Asymptotic Analysis - Volume 64, issue 1-2

Authors: Liess, Otto

Article Type: Research Article

Abstract: The aim of this paper is to study decay estimates for global solutions of the system of linear crystal elasticity in three space dimensions for cubic crystals in the generic nearly isotropic case. Our main result is that for large time, the solutions to the system decay with a decay rate of order t−1/2−1/κ for some κ∈N which is strictly positive (cf. Remark 1.2). We should observe that in the isotropic case, the optimal decay rate is t−1 and that the decay rate for the wave equation in n space variables is t−(n−1)/2 . On the other hand, …the decay rate for solutions of the Maxwell system for optically biaxial crystals is of order t−1/2 . (See O. Liess, Asymptot. Anal. 4 (1991), 61–95.) Show more

Keywords: cubic crystals, decay estimates, nearly isotropic crystals, singular surfaces, uniplanar singularities

DOI: 10.3233/ASY-2009-0932

Citation: Asymptotic Analysis, vol. 64, no. 1-2, pp. 1-27, 2009

Authors: Cassani, Daniele | Tarsi, Cristina

Article Type: Research Article

Abstract: We derive a Pohožaev–Trudinger type embedding for the Lorentz–Sobolev space W1 0 LN,q (Ω), for general domains Ω⊆RN and in particular for Ω=RN . Precisely, we first prove that the corresponding inequality is domain independent and then, by constructing explicit concentrating sequences à la Moser, we establish that the embedding inequality is sharp and we exhibit the best constant.

Keywords: Lorentz–Sobolev spaces, limiting Sobolev embeddings, Trudinger–Moser type inequalities, unbounded domains, best constants

DOI: 10.3233/ASY-2009-0934

Citation: Asymptotic Analysis, vol. 64, no. 1-2, pp. 29-51, 2009

Authors: Betz, Volker | Joye, Alain | Teufel, Stefan

Article Type: Research Article

Abstract: We study the time-dependent scattering of a quantum mechanical wave packet at a barrier for energies larger than the barrier height, in the semi-classical regime. More precisely, we are interested in the leading order of the exponentially small scattered part of the wave packet in the semiclassical parameter when the energy density of the incident wave is sharply peaked around some value. We prove that this reflected part has, to leading order, a Gaussian shape centered on the classical trajectory for all times soon after its birth time. We give explicit formulas and rigorous error bounds for the reflected wave …for all of these times. Show more

Keywords: above barrier scattering, exponential asymptotics, quantum theory

DOI: 10.3233/ASY-2009-0935

Citation: Asymptotic Analysis, vol. 64, no. 1-2, pp. 53-100, 2009

Authors: Bernard, Yann

Article Type: Research Article

Abstract: We consider a model for a free molecular flow in a thin channel bounded by two parallel plates on which Maxwellian boundary conditions with (fixed) accommodation and a generic scattering kernel apply. Using functional analytic tools, we show that as the width of the channel vanishes, and on a suitable temporal scale, the evolution of the density is described by a diffusion problem. We distinguish two classes of temporal scalings (normal and anomalous) and we show that an infinitesimal amount of grazing collisions with the walls of the channel is responsible for the anomalous diffusion. The method employed is adapted …from the original work of F. Golse in Asymptot. Anal. 17 (1998), 1–12. Show more

DOI: 10.3233/ASY-2009-0942

Citation: Asymptotic Analysis, vol. 64, no. 1-2, pp. 101-123, 2009