Resolvent Estimates for the Stokes Operator in a Three-Dimensional Lipschitz Domain
Abstract
By refining the approach developed in \cite{AGH-2015, GS2026a}, we establish resolvent estimates in Lpσ(Ω) for the Stokes operator in a bounded Lipschitz domain Ω in R3 for any (3/2)−e< p≤ ∞, where e>0 depends on Ω. As a consequence, the Stokes operator generates a uniformly bounded analytic semigroup in Lpσ(Ω). The results are particularly surprising, as it is long believed that the Lp resolvent estimates in three-dimensional Lipschitz domains may fail for large p. In the Appendices we prove a theorem on interpolation between L2σ(Ω) and L∞σ(Ω). Given 1< p< ∞ and a bounded Lipschitz domain Ω in Rd, d≥ 2, we show that the resolvent estimate cannot hold in Lp(Ω; Cd) unless the Helmholtz projection is bounded on Lp(Ω; Cd). This explains the counter-example constructed in \cite{Deuring-2001}.
Related
- Quantitative Homogenization and Uniform Lipschitz Estimates for Lamé-Stokes Systems
- Lp-extrapolation via non-local decay estimates: theory and applications to the generalized Stokes operator with rough coefficients
- Weighted Lq-Lr estimates of a generalized Oseen evolution operator in exterior domains, with applications to the Navier-Stokes flow past a rigid body
- Resolvent bounds and eigenvalue estimates of generalized Schrödinger operators with complex potentials on compact manifolds
- Hahn-Type Orthogonality for the Resolvent Delta Operator ℒ=D(I−D)−1
BibTeX
@misc{shen2026resolvent,
title = {{Resolvent Estimates for the Stokes Operator in a Three-Dimensional Lipschitz Domain}},
author = {Zhongwei Shen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02552},
eprint = {2609.02552},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02552},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Resolvent Estimates for the Stokes Operator in a Three-Dimensional Lipschitz Domain
AU - Zhongwei Shen
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.02552
UR - https://arxiv.org/abs/2609.02552
AB - By refining the approach developed in \cite{AGH-2015, GS2026a}, we establish resolvent estimates in L^p_σ(Ω) for the Stokes operator in a bounded Lipschitz domain Ω in R^3 for any (3/2)−e< p≤ ∞, where e>0 depends on Ω. As a consequence, the Stokes operator generates a uniformly bounded analytic semigroup in L^p_σ(Ω). The results are particularly surprising, as it is long believed that the L^p resolvent estimates in three-dimensional Lipschitz domains may fail for large p. In the Appendices we prove a theorem on interpolation between L^2_σ(Ω) and L^∞_σ(Ω). Given 1< p< ∞ and a bounded Lipschitz domain Ω in R^d, d≥ 2, we show that the resolvent estimate cannot hold in L^p(Ω; C^d) unless the Helmholtz projection is bounded on L^p(Ω; C^d). This explains the counter-example constructed in \cite{Deuring-2001}.
ER -