Maximal regularity in exponentially weighted Lebesgue spaces of the Stokes operator in unbounded cylinders

Abstract

We study resolvent estimates 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 ≥ . For a straight cylinder we use exponential weights in the axial direction and Muckenhoupt weights in the cross-section. Next, for cylinders with several exits to in nity we prove that the Stokes operator in L q -spaces with exponential weights generates an exponentially decaying analytic semigroup and has maximal regularity. The proof for straight cylinders uses an operator-valued Fourier multiplier theorem and 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 one-dimensional partial Fourier transform. For general cylinders we use cut-o techniques based on the result for straight cylinders and the case without exponential weight.

Incompressible Navier–Stokes flows with time-dependent prescribed fluxes in domains with cylindrical outlets to infinity

“…note here that the operator is well defined due to the maximal regularity result for the Stokes operator in L q β,σ ( ) (see [18], Theorem 2.5, Remark 2.6) and Corollary 3.5 (2). Moreover, …”

“…Then, based on sharp estimates of nonlinear terms of Navier-Stokes equations in Besov space (Lemma 3.1-Corollary 3.5) and using crucially the maximal L p -regularity of the Stokes operator with exponential weight in , see [18], we get the existence of solutions to the problem (1.2) by Banach's fixed point theorem.…”

“…We know by [18], Theorem 2.4 and Theorem 2.5 that for any q ∈ (1, ∞) the operator −A generates a bounded and exponentially decaying analytic semigroup and maximal L p -regularity in L q β,σ ( ) provided 0 ≤ β < β 0 with some β 0 = β 0 ( ) > 0. Now consider the abstract version of (4.16):…”

Modeling of Fluid Flow in a Flexible Vessel with Elastic Walls

“…Remark 3.2. If γ is C 1,1 then it follows from [18] that the left-hand side in (28) can be replaced by…”

“…where p 0 , p 1 are constants and v * solves the problem (18). In this case the vector function F is evaluated as…”