id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-988	Zhang, Xue; Zhao, Xiaopeng	Strong solutions to density-dependent incompressible smectic-A liquid crystal equations	2025	21	.pdf	application/pdf	7035	319	80	∥ut∥L2 ≤ µ4 28 ∥∇ut∥2L2 + C(∥∇∆φt∥4L2 + ∥∇φt∥4L2 + ∥∆2φ∥4L2 + ∥ut∥4L2 + 1), (3.39) J24 ≤ C∥∇ut∥L2∥∇φt∥L2 ≤ µ4 28 ∥∇ut∥2L2 + C∥∇φt∥2L2 , (3.40) J25 ≤ C∥∆φt∥L2∥∇φ∥2L6∥ut∥L6 ≤ C∥∇∆φt∥1/2L2 ∥∇φt∥1/2L2 ∥∇φ∥2H1∥ut∥H1 ≤ µ4 28 ∥∇ut∥2L2 + C∥∇∆φt∥L2∥∇φt∥L2∥∇φ∥4H1 + C∥∇∆φt∥1/2L2 ∥∇φt∥1/2L2 ∥∇φ∥2H1∥ut∥L2 ≤ µ4 28 ∥∇ut∥2L2 + C(∥ut∥6L2 + ∥∇∆φt∥6L2 + ∥∇φt∥6L2 + ∥∇φ∥6H1 + 1), (3.41) J26 ≤ C∥∇φt∥L2∥∇φ∥L6∥∆φ∥L6∥ut∥L6 ≤ C∥∇φt∥L2∥∇φ∥2H2∥ut∥H1 ≤ µ4 28 ∥∇ut∥2L2 + C∥∇φt∥2L2∥∇φ∥4H2 + C∥∇φt∥L2∥∇φ∥2H2∥ut∥L2 ≤ µ4 28 ∥∇ut∥2L2 + C(∥∇∆φ∥6L2 + ∥∇φ∥6L2 + ∥∇φt∥6L2 + ∥ut∥6L2 + 1). Also, ∫ Ω |u||∇φ||φ|dx ≤ ∥u∥L3∥∇φ∥L2∥φ∥L6 ≤ C∥u∥H1∥φ∥2H1 ≤ C(∥u∥3H1 + ∥φ∥3H1).	cache/ejde-988.pdf	txt/ejde-988.txt