Asymptotic Analysis - Volume 76, issue 3-4

Authors: Ammari, Z. | Breteaux, S.

Article Type: Research Article

Abstract: We study the mean field limit of nonrelativistic quantum many-boson systems with delta potential in one-dimensional space. Such problem is related to the semiclassical limit of a second quantized Hamiltonian with an interaction given by a quartic Wick product. In this framework, we show that the evolution of coherent states is semiclassically given by squeezed coherent states under the action of a time-dependent affine Bogoliubov transformation. Results similar to those stated by Hepp [Comm. Math. Phys. 35 (1974), 265–277] and Ginibre-Velo [Comm. Math. Phys. 66 (1979), 37–76 and 68 (1979), 45–68] are proved. Furthermore, we show propagation of chaos for …Schrödinger dynamics in the mean field limit using the argument of Rodnianski–Schlein [Comm. Math. Phys. 291 (2009), 31–61]. Thus, we provide a derivation of the cubic NLS equation in one dimension. Show more

Keywords: classical limit, coherent state, Fock space, non-linear Schrödinger equation, mean field limit, non-autonomous Schrödinger equation

DOI: 10.3233/ASY-2011-1064

Citation: Asymptotic Analysis, vol. 76, no. 3-4, pp. 123-170, 2012

Authors: Avalishvili, G. | Avalishvili, M. | Miara, B.

Article Type: Research Article

Abstract: In this paper we consider the nonclassical initial and initial-boundary value problems for second-order evolution equations with nonlocal initial conditions. We study the existence and uniqueness, in suitable abstract spaces, of solution of vector-valued distributions for such problems. We also present an algorithm of approximation of the solution of nonclassical problem with nonlocal initial conditions by a sequence of solutions of the corresponding classical problems. As an example we investigate the case of nonclassical hyperbolic equations and systems. In this particular case we show that the existence and uniqueness of solutions rely on algebraic properties of the ratios of values …of the time variable in the nonlocal initial conditions and sizes of the spatial domain. Show more

Keywords: abstract second-order evolution equations, initial-boundary value problems for hyperbolic equations and systems, nonlocal initial conditions

DOI: 10.3233/ASY-2011-1065

Citation: Asymptotic Analysis, vol. 76, no. 3-4, pp. 171-192, 2012

Authors: Chupin, Laurent | Sart, Rémy

Article Type: Research Article

Abstract: In this paper, we are first interested in the compressible Navier–Stokes equations with density dependent viscosities in bounded domains with non-homogeneous Dirichlet conditions. We study the well posedness of such models with density dependent viscosity coefficients in the non-stationary and in the stationary cases. We apply the last result in thin domains context, justifying the compressible Reynolds equations.

Keywords: Navier–Stokes equations, compressible fluids, density dependent viscosities, stationary, steady-state, thin films, lubrication, Reynolds equation

DOI: 10.3233/ASY-2011-1070

Citation: Asymptotic Analysis, vol. 76, no. 3-4, pp. 193-231, 2012

Authors: Truc, Françoise

Article Type: Research Article

Abstract: We consider a Schrödinger operator HA D with a non-vanishing radial magnetic field B=dA and Dirichlet boundary conditions on the unit disk. We assume growth conditions on B near the boundary which guarantee in particular the compactness of the resolvent of this operator. Under some assumptions on an additional radial potential V the operator HA D −V has a discrete negative spectrum and we obtain an upper bound of the number of negative eigenvalues. As a consequence we get an upper bound of the number of eigenvalues of HA D smaller than any positive value λ, which involves …the minimum of B and the square of the L2 -norm of A(r)/r, where A(r) is the specific magnetic potential defined as the flux of the magnetic field through the disk of radius r centered in the origin. Show more

Keywords: eigenvalue estimate, magnetic bottles, radial potential

DOI: 10.3233/ASY-2011-1077

Citation: Asymptotic Analysis, vol. 76, no. 3-4, pp. 233-248, 2012

Article Type: Other

Citation: Asymptotic Analysis, vol. 76, no. 3-4, pp. 249-249, 2012