On the energy equality for the 3D Navier–Stokes equations. (March 2020)

• Record Type: Journal Article
• Authors: Berselli, Luigi C.; Chiodaroli, Elisabetta
• Is Part Of: Nonlinear analysis. Volume 192(2020)
• Publication Date: 2020-03
• Subjects: 35Q30 -- 35D30 -- 76D07; Energy conservation -- Navier–Stokes equations -- Weak and very-weak solutions; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.111704
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we study the problem of energy conservation for the solutions of the initial–boundary value problem associated to the 3D Navier–Stokes equations, with Dirichlet boundary conditions. First, we consider Leray–Hopf weak solutions and we prove some new criteria, involving the gradient of the velocity. Next, we compare them with the existing literature in scaling invariant spaces and with the Onsager conjecture. Then, we consider the problem of energy conservation for very-weak solutions, proving energy equality for distributional solutions belonging to the so-called Shinbrot class. A possible explanation of the role of this classical class of solutions, which is not scaling invariant, is also given.