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

Abstract

We show that the classical Shinbrot's criteria to guarantee that a Leray-Hopf solution satisfies the energy equality follows trivially from the $L^4( (0\,,T)\times\Omega))$ Lions-Prodi particular case. Moreover we extend Shinbrot's result to space coefficients $ r \in (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 $r=\infty$, and just for these two values, the conditions become respectively $u \in L^\infty(L^3)$ and $u \in L^2(L^\infty)$, which imply regularity by appealing to classical Ladyzhenskaya-Prodi-Serrin (L-P-S) type conditions. However, for values $r\in (3,\infty)$ 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 $L^\infty(L^2)$ 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.