id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-367	Tian, Jing; Zhang, Bingsheng	Solution to Navier-Stokes equations for turbulent channel flows	2020	18	.pdf	application/pdf	6752	398	83	Condition (A5) can be replaced by the f weaker condition lim sup x2→±∞ P (x1, x2, x3, t) <∞, (5.10) for any given x1, x3 and t ∈ R. We have found an optimal choice of the function class. Using (5.4) in Lemma 5.4 given in Appendix C, and Liouville’s theorem for harmonic function ∂ ∂x1 P (x1, x2, t), we conclude that ∂ ∂x1 P (x1, x2, t) = p̃1(t), for some function p̃1(t) of time t, so that P = P (x1, x2, t) = p̃0(x2, t) + x1p̃1(t), (5.9) and, by the harmonicity of P (x1, x2, t) in x1 and x2, we have ∂2 ∂x2 2 p̃0(x2, t) = 0.	cache/ejde-367.pdf	txt/ejde-367.txt