Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 1187–1198 | Cite as

Delta Shock Waves in the Shallow Water System

  • C. O. R. SarricoEmail author
  • A. Paiva


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.


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

Mathematics Subject Classification

46F10 35D99 35L67 



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


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

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

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