Advertisement

Delta Shock Waves in the Shallow Water System

Article

Abstract

We consider a Riemann problem for the shallow water system \(u_{t} +\big (v+\textstyle \frac{1}{2}u^{2}\big )_{x}=0\), \(v_{t}+\big (u+uv\big )_{x}=0\) and evaluate all singular solutions of the form \(u(x,t)=l(t)+b(t)H\big (x-\gamma (t)\big )+a(t)\delta \big (x-\gamma (t)\big )\), \(v(x,t)=k(t)+c(t)H\big (x-\gamma (t)\big )\), where \(l,b,a,k,c,\gamma :\mathbb {R}\rightarrow \mathbb {R}\) are \(C^{1}\)-functions of time t, H is the Heaviside function, and \(\delta \) stands for the Dirac measure with support at the origin. A product of distributions, not constructed by approximation processes, is used to define a solution concept, that is a consistent extension of the classical solution concept. Results showing the advantage of this framework are briefly presented in the introduction.

Keywords

Products of distributions Shallow water system Shock waves Delta waves Delta shock waves 

Mathematics Subject Classification

46F10 35D99 35L67 

Notes

Acknowledgements

The present research was supported by FCT, UID/MAT/04561/2013.

References

  1. 1.
    Ali, A., Kalisch, H.: Energy balance for undular bores. C.R. Mec. 338(2), 67–70 (2000)CrossRefMATHGoogle Scholar
  2. 2.
    Bressan, A., Rampazzo, F.: On differential systems with vector valued impulsive controls. Bull. Un. Mat. Ital. 2B(7), 641–656 (1988)MathSciNetMATHGoogle Scholar
  3. 3.
    Colombeau, J.F., Le Roux, A.: Multiplication of distributions in elasticity and hydrodynamics. J. Math. Phys. 29, 315–319 (1988)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Dal Maso, G., LeFloch, P., Murat, F.: Definitions and weak stability of nonconservative products. J. Math. Pure Appl. 74, 483–548 (1995)MathSciNetMATHGoogle Scholar
  5. 5.
    Danilov, V.G., Maslov, V.P., Shelkovich, V.M.: Algebras of singularities of singular solutions to first-order quasi-linear strictly hyperbolic systems. Teoret. Mat. Fiz.114(1), 3–55 (in Russian). Theor. Math. Phys+ 114(1), 1–42 (1988)Google Scholar
  6. 6.
    Danilov, V.G., Mitrovic, D.: Delta shock wave formation in the case of triangular hyperbolic system of conservation laws. J. Differ. Equ. 245, 3704–3734 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Egorov, Y.V.: On the theory of generalized functions. Usp. Mat. Nauk45(5), 3–40 (in Russian). Russ. Math. Surv+ 45(5):1–49 (1990)Google Scholar
  8. 8.
    Hayes, B.T., LeFloch, P.G.: Measure solutions to a strictly hyperbolic system of conservation laws. Nonlinearity 9(6), 1547–1563 (1996)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kalisch, H., Mitrovic, D.: Singular solutions for the shallow-water equations. IMA J. Appl. Math. 77(3), 340–350 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kalisch, H., Mitrovic, D.: Singular solutions of a fully nonlinear \(2\times 2\) system of conservation laws. Proc. Edinb. Math. Soc. (2) 55(3), 711–729 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Maslov, V.P.: Nonstandard characteristics in asymptotical problems. Usp. Mat. Nauk38(6), 3–36 (in Russian). Russ. Math. Surv+ 38(6), 1–42 (1983)Google Scholar
  12. 12.
    Maslov, V.P., Omel’yanov, G.A.: Asymptotic soliton-form solutions of equations with small dispersion. Usp. Mat. Nauk36(3), 63–126 (in Russian). Russ. Math. Surv+ 36(3), 73–149 (1981)Google Scholar
  13. 13.
    Mitrovic, D., Bojkovic, V., Danilov, V.G.: Linearization of the Riemann problem for a triangular system of conservation laws and delta shock wave formation process. Math. Methods Appl. Sci. 33, 904–921 (2010)MathSciNetMATHGoogle Scholar
  14. 14.
    Sarrico, C.O.R.: About a family of distributional products important in the applications. Port. Math. 45(1988), 295–316 (1988)MathSciNetMATHGoogle Scholar
  15. 15.
    Sarrico, C.O.R.: Distributional products and global solutions for nonconservative inviscid Burgers equation. J. Math. Anal. Appl. 281, 641–656 (2003)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Sarrico, C.O.R.: New solutions for the one-dimensional nonconservative inviscid Burgers equation. J. Math. Anal. Appl. 317, 496–509 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Sarrico, C.O.R.: Collision of delta-waves in a turbulent model studied via a distributional product. Nonlinear Anal. Theor. 73, 2868–2875 (2010)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Sarrico, C.O.R.: Products of distributions and singular travelling waves as solutions of advection–reaction equations. Russ. J. Math. Phys. 19(2), 244–255 (2012)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Sarrico, C.O.R.: Products of distributions, conservation laws and the propagation of \(\delta ^{\prime }\)-shock waves. Chin. Ann. Math. Ser. B 33(3), 367–384 (2012)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Sarrico, C.O.R.: The multiplication of distributions and the Tsodyks model of synapses dynamics. Int. J. Math. Anal. 6(21), 999–1014 (2012)MathSciNetMATHGoogle Scholar
  21. 21.
    Sarrico, C.O.R.: A distributional product approach to \(\delta \)-shock wave solutions for a generalized pressureless gas dynamics system. Int. J. Math. 25(1), 1450007 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Sarrico, C.O.R.: The Brio system with initial conditions involving Dirac masses: a result afforded by a distributional product. Chin. Ann. Math. 35B(6), 941–954 (2014). doi: 10.1007/s11401-014-0862-8 MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Sarrico, C.O.R.: New distributional global solutions for the Hunter–Saxton equation. Abstr. Appl. Anal., Art. ID 809095, 9. doi: 10.1155/2014/809095
  24. 24.
    Sarrico, C.O.R.: The Riemann problem for the Brio system: a solution containing a Dirac mass obtained via a distributional product. Russ. J. Math. Phys. 22(4), 518–527 (2015)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Sarrico, C.O.R., Paiva, A.: Products of distributions and collision of a \(\delta \)-wave with a \(\delta ^{\prime }\)-wave in a turbulent model. J. Nonlinear Math. Phys. 22(3), 381–394 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Schwartz, L.: Théorie des Distributions. Hermann, Paris (1965)Google Scholar
  27. 27.
    Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1999)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.CMAF-CIO, Faculdade de Ciências da Universidade de LisboaLisbonPortugal

Personalised recommendations