On the Shinbrot's criteria for energy equality to Newtonian fluids: A simplified proof, and an extension of the range of application

We show that the classical Shinbrot's criteria to guarantee that a Leray-Hopf solution satisfies the energy equality follows trivially from the L-4((0, T) x Omega)) Lions-Prodi particular case. Moreover we extend Shinbrot's result to space coefficients r is an element of (3, 4). In this last case our condition coincides with Shinbrot condition for r = 4, but for r < 4 it is more restrictive than the classical one, 2/p+ 2/ r = 1. It looks significant that in correspondence to the extreme values r = 3 and r = infinity, and just for these two values, the conditions become respectively u is an element of L-infinity(L-3) and u is an element of L-2(L-infinity), which imply regularity by appealing to classical Ladyzhenskaya-Prodi-Serrin (L-P-S) type conditions. However, for values r. (3,8) the L-P-S condition does not apply, even for the more demanding case 3 < r < 4. The proofs are quite trivial, by appealing to interpolation, with L-infinity(L-2) in the first case and with L-2(L-6) in the second case. The central position of this old classical problem in Fluid-Mechanics, together with the simplicity of the proofs (in particular the novelty of the second result) looks at least curious. This may be considered a merit of this very short note. (C) 2020 Elsevier Ltd. All rights reserved.