Vanishing solution of Leray equation in 3-space

This note shows that if {nu, a} is a pair of positive constants and U = (U-1, U-2, U-3), as a weak solution to the Leray equation -nu Delta U + aU + a(y . del)U +(U . del) U + del P = 0 = div U in R-3 , satisfies {UiUj is an element of div (L-1< q < infinity,L-- q < gamma <3(R-3))(3) for i, j is an element of {1, 2, 3}; (q, gamma, U) is an element of (1, infinity) x max{-q, 3 - 2q}, 3] x (W(1,)q/q-1 (R-3))(3 )or (q, gamma) is an element of [2,infinity) x (0, 3], then U vanishes identically. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

The note demonstrates that under certain conditions, the solution U will completely vanish.