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.