Other singularity types or exponents in the rSV equations

Determine whether finite-time blow-up solutions of the regularized Saint–Venant equations (the conservative, non-dispersive Hamiltonian regularization of the shallow water equations studied here) can exhibit singularity types or spatial Hölder regularity exponents different from the C^{3/5} profile established in this work, when considered under settings different from those analyzed in the paper.

Background

The paper proves that solutions to the regularized Saint–Venant (rSV) system can form gradient blow-up with a stable self-similar profile exhibiting sharp C^{3/5} spatial Hölder regularity at the singular time, paralleling analogous results for a scalar analogue (the regularized Burgers equation). This contrasts with the C^{1/3} profiles typical in classical systems such as the compressible Euler and inviscid Burgers equations.

The authors attribute the difference in exponents to structural features of Hamiltonian regularizations that conserve an H^1-type energy and identify the Hunter–Saxton-type leading-order structure in the blow-up regime. Beyond the regimes treated in the analysis, they explicitly raise the question of whether different singularity behaviors or regularity exponents could arise within the rSV framework under other settings.