arXiv subjects

Anatole Gaudin

Publications and source records attributed to Anatole Gaudin.

7 recordsLinked to original sources

arXiv2026

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

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}.

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arXiv2026

The Boussinesq system in 3-dimensional bounded rough domains: Well-posedness in critical spaces and long-time behavior

We study the three-dimensional Boussinesq system in bounded rough domains, including bounded Lipschitz and $\mathrm{C}^{1,α}$ domains, within a critical functional framework. We establish existence and uniqueness results that are global in time for small initial data and local in time for arbitrary initial data. Well-posedness in critical endpoint Besov spaces with third index equal to $\infty$ is obtained in domains with Hölder continuous boundaries, relying on $\mathrm{L}^2$-maximal regularity in time. We also prove well-posedness in critical Besov spaces with third index equal to $1$, using $\mathrm{L}^1$-maximal regularity. In this $\mathrm{L}^1$-in-time setting, the analysis applies to arbitrary bounded Lipschitz domains. In any case, we show that the fluid velocity stabilizes exponentially for large times and that the temperature converges to the initial averaged temperature of the fluid. The linear theory -- fitting the adapted product estimates and vice versa -- is properly established prior to the nonlinear analysis. With this fully prepared linear framework in hand, the nonlinear estimates that follow are then handled in the critical framework with a simplified treatment -- especially in the case where the fluid velocity and the temperature belong to slightly larger spaces than $\mathrm{L}^2(\mathrm{W}^{1,3})$ and $\mathrm{L}^2(\mathrm{L}^{3/2})$ respectively -- when compared with previously known similar results in smooth domains. This approach relies on a robust linear theory and sharp product estimates based on operator-theoretic methods and Besov space techniques. Finally, as part of the analysis, we establish several new results for the underlying linear operators, including refined characterizations for the domains of fractional powers of the Neumann Laplacian and of the Stokes operator in bounded Lipschitz domains.

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arXiv2025

Optimal regularity results for the Stokes--Dirichlet problem

We develop a sharp maximal regularity theory for the resolvent and evolution Stokes equations with no-slip boundary conditions, focusing on bounded domains of low regularity. Our framework covers the full scales of Besov and Sobolev spaces, $B^s_{p,q}$ and $H^{s,p}$, including endpoint cases such as $L^\infty$. Our approach also allows extending the classical $L^p$-theory for $1\leqslant p\leqslant\infty$, giving a complete picture that includes both Bessel potential spaces $H^{s,p}$ and Besov spaces $B^s_{p,q}$, $p,q\in[1,\infty]$.\\ Our first main result establishes resolvent estimates in the half-space encompassing endpoint function spaces, while the second addresses bounded domains of minimal boundary regularity. In both cases we derive resolvent bounds, prove boundedness of the $\mathbf{H}^\infty$-functional calculus for the Stokes--Dirichlet operator, and characterize precisely the domains of its fractional powers.\\ In the half space setting, we work with homogeneous Sobolev and Besov spaces following the notion due to Bahouri, Chemin and Danchin, further refined by the second author. The analysis of solenoidal function spaces provides here a complete toolkit for the study of incompressible fluid flows. As a consequence of our analysis, we obtain an explicit description for the Stokes--Dirichlet operator on $L^\infty(\mathbb R^n_+)$, which seems completely new.\\ For bounded domains, we obtain sharp results for a wide class of rough domains under minimal assumptions on boundary regularity. To this end, we rely on Sobolev multiplier theory. The assumptions coincide with those of Maz'ya--Shaposhnikova, already shown to be optimal in the case of the Laplace equation with Dirichlet boundary conditions.

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arXiv2024

Homogeneous Sobolev and Besov spaces on special Lipschitz domains and their traces

We aim to contribute to the folklore of function spaces on Lipschitz domains. We prove the boundedness of the trace operator for homogeneous Sobolev and Besov spaces on a special Lipschitz domain with sharp regularity. To achieve this, we provide appropriate definitions and properties, ensuring our construction of these spaces is suitable for non-linear partial differential equations and boundary value problems. The trace theorem holds with the sharp range $s \in (\frac{1}{p}, 1 + \frac{1}{p})$. While the case of inhomogeneous function spaces is well-known, the case of homogeneous function spaces appears to be new, even for a smooth half-space. We refine several arguments from a previous paper on function spaces on the half-space and include a treatment for the endpoint cases $p=1$ and $p=+\infty$.

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arXiv2024

On homogeneous Sobolev and Besov spaces on the whole and the half space

In this paper, we propose an elementary construction of homogeneous Sobolev spaces of fractional order on $\mathbb{R}^n$ and $\mathbb{R}^n_+$. This construction completes the construction of homogeneous Besov spaces on $\mathcal{S}'_h(\mathbb{R}^n)$ started by Bahouri, Chemin and Danchin on $\mathbb{R}^n$. We will also extend the treatment done by Danchin and Mucha on $\mathbb{R}^n_+$, and the construction of homogeneous Sobolev spaces of integer orders started by Danchin, Hieber, Mucha and Tolksdorf on $\mathbb{R}^n$ and $\mathbb{R}^n_+$. Properties of real and complex interpolation, duality, and density are discussed. Trace results are also reviewed. Our approach relies mostly on interpolation theory and yields simpler proofs of some already known results in the case of Besov spaces. The lack of completeness on the whole scale will lead to consideration of intersection spaces with decoupled estimates to circumvent this issue. As standard and simple applications, we treat the problems of Dirichlet and Neumann Laplacians in these homogeneous functions spaces.

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arXiv2023

Hodge decompositions and maximal regularities for Hodge Laplacians in homogeneous function spaces on the half-space

In this article, the Hodge decomposition for any degree of differential forms is investigated on the whole space $\mathbb{R}^n$ and the half-space $\mathbb{R}^n_+$ on different scale of function spaces namely homogeneous and inhomogeneous Besov and Sobolev space, $\dot{\mathrm{H}}^{s,p}$, $\dot{\mathrm{B}}^{s}_{p,q}$, ${\mathrm{H}}^{s,p}$ and ${\mathrm{B}}^{s}_{p,q}$, for all $p\in(1,+\infty)$ , $s\in(-1+\frac{1}{p},\frac{1}{p})$. The bounded holomorphic functional calculus, and other functional analytic properties, of Hodge Laplacians is also investigated in the half-space, and yields similar results for Hodge-Stokes and other related operators via the proven Hodge decomposition. As consequences, the homogeneous operator and interpolation theory revisited by Danchin, Hieber, Mucha and Tolksdorf is applied to homogeneous function spaces subject to boundary conditions and leads to various maximal regularity results with global-in-time estimates that could be of use in fluid dynamics. Moreover, the bond between the Hodge Laplacian and the Hodge decomposition will even enable us to state the Hodge decomposition for higher order Sobolev and Besov spaces with additional compatibility conditions, for regularity index $s\in(-1+\frac{1}{p},2+\frac{1}{p})$. In order to make sense of all those properties in desired function spaces, we also give appropriate meaning of partial traces on the boundary in the appendix.La raison d'{ê}tre of this paper lies in the fact that the chosen realization of homogeneous function spaces is suitable for non-linear and boundary value problems, but requires a careful approach to reprove results that are already morally known.

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arXiv2023

Homogeneous Sobolev global-in-time maximal regularity and related trace estimates

In this paper, we prove global-in-time $\dot{\mathrm{H}}^{α,q}$-maximal regularity for a class of injective, but not invertible, sectorial operators on a UMD Banach space X , provided $q\in(1,+\infty) , $α\in(-1+1/q,1/q)$. We also prove the corresponding trace estimate, so that the solution to the canonical abstract Cauchy problem is continuous with values in a not necessarily complete trace space.In order to put our result in perspective, we also provide a short review on L q-maximal regularity which includes some recent advances such as the revisited homogeneous operator and interpolation theory by Danchin, Hieber, Mucha and Tolksdorf. This theory will be used to build the appropriate trace space, from which we want to choose the initial data, and the solution of our abstract Cauchy problem to fall in.

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