On the well-posedness, ill-posedness and norm-inflation for a higher order water wave model on a periodic domain. (March 2020) - Carvajal, X.; Panthee, M.; Pastrán, R.
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 H s ( 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 H s ( 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 H s ( 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 ( T ) norm, with s < 1, for arbitrarily short time.