Asymptotic Analysis - Volume 59, issue 1-2

Authors: Bal, Guillaume | Garnier, Josselin | Motsch, Sébastien | Perrier, Vincent

Article Type: Research Article

Abstract: This paper concerns the homogenization of a one-dimensional elliptic equation with oscillatory random coefficients. It is well known that the random solution to the elliptic equation converges to the solution of an effective medium elliptic equation in the limit of a vanishing correlation length in the random medium. It is also well known that the corrector to homogenization, i.e., the difference between the random solution and the homogenized solution, converges in distribution to a Gaussian process when the correlations in the random medium are sufficiently short-range. Moreover, the limiting process may be written as a stochastic integral with respect to …standard Brownian motion. We generalize the result to a large class of processes with long-range correlations. In this setting, the corrector also converges to a Gaussian random process, which has an interpretation as a stochastic integral with respect to fractional Brownian motion. Moreover, we show that the longer the range of the correlations, the larger is the amplitude of the corrector. Derivations are based on a careful analysis of random oscillatory integrals of processes with long-range correlations. We also make use of the explicit expressions for the solutions to the one-dimensional elliptic equation. Show more

Keywords: homogenization, partial differential equations with random coefficients, long-range memory effects, central limit, Gaussian processes

DOI: 10.3233/ASY-2008-0890

Citation: Asymptotic Analysis, vol. 59, no. 1-2, pp. 1-26, 2008

Authors: Nabongo, Diabate | Boni, Théodore K.

Article Type: Research Article

Abstract: We obtain some conditions under which the positive solution of the numerical approximation for the heat equation ut (x, t)=uxx (x, t), x∈(0, 1), t>0, with the singular boundary condition ux (1, t)=−u−β (1, t), where β>0 quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time and obtain some results on numerical quenching rate and set. Finally we give some numerical results to illustrate our analysis.

Keywords: semidiscretizations, singular boundary condition, quenching, semidiscrete quenching time, convergence, numerical quenching rate, numerical quenching set

DOI: 10.3233/ASY-2008-0889

Citation: Asymptotic Analysis, vol. 59, no. 1-2, pp. 27-38, 2008

Authors: Robinson, James C. | Sadowski, Witold

Article Type: Research Article

Abstract: Current theoretical results for the three-dimensional Navier–Stokes equations only guarantee that solutions remain regular for all time when the initial enstrophy (‖Du0 ‖2 :=∫|curl u0 |2 ) is sufficiently small, ‖Du0 ‖2 ≤χ0 . In fact, this smallness condition is such that the enstrophy is always non-increasing. In this paper we provide a numerical procedure that will verify regularity of solutions for any bounded set of initial conditions, ‖Du0 ‖2 ≤χ1 . Under the assumption that the equations are in fact regular we show that this procedure can be guaranteed to terminate after a finite time.

Keywords: regularity of the Navier–Stokes equations

DOI: 10.3233/ASY-2008-0899

Citation: Asymptotic Analysis, vol. 59, no. 1-2, pp. 39-50, 2008

Authors: Sun, Chunyou | Yang, Meihua

Article Type: Research Article

Abstract: We consider the dynamical behavior of the nonclassical diffusion equation with critical nonlinearity for both autonomous and nonautonomous cases. For the autonomous case, we obtain the existence of a global attractor when the forcing term only belongs to H−1 , this result simultaneously resolves a problem in Acta Mathematicae Applicatae Sinica 18 (2002), 273–276 related to the critical exponent. For the nonautonomous case, assumed that the time-dependent forcing term is translation bounded instead of translation compact, we first prove the asymptotic regularity of solutions, then the existence of a compact uniform attractor together with its structure and regularity has been …obtained; finally, we show the existence of (nonautonomous) exponential attractors. Show more

Keywords: nonclassical diffusion equation, critical exponent, asymptotic regularity, attractor

DOI: 10.3233/ASY-2008-0886

Citation: Asymptotic Analysis, vol. 59, no. 1-2, pp. 51-81, 2008

Authors: Delcroix, Antoine | Marti, Jean-André | Oberguggenberger, Michael

Article Type: Research Article

Abstract: We introduce a new type of local and microlocal asymptotic analysis in algebras of generalized functions, based on the presheaf properties of those algebras and on the properties of their elements with respect to a regularizing parameter. Contrary to the more classical frequential analysis based on the Fourier transform, we can describe a singular asymptotic spectrum which has good properties with respect to nonlinear operations. In this spirit we give several examples of propagation of singularities through nonlinear operators.

Keywords: microlocal analysis, generalized functions, nonlinear operators, presheaf, propagation of singularities, singular spectrum

DOI: 10.3233/ASY-2008-0885

Citation: Asymptotic Analysis, vol. 59, no. 1-2, pp. 83-107, 2008

Authors: Briet, Philippe | Cornean, Horia D. | Louis, Delphine

Article Type: Research Article

Abstract: Consider a charged, perfect quantum gas, in the effective mass approximation, and in the grand-canonical ensemble. We prove in this paper that the generalized magnetic susceptibilities admit the thermodynamic limit for all admissible fugacities, uniformly on compacts included in the analyticity domain of the grand-canonical pressure. The problem and the proof strategy were outlined in MPRF 11 (2005), 177–188. In J. Math. Phys. 47 (2006), 083511 we proved in detail the pointwise thermodynamic limit near z=0. The present paper is the last one of this series, and contains the proof of the uniform bounds on compacts needed in order …to apply Vitali's Convergence Theorem. Show more

Keywords: thermodynamic limit, magnetic field, susceptibilities

DOI: 10.3233/ASY-2008-0884

Citation: Asymptotic Analysis, vol. 59, no. 1-2, pp. 109-123, 2008