On the $\mathcal{R}$-Sectoriality and the Initial Boundary Value Problem for the Viscous Compressible Fluid Flow

Enomoto, Yuko; Shibata, Yoshihiro

doi:10.1619/fesi.56.441

Cited by 69 publications

(72 citation statements)

References 43 publications

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“…Proof We first define

D false(A_{u} false) = {u \in W^{2, q} {({normalΩ}_{F} false(0 false))}^{3} ∣ u = 0 on \partial {normalΩ}_{F} false(0 false)}, A_{u} = \frac{μ}{\bar{ρ}} Δ + \frac{α + μ}{\bar{ρ}} \nabla div,

and

D false(A_{ϑ} false) = \{ϑ \in W^{2, q} ({normalΩ}_{F} false(0 false)) ∣ \frac{\partial ϑ}{\partial n} = 0 on \partial Ω_{F} (0)\}, A_{ϑ} = \frac{κ}{\bar{ρ} c_{v}} Δ .

From [, Theorem 8.2], there exists

γ_{ϑ} \in R

such that

A_{ϑ} - γ_{ϑ}

scriptR

‐sectorial of angle

< π / 2

. Using [, Theorem 2.5], we also obtain the existence of

γ_{u} \in R

such that

A_{u} - γ_{u}

scriptR

‐sectorial of angle

< π / 2

. In particular there exist γ and

…”

Section: Global In Time Existencementioning

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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“…Proof We first define

D false(A_{u} false) = {u \in W^{2, q} {({normalΩ}_{F} false(0 false))}^{3} ∣ u = 0 on \partial {normalΩ}_{F} false(0 false)}, A_{u} = \frac{μ}{\bar{ρ}} Δ + \frac{α + μ}{\bar{ρ}} \nabla div,

and

D false(A_{ϑ} false) = \{ϑ \in W^{2, q} ({normalΩ}_{F} false(0 false)) ∣ \frac{\partial ϑ}{\partial n} = 0 on \partial Ω_{F} (0)\}, A_{ϑ} = \frac{κ}{\bar{ρ} c_{v}} Δ .

From [, Theorem 8.2], there exists

γ_{ϑ} \in R

such that

A_{ϑ} - γ_{ϑ}

scriptR

‐sectorial of angle

< π / 2

. Using [, Theorem 2.5], we also obtain the existence of

γ_{u} \in R

such that

A_{u} - γ_{u}

scriptR

‐sectorial of angle

< π / 2

. In particular there exist γ and

…”

Section: Global In Time Existencementioning

confidence: 99%

“…Hieber and Murata proved local in time existence and uniqueness in a

L^{p}

‐

L^{q}

setting. Let us also mention that an important influence on the methods in this work comes from several recent advances on the

L^{p}

‐

L^{q}

theory of viscous compressible fluids (without structure), see Enomoto and Shibata and Murata and Shibata .…”

Section: Introductionmentioning

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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“…One main tool for the proof of Theorem is the following lemma due to Denk and Schnaubelt , Lemma 2.1 and Enomoto and Shibata ,. Theorem 3.3…”

Section: Analysis In the Whole Spacementioning

confidence: 99%

“…To prove Theorem , we use several properties of uniform C 4 ‐domain, which are stated in the following proposition (cf Enomoto and Shibata , Proposition 6.1, ).…”

Section: Proof Of the Main Theoremsmentioning

confidence: 99%

The existence of ‐bounded solution operators of the thermoelastic plate equation with Dirichlet boundary conditions

Suma'inna

2017

Math Methods in App Sciences

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We consider the linearized thermoelastic plate equation with the Dirichlet boundary condition in a general domain Ω, given by utt+Δ2u+normalΔθ=1ptf1,in3.0235ptnormalΩ×false(0,∞false),θt−normalΔθ−normalΔut=1ptf2,in3.0235ptnormalΩ×false(0,∞false), with the initial condition u|(t=0)=u0, ut|(t=0)=u1, and θ|(t=0)=θ0 in Ω and the boundary condition u=∂νu=θ=0 on Γ, where u=u(x,t) denotes a vertical displacement at time t at the point x=(x1,⋯,xn)∈Ω, while θ=θ(x,t) describes the temperature. This work extends the result obtained by Naito and Shibata that studied the problem in the half‐space case. We prove the existence of scriptR‐bounded solution operators of the corresponding resolvent problem. Then, the generation of C0 analytic semigroup and the maximal Lp‐Lq‐regularity of time‐dependent problem are derived.

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“…The numbers A i appearing in (3)(4)(5)(6)(7)(8)(9) are defined by (3)(4)(5)(6)(7)(8)(9)(10) and P 1 , P 2 , P 3 are constants which will determined later by the boundary conditions. Inserting (3)(4)(5)(6)(7)(8)(9) into the boundary conditions (3-8), we get a linear equation system for the coefficients P i :…”

Section: Definition 12 a Domain ω Is Called A Uniform Cmentioning

confidence: 99%

Maximal regularity for the thermoelastic plate equations with free boundary conditions

Denk

Shibata

2016

J. Evol. Equ.

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We consider the linear thermoelastic plate equations with free boundary conditions in the Lp in time and Lq in space setting. We obtain unique solvability with optimal regularity for the inhomogeneous problem in a uniform C 4 -domain, which includes the cases of a bounded domain and of an exterior domain with C 4 -boundary. Moreover, we prove uniform a priori-estimates for the solution. The proof is based on the existence of R-bounded solution operators of the corresponding generalized resolvent problem which is shown with the help of an operator-valued Fourier multiplier theorem due to Weis.

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On the $\mathcal{R}$-Sectoriality and the Initial Boundary Value Problem for the Viscous Compressible Fluid Flow

Cited by 69 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

The existence of ‐bounded solution operators of the thermoelastic plate equation with Dirichlet boundary conditions

Maximal regularity for the thermoelastic plate equations with free boundary conditions

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