On the well-posedness, ill-posedness and norm-inflation for a higher order water wave model on a periodic domain

Abstract

In this work we are interested in the well-posedness issues for the initial value problem associated with a higher order water wave model posed on a pe\-rio\-dic domain $\mathbb{T}$. We derive some multilinear estimates and use them in the contraction mapping argument to prove local well-posedness for initial data in the periodic Sobolev space $H^s(\mathbb{T})$, $s\geq 1$. With some restriction on the parameters appeared in the model, we use the conserved quantity to obtain global well-posedness for given data with Sobolev regularity $s\geq 2$. Also, we use splitting argument to improve the global well-posedness result in $H^s(\mathbb{T})$ for $1\leq s< 2$. Well-posedness result obtained in this work is sharp in the sense that the flow-map that takes initial data to the solution cannot to be continuous for given data in $H^s(\mathbb{T})$, $s< 1$. Finally, we prove a norm-inflation result by showing that the solution corresponding to a smooth initial data may have arbitrarily large $H^s(\mathbb{T})$ norm, with $s<1$, for arbitrarily short time.