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 periodic domain T. We derive some multilinear estimates and use them in the contraction mapping argument to prove the local well-posedness for initial data in the periodic Sobolev space Hs(T), s≥1. With some restriction on the parameters appeared in the model, we use the conserved quantity to obtain the global well-posedness for given data with Sobolev regularity s≥2. Also, we use splitting argument to improve the global well-posedness result in Hs(T) for 1≤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 be continuous for given data in Hs(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 Hs(T) norm, with s<1, for arbitrarily short time.