id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-400	Fan, Jishan; Zhou,  Yong	Existence and uniqueness for a Ginzburg-Landau system for superconductivity	2020	6	.pdf	application/pdf	2222	136	83	k ∇ψ + ψA ∣∣2 dx, which leads to ‖A‖L∞(0,T ;H1) + ‖A‖L2(0,T ;H2) + ‖∂tA‖L2(0,T ;L2) ≤ C, (2.3) whence ‖φ‖L2(0,T ;H1) ≤ C. (2.4) Multiplying (1.1) by −∆ψ, integrating by parts and taking the real part, using (2.1), (2.3) and (2.4), we obtain η 2 d dt ∫ |∇ψ|2 dx+ 1 k2 ∫ |∆ψ|2 dx ≤ ∣∣Re ∫ iηkφψ ·∆ψ dx ∣∣+ 2 ∣∣Re 1 k ∫ iA∇ψ ·∆ψ dx ∣∣ + Re ∫ A2ψ∆ψ dx+ εRe ∫ (|ψ|2 − 1)ψ ·∆ψ dx ≤ 1 2 1 k2 ∫ |∆ψ|2 dx+ C ∫ |∇φ||∇ψ|dx 4 J. FAN, Y. ZHOU EJDE-2020/17 + C‖A‖2L∞‖∇ψ‖2L2 + C‖A‖L∞‖∇A‖L2‖∇ψ‖L2 + C‖∇ψ‖2L2 , which yields ‖ψ‖L∞(0,T ;H1) + ‖ψ‖L2(0,T ;H2) ≤ C. (2.5) Since∫ T 0 ∫ |ψA|2 dxdt ≤ ‖ψ‖2L3 ∫ T 0 ‖A‖2L6 dt ≤ C, it follows from (2.2) that ‖ψ‖L2(0,T ;H1) ≤ C. (3.6) EJDE-2020/17 GINZBURG-LANDAU SYSTEM FOR SUPERCONDUCTIVITY 5 Testing (1.2) by |A|A and letting u := |A|3/2, using (1.3), (1.8), (1.9), (1.10), (1.11), (3.1) and the vector identities (ν · ∇)A ·A = (A · ∇)A · ν + (curlA× ν) ·A, (3.7) (A · ∇)A · ν = −(A · ∇)ν ·A, (3.8) we arrive at d dt ∫ u2 dx+ C0 ∫ |∇u|2 dx+ C0 ∫ |A||∇A|2 dx ≤ C ∫ ∣∣ i k ∇ψ + ψA ∣∣ |ψ|u4/3 dx+ C ∫ |∇φ|u4/3 dx+ C ∫ ∂Ω u2 dS ≤ C ∥∥ ∣∣ i	cache/ejde-400.pdf	txt/ejde-400.txt