Asymptotic Analysis - Volume 90, issue 3-4

Authors: Kukavica, Igor | Vicol, Vlad

Article Type: Research Article

Abstract: We address a question posed by Glatt-Holtz and Ziane in Advances in Differential Equations 14 (2009), 567–600, regarding moments of strong pathwise solutions to the Navier–Stokes equations in a two-dimensional bounded domain 𝒪. We prove that Eφ(‖u(t)‖H1 (𝒪) 2 )<∞ for any deterministic t>0, where φ(x)=log (1+log (1+x)). Such moment bounds may be used to study statistical properties of the long time behavior of the equation. In addition, we obtain algebraic moment bounds on compact subdomains 𝒪0 of the form Eφε (‖u(t)‖H1 (𝒪0 ) 2 )<∞, where φε (x)=(1+x)(1−ε)/2 , for any deterministic t>0 and any ε>0.

Keywords: stochastic Navier–Stokes equations, moments, pathwise solutions

DOI: 10.3233/ASY-141237

Citation: Asymptotic Analysis, vol. 90, no. 3-4, pp. 189-206, 2014

Authors: Götz, Dario | Shibata, Yoshihiro

Article Type: Research Article

Abstract: In this paper, we consider a generalized resolvent problem for the linearization system of the Navier–Stokes equations describing some free boundary problem of a compressible barotropic viscous fluid flow without taking the surface tension into account. We prove the existence of the ℛ-bounded solution operators, which drives not only the generation of analytic semigroup but also the maximal Lp –Lq regularity by means of Weis' operator valued Fourier multiplier theorem for the corresponding time dependent problem that enable us to prove a local in time existence theorem of the free boundary problem for a compressible barotropic viscous fluid flow …in the Lp in time and Lq space setting (cf. Annali dell Universita di Ferrara 60 (2014), 55–89). The results in this paper were given in the PhD thesis [Three topics in fluid dynamics: viscoelastic, generalized Newtonian, and compressible fluids, 2012, TU Darmstadt] by the first author under supervision of the second author. Here we present a slightly different method of deriving a concrete form of solutions to the model problem. In this paper, one of the essential points is to show the invertibility of a 2×2 Lopatinski matrix function. The corresponding system in [Three topics in fluid dynamics: viscoelastic, generalized Newtonian, and compressible fluids, 2012, TU Darmstadt] is a 3×3 matrix, so that the method presented here is slightly simpler. Show more

Keywords: ℛ-boundedness, Stokes equations, free boundary condition, general domain, analytic semigroup, maximal L_p–L_q regularity

DOI: 10.3233/ASY-141238

Citation: Asymptotic Analysis, vol. 90, no. 3-4, pp. 207-236, 2014

Authors: Alves, Claudianor O. | Moussaoui, Abdelkrim

Article Type: Research Article

Abstract: We show the existence of positive solutions for a class of singular elliptic systems with convection term. The approach combines sub- and supersolution method with the pseudomonotone operator theory and perturbation arguments involving singular terms.

Keywords: pseudomonotone operator, elliptic singular equation, nonlinear equations

DOI: 10.3233/ASY-141245

Citation: Asymptotic Analysis, vol. 90, no. 3-4, pp. 237-248, 2014

Authors: Licht, C. | Orankitjaroen, S.

Article Type: Research Article

Abstract: The effective behavior of a solid made from a periodic distribution of inclusions in a matrix is investigated. Inclusions and matrix are linearly elastic or viscoelastic of Kelvin–Voigt type (and possibly rigid for the inclusions) while the link between them can be pure adhesion or viscous friction with bilateral contact or involve a very thin viscoelastic layer. As long as one constituent is viscoelastic, the effective behavior is no longer of Kelvin–Voigt type but with memory.

Keywords: homogenization, Kelvin–Voigt viscoelasticity, viscoelasticity with memory, interface effects, semi-groups of operators, Laplace transform, integro-differential equations

DOI: 10.3233/ASY-141247

Citation: Asymptotic Analysis, vol. 90, no. 3-4, pp. 249-265, 2014

Authors: Gruais, Isabelle | Poliševski, Dan | Stanescu, Florentina-Alina

Article Type: Research Article

Abstract: We are interested in the asymptotic behaviour of a fluid flow contained in a microscopic periodic distribution of fissures perturbating a porous medium where the Darcy law is valid, when the coupling between both systems is modeled by the Beavers–Joseph interface condition. As the small period of the distribution tends to zero, the interface condition is preserved on a microscopic scale under the additional assumption that the permeability coefficients behave like the squared period of the distribution which is also the squared size of the fissures. Moreover, the resulting pressure is purely macroscopic unlike the velocity field which also depends …on the microscopic variable. Show more

Keywords: fractured porous media, Stokes flow, Beavers–Joseph interface, homogenization, two-scale convergence

DOI: 10.3233/ASY-141248

Citation: Asymptotic Analysis, vol. 90, no. 3-4, pp. 267-280, 2014

Authors: Borthwick, David | Philipp, Pascal

Article Type: Research Article

Abstract: We study the spectral theory of asymptotically hyperbolic manifolds with ends of warped-product type. Our main result is an upper bound on the resonance counting function with a geometric constant expressed in terms of the respective Weyl constants for the core of the manifold and the base manifold defining the ends.

Keywords: spectral theory, resonances, asymptotically hyperbolic

DOI: 10.3233/ASY-141249

Citation: Asymptotic Analysis, vol. 90, no. 3-4, pp. 281-323, 2014

Authors: Strani, Marta

Article Type: Research Article

Abstract: In this paper we describe the metastable behavior of solutions to a class of parabolic systems. In particular, we improve some results contained in SIAM J. Math. Anal. 45(5) (2013), 3084–3113, by using different techniques to describe the slow motion of the internal layers. Numerical simulations illustrate the results.

Keywords: metastability, slow motion, internal layers, reaction–diffusion systems

DOI: 10.3233/ASY-141250

Citation: Asymptotic Analysis, vol. 90, no. 3-4, pp. 325-344, 2014

Authors: Prakash, Ravi | Sili, Ali

Article Type: Research Article

Abstract: We study the behavior of the solutions uε of a degenerating elliptic equation in the framework of the dimension reduction problem coupled with an oscillating boundary. Under a suitable choice of the scalings on the source term and on the diffusivity coefficients, we prove that the limit problem is a nonlocal problem involving only the diffusion through the upper part of the domain.

Keywords: degenerating problem, oscillating boundary, thin domain

DOI: 10.3233/ASY-141251

Citation: Asymptotic Analysis, vol. 90, no. 3-4, pp. 345-365, 2014

Authors: Molica Bisci, Giovanni | Repovš, Dušan

Article Type: Research Article

Abstract: This work is devoted to the study of the existence of at least one weak solution to nonlocal equations involving a general integro-differential operator of fractional type. As a special case, we derive an existence theorem for the fractional Laplacian, finding a nontrivial weak solution of the equation (−Δ)s u=h(x)f(u) in Ω, u=0 in Rn \Ω, where h∈L∞ + (Ω)\{0} and f :R→R is a suitable continuous function. These problems have a variational structure and we find a nontrivial weak solution for them by exploiting a recent local minimum result for smooth functionals defined on a reflexive Banach space. To make …the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. Show more

Keywords: fractional equations, multiple solutions, critical points results

DOI: 10.3233/ASY-141252

Citation: Asymptotic Analysis, vol. 90, no. 3-4, pp. 367-378, 2014

Article Type: Other

Citation: Asymptotic Analysis, vol. 90, no. 3-4, pp. 379-380, 2014