id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-19	Levandosky, Julie L.; Vera, Octavio	Smoothing properties for a coupled Zakharov-Kuznetsov system	2023	41	.pdf	application/pdf	15874	745	86	For ξν as defined in (6.3), α = (α1, α2) such that |α| = β, 4 ≤ β ≤ K, the following holds:∑ |α|=β ∣∣ ∫ t 0 ∫ ξν(∂αu)∂α(uux) ∣∣+ ∣∣ ∫ t 0 ∫ ξν(∂αv)∂α(vvx) ∣∣ ≤ C + C ∑ |α|=β (∫ t 0 ∫ ξν(∂αu)2 ) + C ∑ |α|=β (∫ t 0 ∫ ξν(∂αv)2 ) (6.21) for 0 ≤ t ≤ T , where C depends only on sup 0≤t≤T ∫ ξν(∂γu)2, sup 0≤t≤T ∫ ξν(∂γv)2, (6.22)∫ T 0 ∫ (ξν)x(∂γux)2, ∫ T 0 ∫ (ξν)x(∂γvx)2, (6.23)∫ T 0 ∫ (ξν)x(∂γuy)2, ∫ T 0 ∫ (ξν)x(∂γvy)2 (6.24) for γ = (γ1, γ2) where |γ| ≤ β − 1. EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 35 The proof uses the same ideas as in the proof of Lemma 3.2. In this case, the remainder terms satisfy∣∣ ∫ t 0 ∫ ξuxy(uux)xy ∣∣ EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 9 = ∣∣ ∫ t 0 ∫ ξuxy(2uxuxy + uyuxx + uuxxy) ∣∣ ≤ |ux|L∞ ∫ t 0 ∫ ξu2xy + C|uy|L∞ ∫ t 0 ∫ ξu2xx + |uy|L∞ ∫ t 0 ∫ ξu2xy ≤ C ∫ t 0 ∫ ξ(u2xx + u2xy) where C depends only on ‖u‖H3 .	cache/ejde-19.pdf	txt/ejde-19.txt