Pointwise upper bounds for the solution of the Stokes equation on [formula omitted] and applications

Abstract

Consider the Stokes semigroup T∞ defined on Lσ∞(Ω) where Ω⊂Rn, n≥3, denotes an exterior domain with smooth boundary. It is shown that T∞(z)u0 for u0∈Lσ∞(Ω) and z∈Σθ with θ∈(0,π/2) satisfies pointwise estimates similar to the ones known for G(z)u0 where G denotes the Gaussian semigroup on Rn. In particular, T∞ extends to a bounded analytic semigroup on Lσ∞(Ω) of angle π/2. Moreover, T∞(t) allows Lσ∞(Ω)−C2+α(Ω¯) smoothing for every t>0 and the Stokes semigroups Tp and Tq on Lσp(Ω) and Lσq(Ω) are consistent for all p,q∈(1,∞].