Traces of vector‐valued Sobolev spaces

Scharf, Benjamin; Schmeißer, Hans-J ̈ urgen; Sickel, Winfried

doi:10.1002/mana.201100011

Cited by 17 publications

(21 citation statements)

References 43 publications

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“…For

s \in (0, 1)

and a Banach space U ,

F_{p, q}^{s} false(0, T, U false)

stands for U valued Lizorkin–Triebel space. For precise definition of such spaces we refer to . If

T < \infty

, this spaces can be characterised as follows (see )

\begin{matrix} true \\ right F_{p, q}^{s} (0, T; U) = true{f \in L^{p} (0, T, U) ∣ {false| f false|}_{F_{p, q}^{s} (0, T; U)} < \infty true}, \end{matrix}

where

\begin{matrix} true \\ right {false| f false|}_{F_{p, q}^{s} (0, T; U)} = {(\int_{0}^{T} {(\int_{0}^{T - t} h^{- 1 - s q} {false∥ f (t + h) - f (t) false∥}_{U}^{q} normald h)}^{p / q} normald t)}^{1 / p} . \end{matrix}

These spaces endowed with the natural norm

\begin{matrix} true \\ right {false∥ f false∥}_{F_{p, q}^{s} (0, T; U)} = {false∥ f false∥}_{L^{p} (0, T; U)} + {false| f false|}_{<...>} \end{matrix}

…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

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Mathematical analysis of the motion of a rigid body in a compressible Navier–Stokes–Fourier fluid

Haak

Maity

Takahashi

et al. 2019

Mathematische Nachrichten

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We study an initial and boundary value problem modelling the motion of a rigid body in a heat conducting gas. The solid is supposed to be a perfect thermal insulator. The gas is described by the compressible Navier–Stokes–Fourier equations, whereas the motion of the solid is governed by Newton's laws. The main results assert the existence of strong solutions, in an Lp‐Lq setting, both locally in time and globally in time for small data. The proof is essentially using the maximal regularity property of associated linear systems. This property is checked by proving the scriptR‐sectoriality of the corresponding operators, which in turn is obtained by a perturbation method.

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“…For

s \in (0, 1)

and a Banach space U ,

F_{p, q}^{s} false(0, T, U false)

stands for U valued Lizorkin–Triebel space. For precise definition of such spaces we refer to . If

T < \infty

, this spaces can be characterised as follows (see )

\begin{matrix} true \\ right F_{p, q}^{s} (0, T; U) = true{f \in L^{p} (0, T, U) ∣ {false| f false|}_{F_{p, q}^{s} (0, T; U)} < \infty true}, \end{matrix}

where

\begin{matrix} true \\ right {false| f false|}_{F_{p, q}^{s} (0, T; U)} = {(\int_{0}^{T} {(\int_{0}^{T - t} h^{- 1 - s q} {false∥ f (t + h) - f (t) false∥}_{U}^{q} normald h)}^{p / q} normald t)}^{1 / p} . \end{matrix}

These spaces endowed with the natural norm

\begin{matrix} true \\ right {false∥ f false∥}_{F_{p, q}^{s} (0, T; U)} = {false∥ f false∥}_{L^{p} (0, T; U)} + {false| f false|}_{<...>} \end{matrix}

…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…For ∈ (0, 1) and a Banach space , , (0, , ) stands for valued Lizorkin-Triebel space. For precise definition of such spaces we refer to [22,28]. If < ∞, this spaces can be characterised as follows (see [31])…”

Section: Notationmentioning

confidence: 99%

Mathematical analysis of the motion of a rigid body in a compressible Navier–Stokes–Fourier fluid

Haak

Maity

Takahashi

et al. 2019

Mathematische Nachrichten

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“…The above Proposition 3.6 can be found in [SSS12, Theorem 3.7]. We refer the interested reader to [Sch10] and [SSS12] for a very detailed proof of the theorem.…”

Section: Three Equivalent Norms On Vector-valued Besov Spacesmentioning

confidence: 99%

“…As an initial step we present an expansion of continuous functions with Lipschitz atoms, relaxing the smoothness assumption on the atoms considered in the work of Scharf, Schmeißer and Sickel [SSS12]. For the reader's convenience we use the same notation and definitions as in [SSS12]. The theory and results are first developed for Besov spaces with domain R. The restriction to [0, 1] will be discussed below in Subsection 3.2.…”

Section: Three Equivalent Norms On Vector-valued Besov Spacesmentioning

confidence: 99%

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Characterization of nonlinear Besov spaces

Liu

Prömel

Teichmann

2019

Trans. Amer. Math. Soc.

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The canonical generalizations of two classical norms on Besov spaces are shown to be equivalent even in the case of non-linear Besov spaces, that is, function spaces consisting of functions taking values in a metric space and equipped with some Besov-type topology. The proofs are based on atomic decomposition techniques and metric embeddings. Additionally, we provide embedding results showing how non-linear Besov spaces embed into non-linear p-variation spaces and vice versa. We emphasize that we neither assume the UMD property of the involved spaces nor their separability.

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A Calculus of Abstract Edge Pseudodifferential Operators of Type $\rho,\delta$

Krainer

2015

Elliptic and Parabolic Equations

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Traces of vector‐valued Sobolev spaces

Abstract: We characterize the traces of vector-valued Besov and Lizorkin-Triebel spaces. Therefrom we derive the corresponding assertions for the vector-valued Sobolev spaces W m p (R n , E) . Here we do not assume the UMD property for the Banach space E.

Cited by 17 publications

References 43 publications

Mathematical analysis of the motion of a rigid body in a compressible Navier–Stokes–Fourier fluid

Mathematical analysis of the motion of a rigid body in a compressible Navier–Stokes–Fourier fluid

Characterization of nonlinear Besov spaces

A Calculus of Abstract Edge Pseudodifferential Operators of Type $\rho,\delta$

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