The $\mathrm{L}^1$-Stokes Semigroup

Abstract

We study the Stokes operator with no-slip boundary conditions on the spaces $\mathrm{L}_{σ,n}(Ω)$ and ${\mathrm{L}^1(Ω,\mathbb{C}^d)}/{\nabla \mathrm{W}^{1,1}(Ω,\mathbb{C})}$, where $Ω\subset\mathbb{R}^d$ is an arbitrary bounded $\mathrm{C}^{1,α}$-domain. We show that the Stokes operator on $\mathrm{L}^1_{σ,n}(Ω)$ does not generate a $\mathrm{C}_0$-semigroup, even though the resolvent problem is uniquely solvable. In stark contrast, its realization on ${\mathrm{L}^1(Ω,\mathbb{C}^d)}/{\nabla \mathrm{W}^{1,1}(Ω,\mathbb{C})}$ generates a compact, analytic $\mathrm{C}_0$-semigroup, which leaves $\mathrm{L}^1_{σ,n}(Ω)$ invariant. The key point is that these two realizations, which are canonically identified for $1<p<\infty$ 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 $\mathrm{L}^1$-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 $\mathrm{L}^1$, the celebrated result of Abe and Giga \cite{AG:12} on the Stokes semigroup on $\mathrm{C}_{σ,0}(Ω)$ and the regularity theory refinements for the Stokes operator recently developed by Breit and the second author \cite{BG:25}.