On the maximum modulus theorem for the steady-state Navier–Stokes equations in Lipschitz bounded domains

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.

Keywords:

• stationary Navier–Stokes equations
• maximum modulus theorem
• Lipschitz domains

AMS Subject Classifications::

• 35Q30
• 76D03
• 76D05

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 2. Unless otherwise specified, the symbol c will denote a positive constant whose numerical value is not essential to our purposes.

2. Estimate (6) and related questions have been the object of some recent papers 9–11.

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.