id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-162	Lopes, Juliana Honda; Planas, Gabriela	Existence of solutions for a non-isothermal Navier-Stokes-Allen-Cahn system with thermo-induced coefficients	2022	22	.pdf	application/pdf	8645	414	77	We note that (2.3) implies the existence of some positive constants Ci, i = 1, 2, 3 such that − C1 ≤ F ′′(s), −C2 ≤ F (s) ≤ F ′(s)s+ C3 for all s ∈ R, (2.5) where F (s) = ∫ Then φ1 − φ2 solves (φ1 − φ2)t + u · ∇(φ1 − φ2) = ∇ · (ε(θ)∇(φ1 − φ2))− 1 ε(θ) (F ′(φ1)− F ′(φ2)) together with ∂ ∂η (φ1 − φ2) = 0 on ∂Ω× (0, T ) and (φ1 − φ2)(0) = 0 in Ω. Multiplying the above equation by φ1 − φ2 and integrating in Ω, we see 1 2 d dt ‖φ1 − φ2‖2 + ε0‖∇(φ1 − φ2)‖2 ≤ − ( 1 ε(θ) (F ′(φ1)− F ′(φ2)), φ1 − φ2 ) ≤ C‖F ′(φ1)− F ′(φ2)‖‖φ1 − φ2‖ ≤ C‖φ1 − φ2‖2, EJDE-2022/72 NON-ISOTHERMAL NAVIER-STOKES-ALLEN-CAHN SYSTEM 9 where we used that ∇ · u = 0, the Mean Value Theorem for F ′ and the fact that F ′′ is bounded.	cache/ejde-162.pdf	txt/ejde-162.txt