maslov, Non-standard characteristics in asymptotic problems, Russian Math. Surveys 38 (1983), no. 6, 1–42.
  • 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.
  • 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.
  • C. O. R. Sarrico, About a family of distributional products important in the applications, Portugal. Math. 45 (1988), no. 3, 295–316.
  • ––––, Distributional products and global solutions for nonconservative inviscid Burgers equation, J. Math. Anal. Appl. 281 (2003), no. 2, 641–656.
  • ––––, New solutions for the one-dimensional nonconservative inviscid Burgers equation, J. Math. Anal. Appl. 317 (2006), no. 2, 496–509.
  • ––––, Collision of delta-waves in a turbulent model studied via a distribution product, Nonlinear Anal. 73 (2010), no. 9, 2868–2875.
  • ––––, Products of distributions and singular travelling waves as solutions of advection-reaction equations, Russ. J. Math. Phys. 19 (2012), no. 2, 244–255.
  • ––––, Products of distributions, conservation laws and the propagation of $\delta'$-shock waves, Chin. Ann. Math. Ser. B 33 (2012), no. 3, 367–384.
  • ––––, 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.
  • ––––, 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.
  • ––––, 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.
  • 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.
  • S. Schecter, Existence of Dafermos profiles for singular shocks, J. Differential Equations 205 (2004), no. 1, 185–210.
  • L. Schwartz, Théorie des distributions, Hermann, Paris, 1966.
  • M. Willem, Analyse harmonique réele, Hermann, Paris, 1995.