The L1-Stokes Semigroup
Abstract
We study the Stokes operator with no-slip boundary conditions on the spaces Lσ,n(Ω) and L1(Ω,ℂd)/∇ W1,1(Ω,ℂ), where Ω⊂ℝd is an arbitrary bounded C1,α-domain. We show that the Stokes operator on L1σ,n(Ω) does not generate a C0-semigroup, even though the resolvent problem is uniquely solvable. In stark contrast, its realization on L1(Ω,ℂd)/∇ W1,1(Ω,ℂ) generates a compact, analytic C0-semigroup, which leaves L1σ,n(Ω) invariant. The key point is that these two realizations, which are canonically identified for 1<p<∞ through the Helmholtz decomposition, cease to be equivalent at the endpoint p=1. This leads to genuinely different functional analytic properties. Our result provides the first positive generation theorem for the Stokes operator with no-slip boundary conditions in a pure L1-setting on a bounded domain and settles a problem that had remained open for nearly fifty years; see, e.g., \cite{Koz:01,DHP:01}. In this sense, it completes the theory of the Stokes semigroup across the full scale of solenoidal Lebesgue spaces on (smooth) bounded domains. As an intermediate step, some results on the space of Radon measures are obtained. The proof combines the sun-dual construction with a precise analysis of the failure of the Helmholtz decomposition in L1, the celebrated result of Abe and Giga \cite{AG:12} on the Stokes semigroup on Cσ,0(Ω) and the regularity theory refinements for the Stokes operator recently developed by Breit and the second author \cite{BG:25}.
Related
BibTeX
@misc{binz2026l1stokes,
title = {{The $\mathrmL^1$-Stokes Semigroup}},
author = {Tim Binz and Anatole Gaudin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08715},
eprint = {2609.08715},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08715},
}
RIS
TY - UNPB
TI - The L^1-Stokes Semigroup
AU - Tim Binz
AU - Anatole Gaudin
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08715
UR - https://arxiv.org/abs/2609.08715
AB - We study the Stokes operator with no-slip boundary conditions on the spaces L_σ,n(Ω) and L^1(Ω,ℂ^d)/∇ W^1,1(Ω,ℂ), where Ω⊂ℝ^d is an arbitrary bounded C^1,α-domain. We show that the Stokes operator on L^1_σ,n(Ω) does not generate a C_0-semigroup, even though the resolvent problem is uniquely solvable. In stark contrast, its realization on L^1(Ω,ℂ^d)/∇ W^1,1(Ω,ℂ) generates a compact, analytic C_0-semigroup, which leaves L^1_σ,n(Ω) invariant. The key point is that these two realizations, which are canonically identified for 1<p<∞ through the Helmholtz decomposition, cease to be equivalent at the endpoint p=1. This leads to genuinely different functional analytic properties. Our result provides the first positive generation theorem for the Stokes operator with no-slip boundary conditions in a pure L^1-setting on a bounded domain and settles a problem that had remained open for nearly fifty years; see, e.g., \cite{Koz:01,DHP:01}. In this sense, it completes the theory of the Stokes semigroup across the full scale of solenoidal Lebesgue spaces on (smooth) bounded domains. As an intermediate step, some results on the space of Radon measures are obtained. The proof combines the sun-dual construction with a precise analysis of the failure of the Helmholtz decomposition in L^1, the celebrated result of Abe and Giga \cite{AG:12} on the Stokes semigroup on C_σ,0(Ω) and the regularity theory refinements for the Stokes operator recently developed by Breit and the second author \cite{BG:25}.
ER -