Maximal regularity for the thermoelastic plate equations with free boundary conditions

Denk, Robert; Shibata, Yoshihiro

doi:10.1007/s00028-016-0367-x

Cited by 15 publications

(33 citation statements)

References 22 publications

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“…To prove the

scriptR

‐boundedness of operators

T_{j} false(λ false)

defined above, we use the following lemma that was essentially proved in Shibata and Shimizu , Lemma 5.6 (cf also Denk and Shibata , Lemma 3.6 ).…”

Section: Analysis In the Half‐spacementioning

confidence: 99%

“…Δu − (1 − )Δ ′ u + = g 1 on (0, ∞) × Γ, (Δu − (1 − )Δ ′ u + ) = g 2 on (0, ∞) × Γ, = g 2 on (0, ∞) × Γ was studied by Denk and Shibata. 18 In Denk and Shibata, 18 some localization technique was used to treat the free boundary conditions, because an abstract approach similar to that in Lasiecka and Wilke 17 was not available. The object of this paper is to prove the maximal L p − L q -regularity as well as the generation of C 0 analytic semigroup in a uniform C 4 -domain Ω.…”

Section: Previous Research and Approachesmentioning

confidence: 99%

“…Our analysis is based on the results due to Naito and Shibata 13 that studied the generation of analytic semigroup only in the half-space case, and we use similar argumentation concerning the localization procedure to those in Denk and Shibata. 18 In our approach, we set v = u t and rewrite Equations 1 and 3 as first-order system:…”

Section: Previous Research and Approachesmentioning

confidence: 99%

“…Therefore, the solvability of with can be proved by using an abstract operator‐theoretic idea. Recently, in the case of general domain, the maximal L p ‐ L q ‐regularity of Equation with free boundary conditions:

\begin{matrix} normalΔ u - false(1 - β false) Δ^{'} u + θ & = g_{1} on false(0, \infty false) \times normalΓ, \\ \partial_{ν} false(normalΔ u - false(1 - β false) Δ^{'} u + θ false) & = g_{2} on false(0, \infty false) \times normalΓ, \\ \partial_{ν} θ & = g_{2} on false(0, \infty false) \times normalΓ \end{matrix}

was studied by Denk and Shibata . In Denk and Shibata, some localization technique was used to treat the free boundary conditions, because an abstract approach similar to that in Lasiecka and Wilke was not available.…”

Section: Introductionmentioning

confidence: 99%

“…Namely, our argument depends on the inequality:

\sum_{j = 1}^{\infty} {‖ φ_{j} f ‖}_{L_{q} (Ω)}^{q} \leq C {‖ f ‖}_{L_{q} (Ω)}^{q}

for any f ∈ L q (Ω) (cf Section 5 below). Our analysis is based on the results due to Naito and Shibata that studied the generation of analytic semigroup only in the half‐space case, and we use similar argumentation concerning the localization procedure to those in Denk and Shibata . In our approach, we set v = u t and rewrite Equations and as first‐order system:

U_{t} - A false(D false) U = F in normalΩ \times false(0, T false), B false(D false) U |_{Γ} = 0 U |_{t = 0} = U_{0}

with

U = ((), \begin{matrix} array u \\ array v \\ array θ \end{matrix}), U_{0} = ((), \begin{matrix} array u_{0} \\ array u_{1} \\ array θ_{0} \end{matrix}), A false(D false) = ((), \begin{matrix} array 0 & array 1 & array 0 \\ array - {normalΔ}^{2} & array 0 & array - Δ \\ array 0 & array Δ & array Δ \end{matrix}), B false(D false) = ((), \begin{matrix} array 1 & array 0 & array 0 \\ array \partial_{ν} & array 0 & array 0 \\ array 0 & array 0 & array 1 \end{matrix}) .

Using the Laplace transform to Equation , we obtain the corresponding resolvent problem …”

Section: Introductionmentioning

confidence: 99%

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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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“…To prove the

scriptR

‐boundedness of operators

T_{j} false(λ false)

defined above, we use the following lemma that was essentially proved in Shibata and Shimizu , Lemma 5.6 (cf also Denk and Shibata , Lemma 3.6 ).…”

Section: Analysis In the Half‐spacementioning

confidence: 99%

Section: Previous Research and Approachesmentioning

confidence: 99%

Section: Previous Research and Approachesmentioning

confidence: 99%

\begin{matrix} normalΔ u - false(1 - β false) Δ^{'} u + θ & = g_{1} on false(0, \infty false) \times normalΓ, \\ \partial_{ν} false(normalΔ u - false(1 - β false) Δ^{'} u + θ false) & = g_{2} on false(0, \infty false) \times normalΓ, \\ \partial_{ν} θ & = g_{2} on false(0, \infty false) \times normalΓ \end{matrix}

Section: Introductionmentioning

confidence: 99%

“…Namely, our argument depends on the inequality:

\sum_{j = 1}^{\infty} {‖ φ_{j} f ‖}_{L_{q} (Ω)}^{q} \leq C {‖ f ‖}_{L_{q} (Ω)}^{q}

U_{t} - A false(D false) U = F in normalΩ \times false(0, T false), B false(D false) U |_{Γ} = 0 U |_{t = 0} = U_{0}

with

U = ((), \begin{matrix} array u \\ array v \\ array θ \end{matrix}), U_{0} = ((), \begin{matrix} array u_{0} \\ array u_{1} \\ array θ_{0} \end{matrix}), A false(D false) = ((), \begin{matrix} array 0 & array 1 & array 0 \\ array - {normalΔ}^{2} & array 0 & array - Δ \\ array 0 & array Δ & array Δ \end{matrix}), B false(D false) = ((), \begin{matrix} array 1 & array 0 & array 0 \\ array \partial_{ν} & array 0 & array 0 \\ array 0 & array 0 & array 1 \end{matrix}) .

Using the Laplace transform to Equation , we obtain the corresponding resolvent problem …”

Section: Introductionmentioning

confidence: 99%

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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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Analyticity and stability results for a plate‐membrane type transmission problem

Martínez

Ospino

Monzón

2023

Mathematische Nachrichten

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In this paper, we consider a transmission problem for a system of a thermoelastic plate with (or without) a rotational inertia coupled with a membrane. On the plate a structural damping may or may not act, and on the membrane a damping of Kelvin–Voigt type may or may not be present; free boundary operators for the plate are considered in the coupling with the membrane. We prove well‐posedness of the problem and higher regularity of the solution. Depending on the damping and on the presence of the rotational term, we establish strong stability, exponential stability, lack of exponential stability, polynomial stability, and analyticity of the semigroup associated to the transmission problem.

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Optimal large‐time estimates and singular limits for thermoelastic plate equations with the Fourier law

Chen

Ikehata

2023

Math Methods in App Sciences

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In this paper, we study asymptotic behaviors for classical thermoelastic plate equations with the Fourier law of heat conduction in the whole space , where we introduce a reduction methodology basing on third‐order (in time) differential equations and refined Fourier analysis. We derive optimal growth estimates when , bounded estimates when , and decay estimates when for the vertical displacement in the norm. Particularly, the new critical dimension for distinguishing the decisive role between the plate model and the Fourier law of heat conduction is discovered. Moreover, concerning the small thermal parameter in the temperature equation, we study the singular limit problem. We not only show global (in time) convergence of the vertical displacements between thermoelastic plates and structurally damped plates but also rigorously demonstrate a new second‐order profile of the solution. Our methodology can settle several closely related problems in thermoelasticity.

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Maximal regularity for the thermoelastic plate equations with free boundary conditions

Cited by 15 publications

References 22 publications

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

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

Analyticity and stability results for a plate‐membrane type transmission problem

Optimal large‐time estimates and singular limits for thermoelastic plate equations with the Fourier law

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