id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-383	Chen, Qing; Wu, Guochun; Zhang, Yinghui; Zou,  Lan	Optimal time decay rates for the compressible Navier-Stokes system with and without Yukawa-type potential	2020	25	.pdf	application/pdf	8512	534	83	Furthermore, we have from (2.3) that for k = l − 1, 〈∇l−1N2,∇l−1u〉 = 〈∇l−1 ( − u · ∇u ) ,∇l−1u〉+ 〈 ∇l−1 ( − (P ′(ρ) ρ − P ′(1) ) ∇% ) ,∇l−1u 〉 + 〈 ∇l−1 (µ ρ %∇u ) ,∇lu 〉 + 〈 ∇l−1 ( ∇ (µ ρ % ) · ∇u ) ,∇l−1u 〉 + 〈 ∇l−1 (µ+ ν ρ %div u ) ,∇l−1 div u 〉 + 〈 ∇l−1 ( ∇ (µ+ ν ρ % ) div u ) ,∇l−1u 〉 . δ ( ‖∇l%‖2L2 + ‖∇lu‖2L2 ) , (4.26) EJDE-2020/102 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 15 where (6.9) is used, and for k = l, 〈∇lN2,∇lu〉 = 〈∇l(−u · ∇u),∇lu〉+ 〈 ∇l−1 ((P ′(ρ) ρ − P ′(1) ) ∇% ) ,∇l div u 〉 + 〈 ∇l−1 (µ ρ %∆u ) ,∇l div u 〉 + 〈 ∇l−1 (µ+ ν ρ %∇div u ) ,∇l div u 〉 .	cache/ejde-383.pdf	txt/ejde-383.txt