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

TL;DR

This paper investigates the well-posedness and finite-time derivative blow-up in smooth solutions of a regularized shallow water system, revealing conditions under which solutions remain smooth or develop singularities.

Contribution

It provides the first analysis of local well-posedness and derivative blow-up phenomena for the linearly non-dispersive regularized Saint-Venant equations.

Findings

Smooth solutions conserve energy and do not form shocks.

Certain small-energy solutions develop singularities in derivatives in finite time.

The system admits weakly singular shock-profile solutions that dissipate energy.

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.