Gevrey regularity and finite time singularities for the Kakutani–Matsuuchi model. (February 2022)

• Record Type: Journal Article
• Authors: Bae, Hantaek; Lee, Woojae; Shin, Jaeyong
• Is Part Of: Nonlinear analysis. Volume 63(2022)
• Publication Date: 2022-02
• Subjects: Kakutani–Matsuuchi model -- Gevrey regularity -- Decay rates -- Singularity formation; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2021.103415
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: In this paper, we deal with the Kakutani–Matsuuchi model which describes the surface elevation η 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 ‖ η ( t ) ‖ H ̇ 1 when initial data is sufficiently small in H ̇ 1 . We next show the existence, uniqueness, Gevrey regularity and decay rates of η 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.