Asymptotic Analysis - Volume 130, issue 3-4

Authors: Anderson Cardoso, J. | Carvalho, Jonison Lucas | Medeiros, Everaldo

Article Type: Research Article

Abstract: In this paper we deal with the following class of nonlinear Schrödinger equations − Δ u + V ( | x | ) u = λ Q ( | x | ) f ( u ) , x ∈ R 2 , where λ > 0 is a real parameter, the potential V and the weight Q are radial, which can be singular at the origin, unbounded or decaying at infinity and the nonlinearity f ( s ) …behaves like e α s 2 at infinity. By performing a variational approach based on a weighted Trudinger–Moser type inequality proved here, we obtain some existence and multiplicity results. Show more

Keywords: Nonlinear Schrödinger equation, unbounded or decaying radial potentials, exponential critical growth, weighted Trudinger–Moser type inequality

DOI: 10.3233/ASY-211752

Citation: Asymptotic Analysis, vol. 130, no. 3-4, pp. 297-322, 2022

Authors: Tawri, Krutika

Article Type: Research Article

Abstract: We give an asymptotic upper bound for the k th twisted eigenvalue of the linearized Allen–Cahn operator in terms of the k th eigenvalue of the Jacobi operator, taken with respect to the minimal surface arising as the asymptotic limit of the zero sets of the Allen–Cahn critical points. We use an argument based on the notion of second inner variation developed in Le (On the second inner variations of Allen–Cahn type energies and applications to local minimizers. J. Math. Pures Appl. (9) 103 (2015 ) 1317–1345).

Keywords: Allen-Cahn functional, local minimizer, twisted eigenvalues, inner variations

DOI: 10.3233/ASY-211753

Citation: Asymptotic Analysis, vol. 130, no. 3-4, pp. 323-334, 2022

Authors: Dimassi, Mouez | Fujiié, Setsuro

Article Type: Research Article

Abstract: We study Schrödinger operators H ( h ) = − h 2 Δ + V ( x ) acting in L 2 ( R n ) for non-decaying potentials V . We give a full asymptotic expansion of the spectral shift function for a pair of such operators in the high energy limit. In particular for asymptotically homogeneous potentials W at infinity of degree zero, we also study the semiclassical asymptotics to give a Weyl formula of the spectral shift function above the threshold …max W and Mourre estimates in the range of W except at its critical values. Show more

Keywords: Stark Schrödinger operator, spectral shift function, asymptotic expansions

DOI: 10.3233/ASY-211754

Citation: Asymptotic Analysis, vol. 130, no. 3-4, pp. 335-365, 2022

Authors: Cesbron, L. | Mellet, A. | Puel, M.

Article Type: Research Article

Abstract: We investigate the fractional diffusion approximation of a kinetic equation set in a bounded interval with diffusive reflection conditions at the boundary. In an appropriate singular limit corresponding to small Knudsen number and long time asymptotic, we show that the asymptotic density function is the unique solution of a fractional diffusion equation with Neumann boundary condition. This analysis completes a previous work by the same authors in which a limiting fractional diffusion equation was identified on the half-space, but the uniqueness of the solution (which is necessary to prove the convergence of the whole sequence) could not be established.

Keywords: Kinetic theory, fractional diffusion, boundary conditions

DOI: 10.3233/ASY-221755

Citation: Asymptotic Analysis, vol. 130, no. 3-4, pp. 367-386, 2022

Authors: Mourad, Ayman | Taha, Zahraa

Article Type: Research Article

Abstract: Our aim in this paper is to prove the existence to a Cahn–Hilliard equation with a proliferation term and endowed with Neumann boundary conditions. Such a model has, in particular, applications in biology. We first consider regular nonlinear term then logarithmic one. We finally give some numerical simulations which confirm the theoretical results.

Keywords: Cahn-Hilliard equation, proliferation term, regular nonlinear term, logarithmic nonlinear term, existence, simulations

DOI: 10.3233/ASY-221756

Citation: Asymptotic Analysis, vol. 130, no. 3-4, pp. 387-408, 2022

Authors: Ellouz, Hanen | Feki, Ines | Jeribi, Aref

Article Type: Research Article

Abstract: In this paper, we are concerned with a 3 × 3 block operator matrices acting in a Banach or Hilbert space X 1 × X 2 × X 3 given by A 1 B 1 C 1 A 2 B 2 C 2 A 3 B 3 C …3 , where the linear entries are assumed to be unbounded. We study the closure as well as the self-adjointness in the case where the linear operators A 2 and A 3 are A 1 -bounded, B 1 and B 3 are B 2 -bounded and C 1 and C 2 are C 3 -bounded. These results are of importance to a non-relativistic three-channel potential scattering model. Show more

Keywords: Matrix operator, diagonal dominant, closed operators, self-adjoint operators, bounded from below, non-relativistic quantum mechanics, three-channel Hamiltonians

DOI: 10.3233/ASY-221757

Citation: Asymptotic Analysis, vol. 130, no. 3-4, pp. 409-426, 2022

Authors: Ozawa, Tohru | Tomioka, Kenta

Article Type: Research Article

Abstract: We study the vanishing dispersion limit of strong solutions to the Cauchy problem for the Schrödinger-improved Boussinesq system in a two dimensional domain. We show an explicit representation of limiting profile in terms of the initial data. Moreover, the first approximation is also represented as a pair of solutions of a linear system with coefficients and forcing term given by the limiting profile.

Keywords: Schrödinger-improved Boussinesq system, vanishing dispersion limit

DOI: 10.3233/ASY-221758

Citation: Asymptotic Analysis, vol. 130, no. 3-4, pp. 427-437, 2022

Authors: Zhang, Yang

Article Type: Research Article

Abstract: We construct microlocal solutions of Rayleigh and Stoneley waves in isotropic linear elasticity with the density and the Lamé parameters smooth up to a curved boundary or interface. We compute the direction of the microlocal polarization and show a retrograde elliptical motion of these two type of waves.

Keywords: Rayleigh waves, Stoneley waves, microlocal polarization

DOI: 10.3233/ASY-221759

Citation: Asymptotic Analysis, vol. 130, no. 3-4, pp. 439-475, 2022

Authors: Polyakov, Dmitry M.

Article Type: Research Article

Abstract: Two-term self-adjoint fourth-order differential operator with summable potential on the unit interval is considered. High energy eigenvalue asymptotics and the trace formula for this operator are obtained.

Keywords: Spectrum, eigenvalue asymptotics, fourth-order differential operator, trace formula

DOI: 10.3233/ASY-221760

Citation: Asymptotic Analysis, vol. 130, no. 3-4, pp. 477-503, 2022

Authors: Mel’nyk, Taras A. | Klevtsovskiy, Arsen V.

Article Type: Research Article

Abstract: A steady-state convection-diffusion problem with a small diffusion of order O ( ε ) is considered in a thin three-dimensional graph-like junction consisting of thin cylinders connected through a domain (node) of diameter O ( ε ) , where ε is a small parameter. Using multiscale analysis, the asymptotic expansion for the solution is constructed and justified. The asymptotic estimates in the norm of Sobolev space H 1 as well as in the uniform norm are proved for the difference between the solution and proposed approximations with a …predetermined accuracy with respect to the degree of ε . Show more

Keywords: Asymptotic expansion, convection-diffusion problem, asymptotic estimate, thin graph-like junction

DOI: 10.3233/ASY-221761

Citation: Asymptotic Analysis, vol. 130, no. 3-4, pp. 505-530, 2022

Authors: Cozzi, Elaine

Article Type: Research Article

Abstract: We consider a class of active scalar equations which includes, for example, the 2D Euler equations, the 2D Navier–Stokes equations, and various aggregation equations including the Keller–Segel model. For this class of equations, we establish uniqueness of solutions in the Zygmund space C ∗ 0 . This result improves upon that in (Trans. Amer. Math. Soc. 367 (2015) 3095–3118), where the authors show uniqueness of solutions in BMO . As a corollary of our methods, we establish the uniform in space vanishing viscosity limit of Hölder continuous solutions to the …aggregation equation with Newtonian potential. Show more

Keywords: Active scalars, inviscid limit

DOI: 10.3233/ASY-221762

Citation: Asymptotic Analysis, vol. 130, no. 3-4, pp. 531-551, 2022

Authors: Ghosh, Nibedita | Mahato, Hari Shankar

Article Type: Research Article

Abstract: We study a pore-scale model where two mobile species with different diffusion coefficients react and precipitate in the form of immobile species (crystal) on the surface of the solid parts in a porous medium. The reverse may also happen, i.e. the crystals may dissolute to give mobile species. The mathematical modeling of these processes will give rise to a coupled system of ordinary and partial differential equations. We first prove the existence of a unique nonnegative global weak solution and then upscale the model from microscale to macroscale.

Keywords: Reactive transport, diffusion–reaction–dissolution–precipitation, Langmuir reaction rates, global existence, homogenization, asymptotic analysis, two-scale convergence, boundary unfolding

DOI: 10.3233/ASY-221763

Citation: Asymptotic Analysis, vol. 130, no. 3-4, pp. 553-587, 2022