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

2019/08/20 by Xavier Carvajal, Carvajal, Xavier, Mahendra Panthee +3
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1908.07508

openalex publication_date 2019/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this work we are interested in the well-posedness issues for the initial\nvalue problem associated with a higher order water wave model posed on a\npe -rio -dic domain mathbbT. We derive some multilinear estimates and use\nthem in the contraction mapping argument to prove local well-posedness for\ninitial data in the periodic Sobolev space H^s( mathbbT), s\≥ 1. With\nsome restriction on the parameters appeared in the model, we use the conserved\nquantity to obtain global well-posedness for given data with Sobolev regularity\ns\≥ 2. Also, we use splitting argument to improve the global well-posedness\nresult in H^s( mathbbT) for 1\≤ s< 2. Well-posedness result obtained in\nthis work is sharp in the sense that the flow-map that takes initial data to\nthe solution cannot to be continuous for given data in H^s( mathbbT), s<\n1. Finally, we prove a norm-inflation result by showing that the solution\ncorresponding to a smooth initial data may have arbitrarily large\nH^s( mathbbT) norm, with s<1, for arbitrarily short time.\n

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