On the Uniform Poincaré Inequality

Boulkhemair, Abdesslam; Chakib, Abdelkrim

doi:10.1080/03605300600910241

Cited by 35 publications

(33 citation statements)

References 7 publications

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“…In similar way, we can show that the difference between the Eq. (53) and (12), converges to 0 as h, μ → 0+ This achieve the proof of assertion (ii).…”

Section: On (ϕ) (Iii) Ifsupporting

confidence: 52%

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Approximation and numerical realization of an optimal design welding problem

Chakib

Nachaoui

2013

Numerical Methods Partial

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In this article, we deal with the approximation of an optimal shape design approach for a free boundary problem modeling a welding process. We consider discretization of this problem based on linear finite elements. We prove the existence of discrete optimal solutions. This allows us to show the convergence result of a sequence of discrete solutions to the continuous one. Finally, methods for numerical realization are described and several examples have been carried out to illustrate the efficiency of the proposed approach.

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“…In similar way, we can show that the difference between the Eq. (53) and (12), converges to 0 as h, μ → 0+ This achieve the proof of assertion (ii).…”

Section: On (ϕ) (Iii) Ifsupporting

confidence: 52%

“…Note that T = u + V is solution of (SP ) for each u solution of (12). Thus, if ( , u( )) is solution of (14) then ( , T ( )) is solution of the problem (11).…”

Section: Remarkmentioning

confidence: 94%

Approximation and numerical realization of an optimal design welding problem

Chakib

Nachaoui

2013

Numerical Methods Partial

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“…Proof of Theorem The first estimate obviously follows from . To prove we integrate the density equation over

{normalΩ}_{F} false(t false)

and using the boundary conditions and we obtain

\frac{1}{false| {normalΩ}_{F} (t) false|} \int_{{normalΩ}_{F} false(t false)} ρ false(t, x false) d x = \bar{ρ} false(t ⩾ 0 false) .

Since

dist ({normalΩ}_{S} false(t false), \partial Ω) > ν / 2

and

{normalΩ}_{S} false(t false)

has smooth boundary for every

t \in [0, \infty)

, by Poincaré–Wirtinger inequality we obtain

false∥ ρ (t, x) - \bar{ρ} {false∥}_{L^{q} false(Ω_{F} (t) false)} ⩽ C {∥ \nabla ρ ∥}_{L^{q} false(Ω_{F} (t) false)},

where the constant C can be chosen uniformly with respect to t (see for instance [, Theorem 1]). Thus we have

\begin{matrix} true \\ right true∥ e^{η false(\cdot false)} (ρ - \bar{ρ}) ∥_{W^{1, p} (0, \infty; W^{1, q} false(Ω_{F}} \end{matrix}

…”

Section: Global In Time Existencementioning

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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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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“…Compactness arguments show that the family of surfaces f " .t / I " 2 .0; " 0 I t 2 OE0; T g [ f .t / I t 2 OE0; T g satisfies the uniform cone property needed for Theorem 1 (Section 3) in [3]. Hence, by this Theorem, there holds the following uniform Poincaré-inequality:…”

Section: The Convergence Of the Solutionmentioning

confidence: 85%