Asymptotic Analysis - Volume 89, issue 3-4

Authors: Klein, M. | Rama, J.

Article Type: Research Article

Abstract: In quantum mechanics the temporal decay of certain resonance states is associated with an effective time evolution e−ith(κ) , where h(·) is an analytic family of non-self-adjoint matrices. In general the corresponding resonance states do not decay exponentially in time. Using analytic perturbation theory, we derive asymptotic expansions for e−ith(κ) , simultaneously in the limits κ→0 and t→∞, where the corrections with respect to pure exponential decay have uniform bounds in one complex variable κ2 t. In the Appendix we briefly review analytic perturbation theory, replacing the classical reference to the 1920 book of Knopp [Funktionentheorie II, Anwendungen und …Weiterführung der allgemeinen Theorie, Sammlung Göschen, Vereinigung wissenschaftlicher Verleger Walter de Gruyter, 1920] and its terminology by standard modern references. This might be of independent interest. Show more

Keywords: resonances, exponential decay, long-time corrections, Fermi golden rule, analytic perturbation theory

DOI: 10.3233/ASY-141226

Citation: Asymptotic Analysis, vol. 89, no. 3-4, pp. 189-233, 2014

Authors: Dell'Oro, Filippo | Muñoz Rivera, Jaime E.

Article Type: Research Article

Abstract: We analyze the decay properties of the solution semigroup generated by a linear evolution system modeling a mixture of three interacting continua with frictional damping. In particular, we provide conditions for the strong and exponential stability when the dissipation is contributed only by some equation of the system. Moreover, we study situations where the semigroup decays polynomially or admits trajectories with conserved energy, and we find the optimal polynomial decay rate.

Keywords: ternary mixtures, contraction semigroups, exponential decay, polynomial decay

DOI: 10.3233/ASY-141229

Citation: Asymptotic Analysis, vol. 89, no. 3-4, pp. 235-262, 2014

Authors: Wang, Chao

Article Type: Research Article

Abstract: In this paper, we study the dynamics of a solid body moving in a 2-D fluid flow with slip boundary conditions. The boundaries of the container and the solid belong to C1,1 . First, we prove existence and uniqueness of a strong solution up to collision. Second, we consider the influence of curvature of the solid boundary on the contact problem under slip boundary conditions.

Keywords: 2-D, fluid-solid, well-posedness, slip boundary condition, collision

DOI: 10.3233/ASY-141230

Citation: Asymptotic Analysis, vol. 89, no. 3-4, pp. 263-306, 2014

Authors: Feireisl, Eduard | Jin, Bum Ja | Novotný, Antonín

Article Type: Research Article

Abstract: We consider the motion of a compressible viscous fluid in the asymptotic regime of low Mach and high Reynolds numbers under strong stratification imposed by a conservative external force. Assuming a bi-dimensional character of the flow, we identify the limit system represented by the so-called lake equation – the Euler system supplemented by an anelastic type constraint imposed by the limit density profile. The key ingredient of the proof are new “frequency localized” estimates of Strichartz type.

Keywords: compressible Navier–Stokes system, anelastic approximation, stratified fluid, inviscid incompressible limit

DOI: 10.3233/ASY-141231

Citation: Asymptotic Analysis, vol. 89, no. 3-4, pp. 307-329, 2014

Authors: Dombrowski, Nicolas | Hislop, Peter D. | Soccorsi, Eric

Article Type: Research Article

Abstract: We study two-dimensional magnetic Schrödinger operators with a magnetic field that is equal to b>0 for x>0 and −b for x<0. This magnetic Schrödinger operator exhibits a magnetic barrier at x=0. The unperturbed system is invariant with respect to translations in the y-direction. As a result, the Schrödinger operator admits a direct integral decomposition. We analyze the band functions of the fiber operators as functions of the wave number and establish their asymptotic behavior. Because the fiber operators are reflection symmetric, the band functions may be classified as odd or even. The odd band functions have a unique absolute minimum. …We calculate the effective mass at the minimum and prove that it is positive. The even band functions are monotone decreasing. We prove that the eigenvalues of an Airy operator, respectively, harmonic oscillator operator, describe the asymptotic behavior of the band functions for large negative, respectively positive, wave numbers. We prove a Mourre estimate for perturbations of the magnetic Schrödinger operator and establish the existence of absolutely continuous spectrum in certain energy intervals. We prove lower bounds on magnetic edge currents for states with energies in the same intervals. We also prove that these lower bounds imply stable lower bounds for the asymptotic currents. Because of the unique, non-degenerate minimum of the first band function, we prove that a perturbation by a slowly decaying negative potential creates an infinite number of eigenvalues accumulating at the bottom of the essential spectrum from below. We establish the asymptotic behavior of the eigenvalue counting function for these infinitely-many eigenvalues below the bottom of the essential spectrum. Show more

Keywords: magnetic Schrödinger operators, snake orbits, magnetic field, magnetic edge states, edge conductance

DOI: 10.3233/ASY-141234

Citation: Asymptotic Analysis, vol. 89, no. 3-4, pp. 331-363, 2014

Authors: Egger, Herbert | Schlottbom, Matthias

Article Type: Research Article

Abstract: We provide an asymptotic analysis of linear transport problems in the diffusion limit under minimal regularity assumptions on the domain, the coefficients, and the data. The weak form of the limit equation is derived and the convergence of the solution in the L2 norm is established without artificial regularity requirements. This is important to be able to deal with problems involving realistic geometries and heterogeneous media. In a second step we prove the usual O(ε) convergence rates under very mild additional assumptions. The generalization of the results to convergence in Lp with p≠2 and some limitations are discussed.

Keywords: radiative transfer, neutron transport, diffusion limit, asymptotic analysis

DOI: 10.3233/ASY-141235

Citation: Asymptotic Analysis, vol. 89, no. 3-4, pp. 365-377, 2014

Article Type: Other

Citation: Asymptotic Analysis, vol. 89, no. 3-4, pp. 379-380, 2014