Estimates for the Stokes Operator in Lipschitz Domains

We study the Stokes operator A in a threedimensional Lipschitz domain Ω. Our main result asserts that the domain of A is contained in W 1,p 0 (Ω)∩W (Ω) for some p > 3. Certain L∞-estimates are also established. Our results may be used to improve the regularity of strong solutions of Navier-Stokes equations in nonsmooth domains. In the appendix we provide a simple proof of area integral estimates for solutions of Stokes equations. Introduction. In a recent interesting paper, Deuring and von Wahl [DW] consider strong solutions of the nonstationary Navier-Stokes equations in Ω× (0,T ):   ∂u ∂t = ∆u− (u ·∇)u−∇π+ f, divu = 0, with the initial-Dirichlet condition { u(X,t) = 0 for (X,t) ∈ ∂Ω× (0,T ), u(X,0) = u0(X), X ∈ Ω, where Ω is a bounded Lipschitz domain in R. Based on the functional analytical approach of Fujita and Kato [FK] and the Rellich estimates of Shen [S1], they show that, if u0 ∈ D(A) for some ε ∈ (0, 1 2 ) and f is bounded and locally Hölder continuous, then a solution (u,π) exists for some T > 0 and u ∈ C ( (0,T ],D(A) ) ∩C ( [0,T ],D(A) ) , where A = −P∆ denotes the Stokes operator. The purpose of this note is to describe D(A), the domain of A, in terms of Sobolev’s spaces. In the case of smooth domains, it is well known that D(A) = W (Ω)∩W 1,2 0 (Ω)∩L 2 σ(Ω) 1183 Indiana University Mathematics Journal c ©, Vol. 44, No. 4 (1995) 1184 R. M. Brown & Z. Shen where Lσ(Ω) denotes the space of solenoidal functions in L (Ω) (e.g., see [CF]). One can not expect such results in Lipschitz domains, as the W -estimate, in general, fails in nonsmooth domains. Our main results in this paper assert that (0.1) D(A) ⊂ W 1,p 0 (Ω)∩W (Ω) for some p = p(Ω) > 3 (Theorem 2.17). In particular, it follows from Sobolev’s imbedding that for every t ∈ (0,T ], u(t) ∈ C(Ω̄) for some α = α(Ω) > 0, i.e., the strong solution of the Navier-Stokes equations is Hölder continuous up to the boundary as a function of X. We also obtain the following L∞ estimates: (0.2) ‖u‖L∞(Ω) ≤ C ∥∥∇u∥∥1/2 L2(Ω) ∥∥Au∥∥1/2 L2(Ω) , ‖u‖L∞(Ω) ≤ C ∥∥u∥∥1/4 L2(Ω) ∥∥Au∥∥3/4 L2(Ω) for u ∈ D(A). To establish (0.2), we use the reverse Hölder inequality, (0.1) and some localization techniques. See Theorem 3.1 and Corollary 3.2. Estimates like (0.2) are very useful in the study of Navier-Stokes equations. See [CF] and [H] in the case of smooth domains. To prove (0.1), we shall study the Dirichlet problem for the Stokes equations with a forcing term, and interpolate between the L estimates in [FKV] and the Hölder estimates in [S2]. The following area integral estimate, ∫ Ω |∇u(X)|dist(X,∂Ω) dX ≤ C ∫