Well-posedness and derivative blow-up for a dispersionless regularized shallow water system

Abstract

We study local-time well-posedness and breakdown for solutions of regularized Saint-Venant equations (regularized classical shallow water equations) recently introduced by Clamond and Dutykh. The system is linearly non-dispersive, and smooth solutions conserve an $H^1$-equivalent energy. No shock discontinuities can occur, but the system is known to admit weakly singular shock-profile solutions that dissipate energy. We identify a class of small-energy smooth solutions that develop singularities in the first derivatives in finite time.

28 pages, 1 figure; substantial reorganization, corrected proof of blow-up criteria, new references

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Cited by in corpus (3)

• Global weak solutions of a Hamiltonian regularised Burgers equation
• Hamiltonian regularisation of the unidimensional barotropic Euler equations
• On the blow-up scenario for some modified Serre-Green-Naghdi equations