Um estudo do fenômeno de dissipação anômala em equações de fluidos 2D forçadas

Abstract

In this work, we study the phenomenon of anomalous dissipation of potential enstrophy for the family of Generalized Camassa-Holm equations (GCH) in a two-dimensional periodic domain, which are obtained as an interpolation between second-grade fluid equations and the Camassa-Holm equations. In this context, we first prove the existence and uniqueness of solutions for the GCH equations. Existence is established using the Galerkin method, while for uniqueness, we develop a differential inequality and apply a Grönwall-type inequality.We then demonstrate that for values of β in the range 1/2 < β ≤ 1, there is no anomalous dissipation of potential enstrophy. On the other hand, in the case β = 0, the system exhibits anomalous dissipation of potential enstrophy, specifically infinite dissipation. The analysis of the phenomenon of anomalous dissipation of potential enstrophy was carried out by investigating the inviscid limit of the long-term time averages of the solutions to the GCH equations and subsequently identifying these long-term time averages with stationary statistical solutions in the phase space of potential vorticity.