# American Institute of Mathematical Sciences

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## Large-time existence for one-dimensional Green-Naghdi equations with vorticity

 1 LAMA, Univ Paris Est Creteil, Univ Gustave Eiffel, CNRS, F-94010, Créteil, France 2 Mathématiques, Faculté des sciences I et Laboratoire de mathématiques, École doctorale des sciences et technologie, Université Libanaise, Beyrouth, Liban

* Corresponding author: Raafat Talhouk

Received  June 2020 Revised  March 2021 Published  April 2021

This essay is concerned with the one-dimensional Green-Naghdi equations in the presence of a non-zero vorticity according to the derivation in [5], and with the addition of a small surface tension. The Green-Naghdi system is first rewritten as an equivalent system by using an adequate change of unknowns. We show that solutions to this model may be obtained by a standard Picard iterative scheme. No loss of regularity is involved with respect to the initial data. Moreover solutions exist at the same level of regularity as for first order hyperbolic symmetric systems, i.e. with a regularity in Sobolev spaces of order $s>3/2$.

Citation: Colette Guillopé, Samer Israwi, Raafat Talhouk. Large-time existence for one-dimensional Green-Naghdi equations with vorticity. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021040
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