Gevrey regularity and finite time singularities for the Kakutani-Matsuuchi model

Abstract

In this paper, we deal with the Kakutani-Matsuuchi model which describes the surface elevation eta of the water-waves under the effect of viscosity. We first derive the decay rate of weak solutions. This can be used to obtain the decay rate of parallel to eta(t)parallel to (<(H)over dot>1) when initial data is sufficiently small in <(H)over dot>(1). We next show the existence, uniqueness, Gevrey regularity and decay rates of eta with sufficiently small initial data in B-2,1(1). To do so, we derive a commutator estimate involving Gevrey operator. We then apply our method to the supercritical quasi-geostrophic equations. We finally show the formation of singularities of smooth solutions in finite time for a certain class of initial data . (C) 2021 Elsevier Ltd. All rights reserved.