Asymptotic self-similar blow-up for the regularized Saint-Venant equations

TL;DR

This paper analyzes the formation of singularities in the regularized Saint-Venant equations, establishing the stability of self-similar blow-up profiles and revealing unique regularity properties influenced by Hamiltonian regularization.

Contribution

It rigorously characterizes the structure and stability of self-similar blow-up solutions in the rSV system, connecting them to the Hunter--Saxton and Burgers equations.

Findings

Proves stability of self-similar blow-up profiles in the rSV system.

Identifies $C^{3/5}$ H"older regularity at the singularity.

Shows the same blow-up profile appears in the regularized Burgers equation.

Abstract

We investigate singularity formation in the regularized Saint--Venant (rSV) equations, a conservative, non-dispersive shallow water system that is formally regarded as a Hamiltonian regularization of the isentropic Euler equations. While it is known that smooth solutions to the rSV system can develop gradient blow-up in finite time, the precise structure of such singularities has not been rigorously characterized. In this work, we establish stability of self-similar blow-up profiles of the Hunter--Saxton equation within the rSV framework, using a nonlinear bootstrap argument in dynamically rescaled coordinates. Our analysis captures the detailed space-time dynamics of solutions near the singularity, and proves their sharp $C^{3/5}$ H\"older regularity at the singular time. This regularity differs from the $C^{1/3}$ H\"older regularity of the cubic-root singularities found in the compressible Euler and inviscid Burgers equations. This contrast highlights the structural influence of the Hamiltonian regularization on singularity formation. To illuminate this effect, we also show that the same $C^{3/5}$ blow-up profile emerges in the regularized Burgers equation, a scalar analogue of the rSV system.