Resolvent Estimates and Maximal Regularity in 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 weighted Lebesgue spaces within unbounded cylinders, using advanced Fourier analysis and operator theory techniques.

Contribution

It extends resolvent and regularity results to unbounded cylindrical domains with exponential weights, including complex geometries with multiple exits to infinity.

Findings

Resolves the Stokes operator in weighted spaces for straight cylinders.

Proves exponential decay and maximal regularity for general cylinders.

Uses Fourier multiplier and ${ m R}$-boundedness techniques.

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 ${\mathbb R}^n,n\geq 3$. 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 ${\mathcal R}$-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 result for the case without exponential weight.