Resolvent Estimates and Maximal Regularity in Weighted L q -spaces of the Stokes Operator in an Infinite Cylinder

Abstract

We study resolvent estimate and maximal regularity of the Stokes operator in L q -spaces with exponential weights in the axial directions of unbounded cylinders of R n , n ≥ 3. For straights cylinders we obtain these results in Lebesgue spaces with exponential weights in the axial direction and Muckenhoupt weights in the cross-section. Next, for general cylinders with several exits to infinity we prove that the Stokes operator in L q -spaces with exponential weight along the axial directions generates an exponentially decaying analytic semigroup and has maximal regularity.The proofs for straight cylinders use an operator-valued Fourier multiplier theorem and techniques of unconditional Schauder decompositions based on the R-boundedness of the family of solution operators for a system in the cross-section of the cylinder parametrized by the phase variable of the onedimensional partial Fourier transform. For general cylinders we use cut-off techniques based on the result for straight cylinders and the result for the case without exponential weight.

The Resolvent Problem and $${{H^{\infty}}}$$ -calculus of the Stokes Operator in Unbounded Cylinders with Several Exits to Infinity

“…, (2.9) where c = c (q, r), c = c(q, r) > 0; note that for X = L r ω ( ) the constants c in (2.1) and c in (2.3) are independent of the weight ω, see [19], Remark 5.7, Remark 5.3, and even independent of , which can easily be seen via the extension by 0 of functions on onto R n−1 . Let us recall Khintchine's inequality for complex numbers a j , i.e.,…”

“…For the special case g = 0 this theorem was treated in [19], Theorem 2.1. Therefore, we shall consider only the case f = 0 and assume, due to Lemma 2.9 (3), that g ∈ C ∞ 0 (R; W 1,r ω ( )) ∩ W −1;q,r ω ( ).…”

“…The system (1.2) in an infinite straight cylinder ×R was studied in vector-valued homogeneous Besov spacesḂ s pq (R; L r ( )), s ∈ R, 1 ≤ p, q ≤ ∞, ( [17]) and in L q (R; L r ω ( )) ( [19]), where 1 < q, r < ∞ and ω ∈ A r (R n−1 ) is an arbitrary Muckenhoupt weight. To be more concrete, using the partial Fourier transform F = along the axis of the cylinder × R, the authors obtained estimates for the parametrized Stokes resolvent system on…”

On the Stokes semigroup in some non-Helmholtz domains

“…[5], [6], [7]. (For a cylindrical domain (or an infinite cylinder) it is shown in [4] that S(t) is analytic in C 0,σ (Ω) which is also analytic in L p σ (Ω) [10]. )…”

The L p -Helmholtz projection in finite cylinders

“…Results on resolvent estimates, maximal regularity, and boundedness of the H ∞ -calculus for the Stokes operator in L p σ (Ω) are serialized in [11], [12], and [13]. Again, Ω is assumed to be an infinite layer, an infinite cylinder or the union of finitely many of these with a bounded domain.…”