On the well-posedness, ill-posedness and norm-inflation for a higher order water wave model on a periodic domain. (March 2020)

• Record Type: Journal Article
• Authors: Carvajal, X.; Panthee, M.; Pastrán, R.
• Is Part Of: Nonlinear analysis. Volume 192(2020)
• Publication Date: 2020-03
• Subjects: 35A01 -- 35Q53; KdV equation -- BBM equation -- Initial value problem -- Local and global well-posedness -- Ill-posedness -- Norm-inflation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.111713
• Languages: English
• ISSNs: 0362-546X

Abstract

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.