Analysis of a Family of Shallow Water Waves

We study a family of shallow water wave equations called the b-family equation. Known for having multi-peakon solutions, this family includes the Camassa-Holm equation and the Degasperis-Procesi equation as its most notable and only integrable members (using the perturbative symmetry definition). We show that the periodic and non-periodic Cauchy problem for the b-family equation is well-posed in Sobolev spaces with exponent greater than 3/2. Moreover, we find that the corresponding data-to-solution map is continuous on Sobolev spaces but not uniformly continuous. We prove that this map is not uniformly continuous using approximate solutions together with delicate commutator and multiplier estimates. The novelty of the proof lies in the fact that it makes no use of conserved quantities. Lastly, given a weaker topology, we show that the data-to-solution map is Holder continuous.