id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-4931	Ouya, Jules; Ouedraogo , Arouna	Global Existence of Weak Solutions to 3D Compressible Primitive Equations of Atmospheric Dynamics with Degenerate Viscosity	2023	39	.pdf	application/pdf	13453	681	84	Math, 16 (4) (2023), 2247-2285 2264 − ∫ T 0 ∫ Ω ξuv∂yΦdXdt+ ∫ T 0 ∫ Ω ∇xξ 2 · ΦdXdt+ r ∫ T 0 ∫ Ω ξ|u|uΦdXdt + r1 ∫ T 0 ∫ Ω uΦdXdt+ α ∫ T 0 ∫ Ω ∆xu ·∆xΦdXdt = − ∫ T 0 ∫ Ω 2ξDx(u) : ∇xΦdXdt+ ϵ ∫ T 0 ∫ Ω (∇xξ · ∇xu) ΦdX − r2 ∫ T 0 ∫ × Ω. Due to the lower semi-continuity of convex functions, we can obtain the following energy inequality (116) and B-D entropy (117) by passing to the limits in (100) and (101) as r2, δ −→ 0,∫ Ω ( 1 2 ξu2 + ξ2 + k1|∇x √ ξ|2 ) dxdy + r1 ∫ T 0 ∫ Ω u2dxdydt + r ∫ T 0 ∫ Ω ξ|u|3dxdydt+ ∫ T 0 ∫ Ω 2ξ|Dx(u)|2dxdydt+ g ∫ T 0 ∫ Ω vξ ln ξdxdydt + ∫ T 0 ∫ Ω ξ|∂yu|2dxdydt ≤ ∫ Ω ( 1 2 ξ0u 2 0 + ξ20 + k1|∇x √ ξ0|2 ) ddxdy (116) and∫ Ω (1 2 ξ|u+ 2∇x ln ξ|2 − 2r1 ln ξ ) dxdy + 2 ∫ T 0 ∫ Ω ξ|∂yv|2dxdydt+ r ∫ T 0 ∫ Ω ξ|u|3dxdydt + r1 ∫ T 0 ∫ Ω u2dxdydt+ ∫ T 0 ∫ Ω ξ|∂yu|2dxdy + k1 ∫ T 0 ∫ Ω ξ|∇2 x ln ξ|2dxdydt J. Ouya, A. Ouédraogo / Eur.	cache/ejpam-4931.pdf	txt/ejpam-4931.txt