On some modification Navier–Stokes equations

Abstract This article investigates some modification of the Navier–Stokes equations of type as the modification that was suggested by Lions (Quelques Methodes de Resolution des Problemes aux Limites Non Lineares, DUNOD, Gauthier-Villaris, Paris, 1969) in the form ∂u j ∂t −μ ∑ i=1 n D i (|u j | p i −2 D i u j )+ ∑ i=1 n u i D i u j + ∂p ∂x j =h j (t,x),(t,x)∈Q⊂R n+1, , div u=0, (t,x)∈Q≡(0,T)×Ω, T>0, μ>0, u j (0,x)=0, x∈Ω, u| Γ =0, Γ≡[0,T]×∂Ω, under p i =p ∀i : 1⩽i⩽n, n⩾2 . In this we prove the existence theorem for the different pi⩾max{2,3−2/n} and, on some additional conditions (i.e. of the pi=p⩾4) in the isotropic nonlinearity case, we prove the uniqueness theorem for the considered problem.