Pointwise upper bounds for the solution of the Stokes equation on Lσ∞(Ω) and applications

Consider the Stokes semigroup T∞T∞ defined on Lσ∞(Ω) where Ω⊂RnΩ⊂Rn, n≥3n≥3, denotes an exterior domain with smooth boundary. It is shown that T∞(z)u0T∞(z)u0 for u0∈Lσ∞(Ω) and z∈Σθz∈Σθ with θ∈(0,π/2)θ∈(0,π/2) satisfies pointwise estimates similar to the ones known for G(z)u0G(z)u0 where G   denotes the Gaussian semigroup on RnRn. In particular, T∞T∞ extends to a bounded analytic semigroup on Lσ∞(Ω) of angle π/2π/2. Moreover, T∞(t)T∞(t) allows Lσ∞(Ω)−C2+α(Ω¯) smoothing for every t>0t>0 and the Stokes semigroups TpTp and TqTq on Lσp(Ω) and Lσq(Ω) are consistent for all p,q∈(1,∞]p,q∈(1,∞].