arXiv subjects

Patrick Tolksdorf

Publications and source records attributed to Patrick Tolksdorf.

At least 19 recordsLinked to original sources

arXiv2025

$H^\infty$-calculus for Stokes operators on rough and on unbounded domains

In this article, we give an overview on known as well as new results on the boundedness of the $H^{\infty}$-calculus of the Stokes operator in rough as well as in unbounded (smoother) domains. We present a special case of an abstract comparison principle due to Kunstmann and Weis (\cite{KuW:Hinfty-Stokes}) that serves as the basis for all considerations. Subsequently, we show how this result can be applied to arrive at a bounded $H^{\infty}$-calculus for the Stokes operator. We sketch the proof for no slip boundary conditions in bounded Lipschitz domains which was given in~\cite{KuW:Hinfty-Stokes}. For unbounded domains this approach yields a shorter proof compared to previous arguments. Moreover, we further establish the boundedness of the $H^{\infty}$-calculus for the Stokes operator with Neumann type boundary conditions in bounded convex domains which is entirely new.

math.AP↗

arXiv2025

Strong solutions to the Keller-Segel-Navier-Stokes system in bounded Lipschitz domains

Consider the coupled Keller-Segel-Navier-Stokes or the chemotaxis-consumption-Navier-Stokes system in bounded Lipschitz domains for general coupling terms which, e.g., include buoyancy forces. It is shown that these systems admit local strong as well as global strong solutions for small data in the setting of critical Besov spaces. Moreover, non-trivial equilibria are shown to be exponentially stable. For smoother data, these solutions are shown to be globally bounded and to preserve positivity properties. The approach presented is based on optimal $\mathrm{L}^q$-regularity properties of the Neumann Laplacian and the Stokes operator in bounded Lipschitz domains.

math.AP↗

arXiv2024

On Kato's Square Root Property for the Generalized Stokes Operator

We establish the Kato square root property for the generalized Stokes operator on $\mathbb{R}^d$ with bounded measurable coefficients. More precisely, we identify the domain of the square root of $Au := - \operatorname{div}(μ\nabla u) + \nabla ϕ$, $\operatorname{div}(u) = 0$, with the space of divergence-free $\mathrm{H}^1$-vector fields and further prove the estimate $\|A^{1/2} u \|_{\mathrm{L}^2} \simeq \| \nabla u \|_{\mathrm{L}^2}$. As an application we show that $A^{1/2}$ depends holomorphically on the coefficients $μ$. Besides the boundedness and measurablility as well as an ellipticity condition on $μ$, there are no requirements on the coefficients.

math.AP↗

arXiv2022

The Stokes operator in two-dimensional bounded Lipschitz domains

We consider the Stokes resolvent problem in a two-dimensional bounded Lipschitz domain $Ω$ subject to homogeneous Dirichlet boundary conditions. We prove $\mathrm{L}^p$-resolvent estimates for $p$ satisfying the condition $\lvert 1 / p - 1 / 2 \rvert < 1 / 4 + \varepsilon$ for some $\varepsilon > 0$. We further show that the Stokes operator admits the property of maximal regularity and that its $\mathrm{H}^{\infty}$-calculus is bounded. This is then used to characterize domains of fractional powers of the Stokes operator. Finally, we give an application to the regularity theory of weak solutions to the Navier-Stokes equations in bounded planar Lipschitz domains.

math.AP↗

arXiv2022

Critical regularity issues for the compressible Navier--Stokes system in bounded domains

We are concerned with the barotropic compressible Navier-Stokes system in a bounded domain of $\mathbb{R}^d$ (with $d\geq2$). In a critical regularity setting, we establish local well-posedness for large data with no vacuum and global well-posedness for small perturbations of a stable constant equilibrium state.Our results rely on new maximal regularity estimates - of independent interest - for the semigroup of the Lam\{é} operator, and of the linearized compressible Navier-Stokes equations.

math.AP↗

arXiv2021

Free Boundary Problems via Da Prato-Grisvard Theory

An $\mathrm{L}_1$-maximal regularity theory for parabolic evolution equations inspired by the pioneering work of Da Prato and Grisvard is developed. Besides of its own interest, the approach yields a framework allowing global-in-time control of the change of Eulerian to Lagrangian coordinates in various problems related to fluid mechanics. This property is of course decisive for free boundary problems. This concept is illustrated by the analysis of the free boundary value problem describing the motion of viscous, incompressible Newtonian fluids without surface tension and, secondly, the motion of compressible pressureless gases. For this purpose, an endpoint maximal $\mathrm{L}_1$-regularity approach to the Stokes and Lamé systems is developed. It is applied then to establish global, strong well-posedness results for the free boundary problems described above in the case where the initial domain coincides with the half-space, and the initial velocity is small with respect to a suitable scaling invariant norm.

math.AP↗

arXiv2021

The Kato Square Root Problem follows from an Extrapolation Property of the Laplacian

On a domain $Ω\subseteq \mathbb{R}^d$ we consider second order elliptic systems in divergence form with bounded complex coefficients, realized via a sesquilinear form with domain $V \subseteq H^1(Ω)$. Under very mild assumptions on $Ω$ and $V$ we show that the Kato Square Root Problem for such systems can be reduced to a regularity result for the fractional powers of the negative Laplacian in the same geometric setting. This extends an earlier result of McIntosh to non-smooth coefficients.

math.FA↗

arXiv2021

The Kato Square Root Problem for Mixed Boundary Conditions

We consider the negative Laplacian subject to mixed boundary conditions on a bounded domain. We prove under very general geometric assumptions that slightly above the critical exponent $\frac{1}{2}$ its fractional power domains still coincide with suitable Sobolev spaces of optimal regularity. In combination with a reduction theorem recently obtained by the authors, this solves the Kato Square Root Problem for elliptic second order operators and systems in divergence form under the same geometric assumptions.

math.FA↗

arXiv2021

On off-diagonal decay properties of the generalized Stokes semigroup with bounded measurable coefficients

We investigate off-diagonal decay properties of the generalized Stokes semigroup with bounded measurable coefficients on $\mathrm{L}^2_σ (\mathbb{R}^d)$. Such estimates are well-known for elliptic equations in the form of pointwise heat kernel bounds and for elliptic systems in the form of integrated off-diagonal estimates. On our way to unveil this off-diagonal behavior we prove resolvent estimates in Morrey spaces $\mathrm{L}^{2 , ν} (\mathbb{R}^d)$ with $0 \leq ν< 2$.

math.AP↗

arXiv2021

Extendability of functions with partially vanishing trace

Let $Ω\subseteq \mathbb{R}^d$ be open and $D\subseteq \partialΩ$ be a closed part of its boundary. Under very mild assumptions on $Ω$, we construct a bounded Sobolev extension operator for the Sobolev space $\mathrm{W}^{k , p}_D (Ω)$, $1 \leq p < \infty$, which consists of all functions in $\mathrm{W}^{k , p} (Ω)$ that vanish in a suitable sense on $D$. In contrast to earlier work, this construction is global and \emph{not} using a localization argument, which allows to work with a boundary regularity that is sharp at the interface dividing $D$ and $\partial Ω\setminus D$. Moreover, we provide homogeneous and local estimates for the extension operator. Also, we treat the case of Lipschitz function spaces with a vanishing trace condition on $D$.

math.CA↗

arXiv2020

A non-local approach to the generalized Stokes operator with bounded measurable coefficients

We establish functional analytic properties of the Stokes operator with bounded measurable coefficients on $L^p_σ (\mathbb{R}^d)$, $d \geq 2$, for $\lvert 1 / p - 1 / 2 \rvert < 1 / d$. These include optimal resolvent bounds and the property of maximal $L^q$-regularity. We further give regularity estimates on the gradient of the solution to the Stokes resolvent problem with bounded measurable coefficients. As a key to these results we establish the validity of a non-local Caccioppoli inequality to solutions of the Stokes resolvent problem.

math.AP↗

arXiv2020

Lorentz spaces in action on pressureless systems arising from models of collective behavior

We are concerned with global-in-time existence and uniqueness results for models of pressureless gases that come up in the description of phenomena in astrophysics or collective behavior. The initial data are rough: in particular, the density is only bounded. Our results are based on interpolation and parabolic maximal regularity, where Lorentz spaces play a key role. We establish a novel maximal regularity estimate for parabolic systems in $L_{q,r}(0,T;L_p(Ω))$ spaces.

math.AP↗

arXiv2020

$\mathrm{L}^p$-extrapolation of non-local operators: Maximal regularity of elliptic integrodifferential operators with measurable coefficients

The aim of this article is to deepen the understanding of the derivation of $\mathrm{L}^p$-estimates of non-local operators. We review the $\mathrm{L}^p$-extrapolation theorem of Shen which builds on a real variable argument of Caffarelli and Peral and adapt this theorem to account for non-local weak reverse Hölder estimates. These non-local weak reverse Hölder estimates appear for example in the investigation of non-local elliptic integrodifferential operators. This originates from the fact that here only a non-local Caccioppoli inequality is valid, see Kuusi, Mingione, and Sire. As an application, we prove resolvent estimates and maximal regularity properties in $\mathrm{L}^p$-spaces of non-local elliptic integrodifferential operators.

math.AP↗

arXiv2019

The Stokes resolvent problem: Optimal pressure estimates and remarks on resolvent estimates in convex domains

The Stokes resolvent problem $λu - Δu + \nabla ϕ= f$ with $\mathrm{div}(u) = 0$ subject to homogeneous Dirichlet or homogeneous Neumann-type boundary conditions is investigated. In the first part of the paper we show that for Neumann-type boundary conditions the operator norm of $\mathrm{L}^2_σ (Ω) \ni f \mapsto π\in \mathrm{L}^2 (Ω)$ decays like $\lvert λ\rvert^{- 1 / 2}$ which agrees exactly with the scaling of the equation. In comparison to that, we show that the operator norm of this mapping under Dirichlet boundary conditions decays like $\lvert λ\rvert^{- α}$ for $0 \leq α< 1 / 4$ and we show that this decay rate cannot be improved to any exponent $α> 1 / 4$, thereby, violating the natural scaling of the equation. In the second part of this article, we investigate the Stokes resolvent problem subject to homogeneous Neumann-type boundary conditions if the underlying domain $Ω$ is convex. We establish optimal resolvent estimates and gradient estimates in $\mathrm{L}^p (Ω; \mathbb{C}^d)$ for $2d / (d + 2) < p < 2d / (d - 2)$ (with $1 < p < \infty$ if $d = 2$). This interval is larger than the known interval for resolvent estimates subject to Dirichlet boundary conditions on general Lipschitz domains and is to the best knowledge of the author the first result that provides $\mathrm{L}^p$-estimates for the Stokes resolvent subject to Neumann-type boundary conditions on general convex domains.

math.AP↗

arXiv2019

The Navier--Stokes equations in exterior Lipschitz domains: $\mathrm{L}^p$-theory

We show that the Stokes operator defined on $\mathrm{L}^p_σ (Ω)$ for an exterior Lipschitz domain $Ω\subset \mathbb{R}^n$ $(n \geq 3)$ admits maximal regularity provided that $p$ satisfies $| 1/p - 1/2| < 1/(2n) + \varepsilon$ for some $\varepsilon > 0$. In particular, we prove that the negative of the Stokes operator generates a bounded analytic semigroup on $\mathrm{L}^p_σ(Ω)$ for such $p$. In addition, $\mathrm{L}^p$-$\mathrm{L}^q$-mapping properties of the Stokes semigroup and its gradient with optimal decay estimates are obtained. This enables us to prove the existence of mild solutions to the Navier--Stokes equations in the critical space $\mathrm{L}^{\infty} (0 , T ; \mathrm{L}^3_σ (Ω))$ (locally in time and globally in time for small initial data).

math.AP↗

arXiv2018

On the $\mathrm{L}^p$-theory of the Navier--Stokes equations on three-dimensional bounded Lipschitz domains

On a bounded Lipschitz domain $Ω\subset \mathbb{R}^d$, $d \geq 3$, we continue the study of Shen and of Kunstmann and Weis of the Stokes operator on $\mathrm{L}^p_σ (Ω)$. We employ their results in order to determine the domain of the square root of the Stokes operator as the space $\mathrm{W}^{1 , p}_{0 , σ} (Ω)$ for $\lvert \frac{1}{p} - \frac{1}{2} \rvert < \frac{1}{d} + \varepsilon$ and some $\varepsilon > 0$. This characterization provides gradient estimates as well as $\mathrm{L}^p$-$\mathrm{L}^q$-mapping properties of the corresponding semigroup. In the three-dimensional case this provides a means to show the existence of solutions to the Navier--Stokes equations in the critical space $\mathrm{L}^{\infty} (0 , \infty ; \mathrm{L}^3_σ (Ω))$ whenever the initial velocity is small in the $\mathrm{L}^3$-norm. Finally, we present a different approach to the $\mathrm{L}^p$-theory of the Navier--Stokes equations by employing the maximal regularity proven by Kunstmann and Weis.

math.AP↗

arXiv2017

Nematic Liquid Crystals in Lipschitz domains

We consider the simplified Ericksen-Leslie model in three dimensional bounded Lipschitz domains. Applying a semilinear approach, we prove local and global well-posedness (assuming a smallness condition on the initial data) in critical spaces for initial data in $L^3_σ$ for the fluid and $W^{1,3}$ for the director field. The analysis of such models, so far, has been restricted to domains with smooth boundaries.

math.AP↗

arXiv2017

Strong Time Periodic Solutions to the Bidomain Equations with FitzHugh-Nagumo Type Nonlinearities

Consider the bidomain equations subject to ionic transport described by the models of FitzHugh-Nagumo, Aliev-Panfilov, or Rogers-McCulloch. It is proved that this set of equations admits a unique, strong T-periodic solution provided it is innervated by T-periodic intra- and extracellular currents. The approach relies on a new periodic version of the classical Da Prato-Grisvard theorem on maximal L^p-regularity in real interpolation spaces.

math.AP↗