Asymptotic Analysis - Volume 97, issue 1-2

Authors: Dittrich, Jaroslav | Exner, Pavel | Kühn, Christian | Pankrashkin, Konstantin

Article Type: Research Article

Abstract: Let S ⊂ R 3 be a C 4 -smooth relatively compact orientable surface with a sufficiently regular boundary. For β ∈ R + , let E j ( β ) denote the j th negative eigenvalue of the operator associated with the quadratic form H 1 ( R 3 ) ∋ u ↦ ∭ R 3 | ∇ u | 2 …d x − β ∬ S | u | 2 d σ , where σ is the two-dimensional Hausdorff measure on S . We show that for each fixed j one has the asymptotic expansion E j ( β ) = − β 2 4 + μ j D + o ( 1 ) as  β → + ∞ , where μ j D is the j th eigenvalue of the operator − Δ S + K − M 2 on L 2 ( S ) , in which K and M are the Gauss and mean curvatures, respectively, and − Δ S is the Laplace–Beltrami operator with the Dirichlet condition at the boundary of S . If, in addition, the boundary of S is C 2 -smooth, then the remainder estimate can be improved to O ( β − 1 log β ) . Show more

Keywords: singular Schrödinger operator, δ-interaction, strong coupling, eigenvalue

DOI: 10.3233/ASY-151341

Citation: Asymptotic Analysis, vol. 97, no. 1-2, pp. 1-25, 2016

Authors: Schimperna, Giulio | Segatti, Antonio | Zelik, Sergey

Article Type: Research Article

Abstract: In this paper we analyze a singular heat equation of the form ϑ t + Δ ϑ − 1 = f . The singular term ϑ − 1 gives rise to very fast diffusion effects. The equation is settled in a smooth bounded domain Ω ⊂ R 3 and complemented with a general dynamic boundary condition of the form α ϑ t − β Δ Γ ϑ = ∂ n …ϑ − 1 , where Δ Γ is the Laplace–Beltrami operator and α and β are non-negative coefficients (in particular, the homogeneous Neumann case given by α = β = 0 is included). For this problem, we first introduce a suitable weak formulation and prove a related existence result. For more regular initial data, we show that there exists at least one weak solution satisfying instantaneous regularization effects which are uniform with respect to the time variable. In this improved regularity class, uniqueness is also shown to hold. Show more

Keywords: very fast diffusion, Moser iterations, dynamic boundary conditions

DOI: 10.3233/ASY-151342

Citation: Asymptotic Analysis, vol. 97, no. 1-2, pp. 27-59, 2016

Authors: Klein, Markus | Rosenberger, Elke

Article Type: Research Article

Abstract: We analyze a general class of difference operators H ε = T ε + V ε on ℓ 2 ( ( ε Z ) d ) , where V ε is a multi-well potential and ε is a small parameter. We construct approximate eigenfunctions in neighbourhoods of the different wells and give weighted ℓ 2 -estimates for the difference of these and the exact eigenfunctions of the associated …Dirichlet-operators. Show more

Keywords: semi-classical difference operator, tunneling, WKB-expansions, Dirichlet eigenfunctions, asymptotic expansion, multi-well potential, Agmon estimates

DOI: 10.3233/ASY-151343

Citation: Asymptotic Analysis, vol. 97, no. 1-2, pp. 61-89, 2016

Authors: Hille, Sander | Horbacz, Katarzyna | Szarek, Tomasz | Wojewódka, Hanna

Article Type: Research Article

Abstract: The law of the iterated logarithm for some Markov operators, which converge exponentially to the invariant measure, is established. The operators correspond to iterated function systems which, for example, may be used to generalize the cell cycle model examined by Lasota and Mackey [J. Math. Biol. 38 (1999), 241–261].

Keywords: law of the iterated logarithm, Markov operators

DOI: 10.3233/ASY-151344

Citation: Asymptotic Analysis, vol. 97, no. 1-2, pp. 91-112, 2016

Authors: Dell’Antonio, Gianfausto | Michelangeli, Alessandro

Article Type: Research Article

Abstract: We discuss the general model of a Schrödinger quantum particle constrained on a straight half-line with given self-adjoint boundary condition at the origin and an interaction potential supported around the origin. We study the limit when the range of the potential scales to zero and its magnitude blows up. We show that in the limit the dynamics is generated by a self-adjoint negative Laplacian on the half-line, with a possible preservation or modification of the boundary condition at the origin, depending on the magnitude of the scaling and of the strength of the potential.

Keywords: point interaction, self-adjoint boundary conditions, singular scaling limits in Schrödinger operators, resolvent convergence, Konno–Kuroda resolvent formula

DOI: 10.3233/ASY-151349

Citation: Asymptotic Analysis, vol. 97, no. 1-2, pp. 113-138, 2016

Authors: Albanez, Débora A.F. | Nussenzveig Lopes, Helena J. | Titi, Edriss S.

Article Type: Research Article

Abstract: Motivated by the presence of a finite number of determining parameters (degrees of freedom) such as modes, nodes and local spatial averages for dissipative dynamical systems, we present here a continuous data assimilation algorithm for three-dimensional viscous hydrodynamic models. However, to validate the convergence of this algorithm our proofs require the existence of uniform global bounds on the gradients of the solutions of the underlying system in terms of certain combinations of the physical parameters (such as kinematic viscosity, the size of the domain and the forcing term). Therefore our proofs cannot be applied to the three-dimensional Navier–Stokes equations; instead …we demonstrate the implementation of this algorithm, for instance, in the context of the three-dimensional Navier–Stokes-α equations. This algorithm consists of introducing a nudging process through a general type of approximation interpolation operator (which is constructed from observational measurements) that synchronizes the large spatial scales of the approximate solutions with those of unknown solutions of the Navier–Stokes-α equations corresponding to these measurements. Our main result provides conditions on the finite-dimensional spatial resolution of the collected data, sufficient to guarantee that the approximating solution, which is obtained from this collected data, converges to the unknown reference solution over time. These conditions are given in terms of the physical parameters. Show more

Keywords: continuous data assimilation, three-dimensional Navier–Stokes-α equations, determining modes, volume elements and nodes

DOI: 10.3233/ASY-151351

Citation: Asymptotic Analysis, vol. 97, no. 1-2, pp. 139-164, 2016

Authors: Yamazaki, Taeko

Article Type: Research Article

Abstract: This paper is concerned with an abstract dissipative hyperbolic equation with time-dependent coefficient. Under an assumption which ensures that the energy does not decay, this paper provides a condition on the coefficient, which is necessary and sufficient so that the solutions tend to the solutions of the free wave equation.

Keywords: abstract linear hyperbolic equation, asymptotic behavior, asymptotically free property

DOI: 10.3233/ASY-151354

Citation: Asymptotic Analysis, vol. 97, no. 1-2, pp. 165-187, 2016