On the Shinbrot's criteria for energy equality to Newtonian fluids: A simplified proof, and an extension of the range of application. (July 2020) - Beirão da Veiga, Hugo; Yang, Jiaqi
Abstract: 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 ) × Ω ) ) Lions–Prodi particular case. Moreover we extend Shinbrot's result to space coefficients r ∈ ( 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 = ∞, and just for these two values, the conditions become respectively u ∈ L ∞ ( L 3 ) and u ∈ L 2 ( L ∞ ), which imply regularity by appealing to classical Ladyzhenskaya–Prodi–Serrin (L–P–S) type conditions. However, for values r ∈ ( 3, ∞ ) 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 ∞ ( 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.