Properties of the Boundary Flux of a Singular Diffusion Process

Consider a nonhomogeneous fluid flowing through a homogeneous porous medium occupyinga bounded domain ΩR~n.Let u=u(x,t)represent the density of the fluid at the positionx∈Ω and time t,which satisfies the following singular diffusion equation     

Young Measure Solutions of a Class of Singular Diffusion Equations

Consider the first initial-boundary value problem     

Young Measure Solutions of a Class of Singular Diffusion Equations

Ｃｏｎｓｉｄｅｒｔｈｅｆｉｒｓｔｉｎｉｔｉａｌ ｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍ ｕ ｔ=ｄｉｖ ｑ( ｕ) ,　 (ｘ,ｔ) ∈ＱＴ,(1 )ｕ(ｘ,ｔ) =0 ,　 (ｘ,ｔ) ∈ Ω× (0 ,Ｔ) ,(2 )ｕ(ｘ,0 ) =ｕ0 (ｘ) ,　ｘ∈Ω ,(3 )ｗｈｅｒｅΩｉｓａｂｏｕｎｄｅｄｄｏｍａｉｎｉｎＲＮｗｉｔｈｓｍｏｏｔｈｂｏｕｎｄａｒｙ Ω ,ＱＴ=Ω× (0 ,Ｔ) , ｑ = φ ,φ∈Ｃ1(ＲＮ) ,ａｎｄ φ , ｑｓａｔｉｓｆｙｔｈｅｓｔｒｕｃｔｕｒｅｃｏｎｄｉｔｉｏｎｓ(λ｜ξ｜1+δ-1 ) +≤ φ(ξ) ≤Λ｜ξ｜1+δ+ 1 ,　 ξ∈ＲＮ,(4 )｜ ｑ(ξ) …     

Global Existence and Blow-up of Solutions to Multi-dimensional （n ≤ 5） Viscous Cahn-Hilliard Equation

In this paper we consider the viscous Cahn-Hilliard equation with spatial dimension n ≤5, and established global existence of weak solutions for small initial value and blow-up of solutions for any nontrivial initial data.