Asymptotic Analysis - Volume 59, issue 3-4

Authors: Gentil, Ivan | Imbert, Cyril

Article Type: Research Article

Abstract: In this paper, we study a Fokker–Planck equation of the form ut =ℐ[u]+div(xu), where the operator ℐ, which is usually the Laplacian, is replaced here with a general Lévy operator. We prove by the entropy production method the exponential decay in time of the solution to the steady state of the associated stationary equation.

Keywords: Fokker–Planck equation, Lévy operator, Φ-entropy inequalities, entropy production method, logarithmic Sobolev inequalities, fractional Laplacian

DOI: 10.3233/ASY-2008-0887

Citation: Asymptotic Analysis, vol. 59, no. 3-4, pp. 125-138, 2008

Authors: Zheng, Chuang

Article Type: Research Article

Abstract: In this paper we study the controllability of an Euler Implicit time discrete heat equation in a bounded domain with a local internal controller. We prove that, based on Lebeau–Robbiano's time iteration method, the projection in appropriate filtered space is null controllable with uniformly bounded control. In this way, the well-known null-controllability property of the heat equation can be proven as the limit, as ▵t→0, of the controllability of projections of the time-discrete one. Consequently we prove the uniform approximate controllability after filtering with bounded control. A further study is made and analogous results are obtained for other discrete schemes, …i.e. Euler Explicit schemes, θ-method schemes. We also discuss the null controllability of the Euler Implicit time discrete parabolic equation of fractional order. Show more

Keywords: null controllability, heat equation, time discretization, time iteration method, filtering

DOI: 10.3233/ASY-2008-0888

Citation: Asymptotic Analysis, vol. 59, no. 3-4, pp. 139-177, 2008

Authors: Aloui, L.

Article Type: Research Article

Abstract: We prove, under the geometric control condition, a smoothing effect for solutions of a regularized Schrödinger equation on bounded domains with boundary. This result is derived from the resolvent properties.

Keywords: Schrödinger operator, resolvent, smoothing effect

DOI: 10.3233/ASY-2008-0892

Citation: Asymptotic Analysis, vol. 59, no. 3-4, pp. 179-193, 2008

Authors: Korotyaev, Evgeny

Article Type: Research Article

Abstract: We consider the first order periodic systems perturbed by a 2N×2N matrix-valued periodic potential on the real line. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define the Lyapunov function, which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the scalar case. The Lyapunov function has branch points, which we call resonances. We prove the existence of real or complex resonances. We determine the asymptotics of the periodic, anti-periodic spectrum and of the resonances at high energy …(in terms of the Fourier coefficients of the potential). We show that there exist two types of gaps: (i) stable gaps, i.e., the endpoints are periodic and anti-periodic eigenvalues, (ii) unstable (resonance) gaps, i.e., the endpoints are resonances (real branch points). Moreover, we determine various new trace formulae for potentials and the Lyapunov exponent. Show more

Keywords: periodic systems, spectrum, high energy asymptotics

DOI: 10.3233/ASY-2008-0893

Citation: Asymptotic Analysis, vol. 59, no. 3-4, pp. 195-225, 2008

Authors: Freidlin, Mark | Spiliopoulos, Konstantinos

Article Type: Research Article

Abstract: Second initial boundary problem in narrow domains of width ε�1 for linear second-order differential equations with nonlinear boundary conditions is considered in this paper. Using probabilistic methods we show that the solution of such a problem converges as ε↓0 to the solution of a standard reaction–diffusion equation in a domain of reduced dimension. This reduction allows to obtain some results concerning wave front propagation in narrow domains. In particular, we describe conditions leading to jumps of the wave front.

Keywords: reaction–diffusion equations, narrow domains, wave front propagation, instantaneous reflection

DOI: 10.3233/ASY-2008-0894

Citation: Asymptotic Analysis, vol. 59, no. 3-4, pp. 227-249, 2008

Article Type: Other

Citation: Asymptotic Analysis, vol. 59, no. 3-4, pp. 251-252, 2008