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

Author(s)

Carvajal, X. ^[1] ; Panthee, M. ^[2] ; Pastran, R. ^[3]

Affiliation

^[1] Univ Fed Rio de Janeiro, Inst Matemat, BR-21941909 Rio De Janeiro, RJ - Brazil

^[2] IMECC UNICAMP, Dept Math, BR-13083859 Sao Paulo, SP - Brazil

^[3] Univ Nacl Colombia, Dept Math, AK 30 45-03, Bogota 304503, AK - Colombia

• Document type: Journal article
• Source: NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS; v. 192, MAR 2020.
• Web of Science Citations: 0

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. (C) 2019 Elsevier Ltd. All rights reserved. (AU)

• FAPESP's process: 16/25864-6 - Nonlinear Evolution Equations of Dispersive Type
• Grantee: Mahendra Prasad Panthee
• Support Opportunities: Regular Research Grants