Asymptotic Analysis - Volume 131, issue 3-4

Authors: Moser, Maximilian

Article Type: Research Article

Abstract: We consider the sharp interface limit for the scalar-valued and vector-valued Allen–Cahn equation with homogeneous Neumann boundary condition in a bounded smooth domain Ω of arbitrary dimension N ⩾ 2 in the situation when a two-phase diffuse interface has developed and intersects the boundary ∂ Ω . The limit problem is mean curvature flow with 90°-contact angle and we show convergence in strong norms for well-prepared initial data as long as a smooth solution to the limit problem exists. To this end we assume that the limit problem has a smooth solution on [ 0 …, T ] for some time T > 0 . Based on the latter we construct suitable curvilinear coordinates and set up an asymptotic expansion for the scalar-valued and the vector-valued Allen–Cahn equation. In order to estimate the difference of the exact and approximate solutions with a Gronwall-type argument, a spectral estimate for the linearized Allen–Cahn operator in both cases is required. The latter will be shown in a separate paper, cf. (Moser (2021 )). Show more

Keywords: Sharp interface limit, mean curvature flow, contact angle, Allen–Cahn equation, vector-valued Allen–Cahn equation

DOI: 10.3233/ASY-221775

Citation: Asymptotic Analysis, vol. 131, no. 3-4, pp. 297-383, 2023

Authors: Gómez, Delfina | Nazarov, Sergei A. | Orive-Illera, Rafael | Pérez-Martínez, María-Eugenia

Article Type: Research Article

Abstract: We examine the band-gap structure of the spectrum of the Neumann problem for the Laplace operator in a strip with periodic dense transversal perforation by identical holes of a small diameter ε > 0 . The periodicity cell itself contains a string of holes at a distance O ( ε ) between them. Under assumptions on the symmetry of the holes, we derive and justify asymptotic formulas for the endpoints of the spectral bands in the low-frequency range of the spectrum as ε → 0 . We demonstrate that, for ε small …enough, some spectral gaps are open. The position and size of the opened gaps depend on the strip width, the perforation period, and certain integral characteristics of the holes. The asymptotic behavior of the dispersion curves near the band edges is described by means of a ‘fast Floquet variable’ and involves boundary layers in the vicinity of the perforation string of holes. The dependence on the Floquet parameter of the model problem in the periodicity cell requires a serious modification of the standard justification scheme in homogenization of spectral problems. Some open questions and possible generalizations are listed. Show more

Keywords: Band-gap structure, spectral perturbations, homogenization, perforated media, Neumann-Laplace operator, waveguide

DOI: 10.3233/ASY-221776

Citation: Asymptotic Analysis, vol. 131, no. 3-4, pp. 385-441, 2023

Authors: Charve, Frédéric

Article Type: Research Article

Abstract: The aim of this article is to extend previous works about the asymptotics of an ill-prepared fast rotating, highly stratified incompressible Navier–Stokes system. Thanks to improved Strichartz and a priori estimates, we are able not only to cover a case which was unanswered in our previous work (allowing bigger ill-prepared initial data) but also to improve the convergence rates and reduce some assumptions on the initial data. In passing we also widen the range of some parameters and compare two methods to obtain dispersive estimates.

Keywords: Stratified rotating Navier-Stokes system, 3-dimensional quasi-geostrophic system, Strichartz estimates, Besov spaces

DOI: 10.3233/ASY-221777

Citation: Asymptotic Analysis, vol. 131, no. 3-4, pp. 443-470, 2023

Authors: Abels, Helmut | Ameismeier, Tobias

Article Type: Research Article

Abstract: We consider the dynamical evolution of a thin rod described by an appropriately scaled wave equation of nonlinear elasticity. Under the assumption of well-prepared initial data and external forces, we prove that a solution exists for arbitrarily large times, if the diameter of the cross section is chosen sufficiently small. The scaling regime is such that the limiting equations are linear.

Keywords: Wave equation, von Kármán equation, nonlinear elasticity, long time existence

DOI: 10.3233/ASY-221778

Citation: Asymptotic Analysis, vol. 131, no. 3-4, pp. 471-512, 2023

Authors: Feizmohammadi, Ali | Kian, Yavar

Article Type: Research Article

Abstract: We consider the formally determined inverse problem of recovering an unknown time-dependent potential function from the knowledge of the restriction of the solution of the wave equation to a small subset, subject to a single external source. We show that one can determine the potential function, up to the natural obstruction for the problem, by using a single source placed in the exterior of the spacetime domain and subsequently measuring the solution in a small neighborhood outside of the spacetime domain. The approach is based on considering a dense collection of light rays and constructing a source function that combines …a countable collection of sources that each generates a wave packet near a light ray in the collection. We show that measuring the solution corresponding to that single source simultaneously determines the light ray transform along all the light rays in the collection. The result then follows from injectivity of the light ray transform. Our proof also provides a reconstruction algorithm. Show more

Keywords: Inverse problem, Fourier analysis, geometric optics, light ray transform, wave equation

DOI: 10.3233/ASY-221779

Citation: Asymptotic Analysis, vol. 131, no. 3-4, pp. 513-539, 2023

Authors: Touati, Arthur

Article Type: Research Article

Abstract: We study the propagation of a compactly supported high-frequency wave through a semi-linear wave equation with a null structure. We prove that the self-interaction of the wave creates harmonics which remain close to the light-cone in the evolution. By defining a well-chosen ansatz, we describe precisely those harmonics. Moreover, by applying the vector field method to the equation for the remainder in the ansatz, we prove that the solution exists globally. The interaction between the dispersive decay of waves and their high-frequency behaviour is the main difficulty, and the latter is not compensated by smallness of the initial data, allowing …us to consider the high-frequency limit where the wavelength tends to 0. Show more

Keywords: Wave equation, null structure, high-frequency solutions, non-linear partial differential equations, vector field method

DOI: 10.3233/ASY-221780

Citation: Asymptotic Analysis, vol. 131, no. 3-4, pp. 541-582, 2023

Authors: Li, Li

Article Type: Research Article

Abstract: We consider a time-independent variable coefficients fractional porous medium equation and formulate an associated inverse problem. We determine both the conductivity and the absorption coefficient from exterior partial measurements of the Dirichlet-to-Neumann map. Our approach relies on a time-integral transform technique as well as the unique continuation property of the fractional operator.

Keywords: Inverse problem, fractional porous medium equation, unique continuation property

DOI: 10.3233/ASY-221781

Citation: Asymptotic Analysis, vol. 131, no. 3-4, pp. 583-594, 2023