Improved local well-posedness for the periodic "good" Boussinesq equation

Details Improved local well-posedness for the periodic "good" Boussinesq equation Improved local well-posedness for the periodic "good" Boussinesq equation

2012-01-09

arxiv.org/abs/1201.1942v1

Mathematics

Analysis of PDEs

We prove the local well-posedness of the 1D "good" Boussinesq equation in the periodic case for the initial data in $H^{-3/8+}

Scientific paper

We prove that the "good" Boussinesq model with the periodic boundary condition is locally well-posed in the space $H^{s}\times H^{s-2}$ for $s > -3/8$. In the proof, we employ the normal form approach, which allows us to explicitly extract the rougher part of the solution. This also leads to the conclusion that the remainder is in a smoother space $C([0,T], H^{s+a}), where $0 <= a < \min (2s+1, 1/2)$. If we have a mean-zero initial data, this implies a smoothing effect of this order for the non-linearity. This is new even in the previously considered cases $s > -1/4$.

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