Maximal Regularity in Exponentially Weighted Lebesgue Spaces of the Stokes Operator in Unbounded Cylinders

TL;DR

This paper establishes resolvent estimates and maximal regularity for the Stokes operator in exponentially weighted Lebesgue spaces within unbounded cylinders, demonstrating decay properties and analytic semigroup generation.

Contribution

It extends maximal regularity results for the Stokes operator to unbounded cylindrical domains with exponential weights, including complex geometries with multiple exits to infinity.

Findings

Stokes operator generates an exponentially decaying analytic semigroup in weighted spaces.

Maximal regularity holds for the Stokes operator in these unbounded cylindrical domains.

The approach combines Fourier multiplier theorems, Schauder decompositions, and cut-off techniques.

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 $\R^n,n\geq 3$. 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 infinity 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 ${\mathcal R}$-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-off techniques based on the result for straight cylinders and the case without exponential weight.