Well-posedness of the Green–Naghdi model for an uneven bottom in presence of the Coriolis effect and surface tension

Abstract

The objective of this work is to derive and analyze a Green–Naghdi model with Coriolis effect and surface tension in nonflat bottom geometry. Gui et al. derive a Green–Naghdi-type model in flat bottom geometry under the gravity and Coriolis effect. Chen et al. proved the existence and uniqueness of solution in Sobolev space under a condition depending on the initial velocity and the Coriolis effect. In this paper, we provide a rigorous derivation of Green–Naghdi model under the influence of the two mentioned effects, with nonflat bottom. After that, the existence and construction of solutions for the derived model will be proved under two alternative conditions: the first one is the same condition as in Chen et al. and Berjawi et al. and the second one concerns only the Coriolis coefficient $\Omega$ that supposed to be only of order $O\left(\sqrt{\mu }\right)$. This existence and uniqueness result ameliorate the result of Chen et al. and Berjawi et al. in the sense that no condition on the velocity is needed. We also prove the continuity of the associated flow map.