Equivalence between entropy and renormalized solutions for parabolic equations

摘要: If Ω is an open bounded set in RN, N≥2, with a connected Lipschitz boundary ∂Ω, a(x,ξ) operator of Leray–Lions type, β and γ are non decreasing continuous real functions, β(0)=γ(0)=0, then for every (f,g)∈L1(]0,T[×RN)×L1(]0,T[×∂Ω),(u0,v0)∈L1(RN)×L1(∂Ω), we prove that the entropy solution coincides renormalized to following problem: {u′−div[a(.,∇u)]+β(u)=fon ]0,T[×(RN∖∂Ω),(τu)′+[∂u∂νa]+γ(τu)=g [u]=0on ]0,T[×∂Ω,(u(0,.),τu(0,.))=(u0,v0)a.e. on RN×∂Ω, where [u] [∂u∂νa] respectively jump across ∂Ω u normal derivative ∂u∂νa related a.