On the boundedness of the Stokes semigroup in two-dimensional exterior domains

The L ∞-Stokes semigroup in exterior domains

“…It is well-known that the Stokes semigroup S (t) is a bounded analytic semigroup in L r σ [11], [33], [12] in the sense that both ||S (t)|| L and t||dS (t)/dt|| L are bounded in (0, ∞), where X = L r σ for r ∈ (1, ∞). Our estimate (1.6) here gives a local-in-time bound for S (t).…”

Remark on the Lq− L∞estimate of the Stokes semigroup in a 2-dimensional exterior domain

“…And then, by combining several known results concerning the estimates of Stokes resolvent in R n and the asymptotic behavior of Stokes resolvent in the exterior domain near the boundary which was obtained in [7], we will be able to show Theorem 1.2. We would like to note that if we apply the known estimations of the Stokes resolvent to the representation formula due to Borchers and Varnhorn [4] we can also prove Theorem 1.2. Therefore, the proof itself is not so surprizing if we know how to prove the theorem, but we believe that it is worth while giving the proof of Theorem 1.2, because the result itself is very important.…”

“…According to the result of [4], we know that −A generates an analytic semigroup e −tA in a 2-dimensional exterior domain.…”

“…And therefore, his proof does not seem to be applied directly to the 2-dimensional case, because the 2-dimensional Stokes resolvent has the logarithmic singularity at the origin. Borchers and Varnhorn [4] overcame this difficulty first by using the Stokes potentials to show the L p boundedness of Stokes semigroup in 2-dimensional exterior domain.…”

“…For simplicity, we write: P = P Ω , A = A Ω . It is known that −A generates an analytic semigroup e −tA in J q (Ω) [9,4,25]. To denote various constants we use the same letter C, and by C A,B,··· we denotes the constant depending on the quantities A, B, · · · .…”