Weakly Singular Waves and Blow-up for a Regularization of the Shallow-water System

This thesis studies a regularization of the classical Saint-Venant (shallow-water) system, namely the regularized shallow-water (Airy or Saint-Venant) system, recently
introduced by D. Clamond and D. Dutykh. This regularization is non-dispersive and formally conserves mass, momentum and energy. We show that for every classical shock wave, this system admits a corresponding
non-oscillatory traveling wave solution which is continuous and piecewise smooth, having a weak singularity at a single point where the energy is dissipated as it is for the classical
shock. This system also admits cusped solitary waves of both elevation and depression. The Hs (s > 2) large time existence with respect to the scaling of initial data and
uniqueness are established using an iteration scheme. The solution exists so long as the first derivatives are bounded in L1. When the energy is small, the height of water admits a positive lower bound dictated by the smallness of the energy. Lastly, we show that there exists smooth initial data with which the L1 norm of the first derivatives go unbounded in finite amount of time. This is proved by a Riccati-type
analysis with the help from Landau-Kolmogorov inequality that addresses the nonlocal part.