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

Yue Pu
This thesis studies a regularization of the classical Saint-Venant (shallow-water) system, namely the regularized shallow-water (Airy or Saint-Venant) system, recentlyintroduced 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 correspondingnon-oscillatory traveling wave solution which is continuous and piecewise smooth, having a weak singularity at a single point
more » ... where the energy is dissipated as it is for the classicalshock. 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 anduniqueness 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-typeanalysis with the help from Landau-Kolmogorov inequality that addresses the nonlocal part.
doi:10.1184/r1/7189049 fatcat:ifdodpj7b5dlfgpi45uum7popy