Asymptotic Analysis - Volume 135, issue 1-2

Authors: Freitas, M.M. | Almeida Júnior, D.S. | Miranda, L.G.R. | Ramos, A.J.A. | Caljaro, R.Q.

Article Type: Research Article

Abstract: This paper is concerned with the study of global attractors for a new semilinear Timoshenko–Ehrenfest type system. Firstly we establish the well-posedness of the system using Faedo–Galerkin method. By considering only one damping term acting on the vertical displacement, we prove the existence of a smooth finite dimensional global attractor using the recent quasi-stability theory. Our results holds for any parameters of the system.

Keywords: Timoshenko-type systems, gradient systems, well-posedness, quasi-stability, global attractors

DOI: 10.3233/ASY-231843

Citation: Asymptotic Analysis, vol. 135, no. 1-2, pp. 1-23, 2023

Authors: Dor, Dieunel

Article Type: Research Article

Abstract: In this paper, we consider the hyperbolic Cahn–Hilliard equation with a proliferation term, which has applications in biology. First, we study the well-posedness and the regularity of the solutions, which then allow us to study the dissipativity and the high-order dissipativity and finally the existence of the exponential attractor with Dirichlet boundary conditions. Finally, we give numerical simulations that confirm the results.

Keywords: Cahn-Hilliard equation, relaxation term, mass source, well posedness, tumor growth, global attractor, exponential attractor, simulations

DOI: 10.3233/ASY-231844

Citation: Asymptotic Analysis, vol. 135, no. 1-2, pp. 25-53, 2023

Authors: Briant, Marc | Grec, Bérénice

Article Type: Research Article

Abstract: We present the arising of the Fick cross-diffusion system of equations for fluid mixtures from the multi-species Boltzmann equation in a rigorous manner in Sobolev spaces. To this end, we formally show that, in a diffusive scaling, the hydrodynamical limit of the kinetic system is the Fick model supplemented with a closure relation and we give explicit formulae for the macroscopic diffusion coefficients from the Boltzmann collision operator. Then, we provide a perturbative Cauchy theory in Sobolev spaces for the constructed Fick system, which turns out to be a dilated parabolic equation. We finally prove the stability of the system …in the Boltzmann equation, ensuring a rigorous derivation between the two models. Show more

Keywords: Multispecies Boltzmann equation, gaseous and fluid mixture, Fick’s equation, perturbative theory, hydrodynamical limit, Knudsen number

DOI: 10.3233/ASY-231847

Citation: Asymptotic Analysis, vol. 135, no. 1-2, pp. 55-80, 2023

Authors: Fahs, Rayan

Article Type: Research Article

Abstract: The aim of this paper is to establish the asymptotic expansion of the eigenvalues of the Stark Hamiltonian, with a strong uniform electric field and Dirichlet boundary conditions on a smooth bounded domain of R N , N ⩾ 2 . This work aims at generalizing the recent results of Cornean, Krejčiřik, Pedersen, Raymond, and Stockmeyer in dimension 2. More precisely, in dimension N , in the strong electric field limit, we derive, under certain local convexity conditions, a full asymptotic expansion of the low-lying eigenvalues. To establish our main result, we perform …the construction of quasi-modes. The “optimality” of our constructions is then established thanks to a reduction to model operators and localization estimates. Show more

Keywords: Stark effects, strong electric field, asymptotics of eigenvalues, Dirichlet boundary conditions, Airy function, harmonic oscillator, Weingarten map

DOI: 10.3233/ASY-231848

Citation: Asymptotic Analysis, vol. 135, no. 1-2, pp. 81-113, 2023

Authors: Akil, Mohammad | Ghader, Mouhammad | Hajjej, Zayd | Sammoury, Mohamad Ali

Article Type: Research Article

Abstract: In this paper, we investigate the stability of the transmission problem for Rayleigh beam model with heat conduction. First, we reformulate our system into an evolution equation and prove our problem’s well-posedness. Next, we demonstrate the resolvent of the operator is compact in the energy space, then by using the general criteria of Arendt–Batty, we prove that the thermal dissipation is enough to stabilize our model. Finally, a polynomial energy decay rate has been obtained which depends on the mass densities and the moments of inertia of the Rayleigh beams.

Keywords: Rayleigh beam, heat conduction, C0-semigroup, polynomial stability

DOI: 10.3233/ASY-231849

Citation: Asymptotic Analysis, vol. 135, no. 1-2, pp. 115-156, 2023

Authors: Bittencourt Moraes, G.E. | de Camargo, S.J. | Jorge Silva, M.A.

Article Type: Research Article

Abstract: This is the second paper of a trilogy intended by the authors in what concerns a unified approach to the stability of thermoelastic arched beams of Bresse type under Fourier’s law. Differently of the first one, where the thermal couplings are regarded on the axial and bending displacements, here the thermal couplings are taken over the shear and bending forces. Such thermal effects still result in a new prototype of partially damped Bresse system whose stability results demand a proper approach. Combining a novel path of local estimates by means of the resolvent equation along with a control-observability analysis developed …for elastic non-homogeneous systems of Bresse type proposed in trilogy’s first paper, we are able to provide a unified methodology of the asymptotic stability results, by proving the pattern of them with respect to boundary conditions and the action of temperature couplings, which is in compliance with our previous and present goal. Show more

Keywords: Bresse system, thermoelasticity, stability, optimality

DOI: 10.3233/ASY-231850

Citation: Asymptotic Analysis, vol. 135, no. 1-2, pp. 157-183, 2023

Authors: Aslan, Halit Sevki | Rempel Ebert, Marcelo

Article Type: Research Article

Abstract: In the present paper, we study the influence of oscillations of the time-dependent damping term b ( t ) u t on the asymptotic behavior of the energy for solutions to the Cauchy problem for a σ -evolution equation u t t + ( − Δ ) σ u + b ( t ) u t = 0 , ( t , x ) ∈ [ 0 , ∞ ) × R n , u ( …0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) , x ∈ R n , where σ > 0 and b is a continuous and positive function. Mainly we consider damping terms that are perturbations of the scale invariant case b ( t ) = β ( 1 + t ) − 1 , with β > 0 , and we discuss the influence of oscillations of b on the energy estimates according to the size of β . Show more

Keywords: Damped σ-evolution equation, time-dependent damping, energy estimates, effective dissipation, non-effective dissipation

DOI: 10.3233/ASY-231851

Citation: Asymptotic Analysis, vol. 135, no. 1-2, pp. 185-207, 2023

Authors: Lafitte, Olivier | Runborg, Olof

Article Type: Research Article

Abstract: In this work we show an error estimate for a first order Gaussian beam at a fold caustic, approximating time-harmonic waves governed by the Helmholtz equation. For the caustic that we study the exact solution can be constructed using Airy functions and there are explicit formulae for the Gaussian beam parameters. Via precise comparisons we show that the pointwise error on the caustic is of the order O ( k − 5 / 6 ) where k is the wave number in Helmholtz.

Keywords: High frequency wave asymptotics, Gaussian beams, caustics, error estimates

DOI: 10.3233/ASY-231852

Citation: Asymptotic Analysis, vol. 135, no. 1-2, pp. 209-255, 2023

Authors: Jleli, Mohamed | Samet, Bessem

Article Type: Research Article

Abstract: We consider hyperbolic inequalities with Hardy potential u t t − Δ u + λ | x | 2 u ⩾ | x | − a | u | p in  ( 0 , ∞ ) × B 1 ∖ { 0 } , u ( t , x ) ⩾ f ( x ) on  ( 0 , ∞ ) × ∂ B 1 , …where B 1 is the unit ball in R N , N ⩾ 3 , λ > − ( N − 2 2 ) 2 , a ⩾ 0 , p > 1 and f is a nontrivial L 1 -function. We study separately the cases: λ = 0 , − ( N − 2 2 ) 2 < λ < 0 and λ > 0 . For each case, we obtain an optimal criterium for the nonexistence of weak solutions. Our study yields naturally optimal nonexistence results for the corresponding stationary problem. The novelty of this work lies in two facts: (i) To the best of our knowledge, in all previous works dealing with nonexistence results for evolution equations with Hardy potential in a bounded domain, only the parabolic case has been investigated, making use of some comparison principles. (ii) To the best of our knowledge, in all previous works, the issue of nonexistence has been studied only in the case of positive solutions. In this paper, there is no restriction on the sign of solutions. Show more

Keywords: Hyperbolic inequalities, Hardy potential, bounded domain, weak solution, nonexistence

DOI: 10.3233/ASY-231854

Citation: Asymptotic Analysis, vol. 135, no. 1-2, pp. 257-275, 2023

Authors: Hassine, Maatoug | Ouni, Marwa

Article Type: Research Article

Abstract: This work is devoted to a topological asymptotic expansion for the nonlinear Navier–Stokes operator. We consider the 3D Navier–Stokes equations as a model problem and we derive a topological sensitivity analysis for a design function with respect to the insertion of a small obstacle inside the fluid flow domain. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is examined. The performed mathematical framework can be applied for a large class of design functions and arbitrarily shaped geometric perturbations. The obtained asymptotic formula can serve as a useful tool for solving a variety of topology …optimization problems in fluid mechanics. Show more

Keywords: Topological asymptotic expansion, 3D Navier–Stokes equations, analysis of sensitivity, topological gradient based methods, fluid mechanics

DOI: 10.3233/ASY-231855

Citation: Asymptotic Analysis, vol. 135, no. 1-2, pp. 277-304, 2023