Asymptotic Analysis - Volume 14, issue 2

Authors: Gasser, Ingenuin | Markowich, Peter A.

Article Type: Research Article

Abstract: We analyse the classical limit of the quantum hydrodynamic equations as the Planck constant tends to zero. The equations have the form of an Euler system with a constant pressure and a dispersive regularisation term, which (formally) tends to zero in the classical limit. The main tool of the analysis is the exploitation of a kinetic equation, which lies behind the quantum hydrodynamic system. The presented analysis can also be interpreted as an alternative approach to the geometrical optics (WKB)-analysis of the Schrödinger equation.

Keywords: Quantum hydrodynamics, Wigner transform, classical limit, WKB-asymptotics of the Schrödinger equation

DOI: 10.3233/ASY-1997-14201

Citation: Asymptotic Analysis, vol. 14, no. 2, pp. 97-116, 1997

Authors: Kondratiev, V.A. | Véron, L.

Article Type: Research Article

Abstract: We study the asymptotic behaviour of the solutions of the parabolic equation (1) ∂u/∂t−Lu+a(x)|u|q−1 u=0 or the elliptic equation (2) ∂2 u/∂t2 +Lu−a(x)|u|q−1 u=0 in Ω×(0,∞) when Ω is bounded, u satisfies the Neumann boundary condition in ∂Ω×(0,∞),L is a linear strongly elliptic operator in Ω,q is bigger than 1 and a(x)≥0. We also study the vanishing property of t$\mapsto $ u(x,t) when 0 < q < 1.

DOI: 10.3233/ASY-1997-14202

Citation: Asymptotic Analysis, vol. 14, no. 2, pp. 117-156, 1997

Authors: Burq, N.

Article Type: Research Article

Abstract: On généralise les résultats de C. Bardos, G. Lebeau et J. Rauch sur la contrôlabilité exacte de l'équation des ondes avec conditions de Dirichlet au cas où l'ouvert considéré est de classe C3 et les coefficients du Laplacien de classe C2 . On obtient, en utilisant des H-mesures ou mesures de défaut microlocales, des conditions nécessaires et des conditions suffisantes fortes pour la contrôlabilité exacte.

DOI: 10.3233/ASY-1997-14203

Citation: Asymptotic Analysis, vol. 14, no. 2, pp. 157-191, 1997