arXiv subjects

Reinhard Farwig

Publications and source records attributed to Reinhard Farwig.

6 recordsLinked to original sources

arXiv2026

Time periodic problem of the Navier-Stokes equations in an exterior domain with periodically moving boundary

In this paper we consider the Navier-Stokes equations in exterior domains of $\mathbb{R}^n$, $n\geq 3$, with a periodically in time moving boundary $\partialΩ(t)$ and external force $f(t)$. For this case we prove the existence of a locally unique mild time periodic solution in weighted function spaces with radially symmetric Muckenhoupt weights. The solutions split into a stationary part controlled by potential theoretic estimates and a purely oscillatory part constructed as mild solution via analytic semigroup theory. To deal with perturbation terms of even second order - coming from a coordinate transform and the moving boundary - in weighted, homogeneous Sobolev spaces a maximal $L^1$ type regularity estimate will be used in weighted Lorentz spaces. To control the convective term an $\mathcal H^\infty$-calculus in weighted spaces of the Stokes operator, its $BIP$ property and embedding estimates of fractional powers are exploited, see a recent paper by the authors: The Stokes operator on exterior domains in homogeneous weighted function spaces: From weak theory to $\mathscr H^\infty$-calculus to fractional domains (2025).

math.AP↗

arXiv2026

The Stokes Operator on Exterior Domains in Homogeneous Weighted Function Spaces:From Weak Theory to $\mathscr H^\infty$-calculus to Fractional Domains

We consider the Stokes operator $A$ on smooth exterior domains $Ω$ of $\mathbb{R}^n$ in homogeneous Sobolev spaces $\widehat H^{κ,q}_w(Ω)$ with radially symmetric Muckenhoupt weights $w\in \mathscr A_q$. A fundamental property is the existence of a bounded $\mathscr H^\infty$-calculus of the Stokes operator on weighted nonhomogeneous and homogeneous $L^q$ Sobolev spaces. This property implies the existence of uniformly bounded purely imaginary powers $A^{it}$, $t\in\mathbb{R}$, and the characterization of domains of fractional powers $A^θ$ equipped with nonhomogeneous ($\| u\|_{L^q_w} + \|A^θu\|_{L^q_w}$) as well as homogeneous norm ($\|A^θu\|_{L^q_w}$) as complex interpolation spaces. The final aim is the identification with homogeneous spaces $\widehat{\mathcal D}((-Δ_{q,w})^θ) = [L^q_{w},\widehat{\mathcal D}(-Δ_{q,w})]_θ$ intersected by a space of solenoidal vector fields. Moreover, we obtain weighted variational inequalities for weak solutions of the Stokes equations, weighted $L^q$-$L^r$ decay estimates of the Stokes semigroup and $L^p$-maximal regularity on $L^q_{σ,w}(Ω)$.

math.AP↗

arXiv2026

The time periodic problem for the Navier-Stokes equations in exterior domains in weighted spaces

The paper considers the time periodic problem of the Navier-Stokes system in an exterior domain under time periodic external forces. Existence of periodic mild solutions is obtained in the critical scale invariant space $C(\mathbb{R};L^n)$ $(n \geq 4)$ if the external force is small without exploiting any divergence form as in the study of Okabe and Tsutsui (2017) for the whole space case in Lorentz spaces. Previous studies mainly rely on either potential theoretical estimates or time-space integral estimates in Lorentz spaces introduced by Yamazaki (Math. Ann.(2000)). To the best of our knowledge, there are no results using Muckenhoupt weights in $L^q$ class for $1< q <\infty$ to construct time periodic solutions of the Navier-Stokes equations in the exterior domain case. In this article, a new method based on radially symmetric Muckenhoupt weights in space is used. To apply these weights, we reconsider weighted $L^p$-$L^q$ decay estimates for the Stokes semigroup. This important result was announced by Kobayashi and Kubo (2012-2015) about ten years ago with a sketch of the proof by Kubo. In this paper, we give a rigorous proof of the result and, as an important application, solve the time periodic problem for the Navier-Stokes equations on an exterior domain.

math.AP↗

arXiv2022

Sobolev spaces on arbitrary domains and semigroups generated by fractional Laplacian

We describe a procedure to introduce Sobolev spaces and the semigroup generated by the fractional Dirichlet Laplacian on an arbitrary domain of $\R^d$. In particular, the well-definedness of the spaces of both non-homogeneous and homogeneous type together with their duality properties, embeddings, and Gagliardo-Nirenberg inequalities will be discussed. We also show the continuity and the smoothing property of the semigroup.

math.FA↗

arXiv2020

Maximal $L^1$-regularity of generators for bounded analytic semigroups in Banach spaces

In this paper, we prove that the generator of any bounded analytic semigroup in $(θ,1)$-type real interpolation of its domain and underlying Banach space has maximal $L^1$-regularity, using a duality argument combined with the result of maximal continuous regularity. As an application, we consider maximal $L^1$-regularity of the Dirichlet-Laplacian and the Stokes operator in inhomogeneous $B^s_{q,1}$-type Besov spaces on domains of $\mathbb R^n$, $n\geq 2$.

math.FA↗

arXiv2014

Maximal Regularity in Exponentially Weighted Lebesgue Spaces of the Stokes Operator in Unbounded Cylinders

We study resolvent estimates and maximal regularity of the Stokes operator in $L^q$-spaces with exponential weights in the axial directions of unbounded cylinders of $\R^n,n\geq 3$. For a straight cylinder we use exponential weights in the axial direction and Muckenhoupt weights in the cross-section. Next, for cylinders with several exits to infinity we prove that the Stokes operator in $L^q$-spaces with exponential weights generates an exponentially decaying analytic semigroup and has maximal regularity. The proof for straight cylinders uses an operator-valued Fourier multiplier theorem and unconditional Schauder decompositions based on the ${\mathcal R}$-boundedness of the family of solution operators for a system in the cross-section of the cylinder parametrized by the phase variable of the one-dimensional partial Fourier transform. For general cylinders we use cut-off techniques based on the result for straight cylinders and the case without exponential weight.

math-ph↗