Abstract
We prove that the steady-state Navier–Stokes problem in a Lipschitz bounded three-dimensional domain has a solution which satisfies a maximum modulus estimate, provided the fluxes of the boundary datum through the boundary of any interior connected component are less then a computable positive constant.
Acknowledgements
The author wish to thank the anonymous referee for his careful reading of the manuscript and for his useful suggestions.
Notes
Notes
1. u : Ω → ℝ3 and p : Ω → ℝ are the (unknown) kinetic field, and pressure field respectively; (∇u)u is the vector with components u j ∂ j u i . If V is a linear subspace of , then the symbol V σ(Ω) denotes the set of all fields u ∈ V that are weakly divergence free in Ω Citation2. Unless otherwise specified, the symbol c will denote a positive constant whose numerical value is not essential to our purposes.
2. Estimate (Equation6) and related questions have been the object of some recent papers Citation9–11.
3. Since Ω is Lipschitz, there exists a family of finite (congruent) cone such that γ(ξ)∖{ξ} ⊂ Ω for all ξ ∈ ∂Ω. We say that a function f(x) in Ω tends nontangentially to a function g defined almost everywhere on ∂Ω if for almost all ξ ∈ ∂Ω.