Asymptotic Analysis - Volume 126, issue 3-4

Authors: Maione, Alberto | Salort, Ariel M. | Vecchi, Eugenio

Article Type: Research Article

Abstract: In this note we prove the validity of the Maz’ya–Shaposhnikova formula in magnetic fractional Orlicz–Sobolev spaces. This complements a previous asymptotic study performed by the second author in (Studia Mathematica (2020 )).

Keywords: Asymptotic behaviours, Orlicz spaces, fractional Sobolev norms

DOI: 10.3233/ASY-211677

Citation: Asymptotic Analysis, vol. 126, no. 3-4, pp. 201-214, 2022

Authors: Petrov, Pavel S. | Ehrhardt, Matthias | Trofimov, Mikhail

Article Type: Research Article

Abstract: Recently, it was shown that the solution of the Helmholtz equation can be approximated by a series over the solutions of iterative parabolic equations (IPEs). An expansion of the fundamental solution of the Helmholtz equation over solutions of IPEs is considered. It is shown that the resulting Taylor-like series can be easily transformed into a Padé-type approximation. In practical propagation problems such iterative Padé approximations exhibit improved wide-angle capabilities and faster convergence to the solution of the Helmholtz equation in comparison to Taylor-like expansion over IPEs solutions. A Gaussian smoothing of the expansion terms gives insight into the derivation of …initial conditions consistent for IPEs, which can be used for point source simulation. A correct point source model consistent with the wide-angle one-way propagation equations is important in many practical applications of the parabolic equations theory. Show more

Keywords: Helmholtz equation, paraxial approximation, multiple scales, iterative parabolic equations, wide-angle parabolic equations, self-starter, pattern solver tool

DOI: 10.3233/ASY-211679

Citation: Asymptotic Analysis, vol. 126, no. 3-4, pp. 215-228, 2022

Authors: Chen, Zhuo | Ji, Chao

Article Type: Research Article

Abstract: In this paper, by using variational methods, we study the existence and concentration of ground state solutions for the following fractional Schrödinger equation ( − Δ ) α u + V ( x ) u = A ( ϵ x ) f ( u ) , x ∈ R N , where α ∈ ( 0 , 1 ) , ϵ is a positive parameter, N > 2 α , ( − Δ ) α …stands for the fractional Laplacian, f is a continuous function with subcritical growth, V ∈ C ( R N , R ) is a Z N -periodic function and A ∈ C ( R N , R ) satisfies some appropriate assumptions. Show more

Keywords: Concentration of solutions, ground state, fractional Schrödinger equations, periodic potential, Nehari manifold

DOI: 10.3233/ASY-211680

Citation: Asymptotic Analysis, vol. 126, no. 3-4, pp. 229-253, 2022

Authors: Chauleur, Quentin

Article Type: Research Article

Abstract: We construct global dissipative solutions on the torus of dimension at most three of the defocusing isothermal Euler–Langevin–Korteweg system, which corresponds to the Euler–Korteweg system of compressible quantum fluids with an isothermal pressure law and a linear drag term with respect to the velocity. In particular, the isothermal feature prevents the energy and the BD-entropy from being positive. Adapting standard approximation arguments we first show the existence of global weak solutions to the defocusing isothermal Navier–Stokes–Langevin–Korteweg system. Introducing a relative entropy function satisfying a Gronwall-type inequality we then perform the inviscid limit to obtain the existence of dissipative solutions of …the Euler–Langevin–Korteweg system. Show more

Keywords: Euler–Langevin–Korteweg system, Navier–Stokes–Langevin–Korteweg system, relative entropy estimates, dissipative solutions, augmented systems

DOI: 10.3233/ASY-211681

Citation: Asymptotic Analysis, vol. 126, no. 3-4, pp. 255-283, 2022

Authors: Santra, Sanjiban

Article Type: Research Article

Abstract: We prove the existence and the limit profile of the least energy solution of a half Laplacian equation with competing powers.

Keywords: Fractional Laplacian, existence, blow-up analysis

DOI: 10.3233/ASY-211682

Citation: Asymptotic Analysis, vol. 126, no. 3-4, pp. 285-302, 2022

Authors: Hernández-Llanos, Pedro

Article Type: Research Article

Abstract: In this article we obtain a 1-dimensional asymptotic model for a junction of thin hyperelastic rods as the thickness goes to zero. We show, under appropriate hypotheses on the loads, that the deformations that minimize the total energy weakly converge in a Sobolev space towards the minimum of a 1 D -dimensional energy for elastic strings by using techniques from Γ-convergence.

Keywords: Junctions, thin structures, hyperelasticity, nonlinear elasticity, thin rods, asymptotic analysis

DOI: 10.3233/ASY-211683

Citation: Asymptotic Analysis, vol. 126, no. 3-4, pp. 303-322, 2022

Authors: Gendron, Germain

Article Type: Research Article

Abstract: In this paper, we study an inverse Steklov problem in a class of n -dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric. Precisely, we aim at studying the continuous dependence of the warping function defining the warped product with respect to the Steklov spectrum. We first show that the knowledge of the Steklov spectrum up to an exponential decreasing error is enough to determine uniquely the warping function in a neighbourhood of the boundary. Second, when the warping functions are symmetric with respect to 1/2, we prove a log-type stability estimate in …the inverse Steklov problem. As a last result, we prove a log-type stability estimate for the corresponding Calderón problem. Show more

Keywords: Inverse Calderón problem, Steklov spectrum, Weyl–Titchmarsh functions, Nevanlinna theorem, Müntz–Jackson’s theorem

DOI: 10.3233/ASY-211684

Citation: Asymptotic Analysis, vol. 126, no. 3-4, pp. 323-377, 2022

Authors: Gariboldi, Claudia | Takahashi, Takéo

Article Type: Research Article

Abstract: We consider an optimal control problem for the Navier–Stokes system with Navier slip boundary conditions. We denote by α the friction coefficient and we analyze the asymptotic behavior of such a problem as α → ∞ . More precisely, we prove that if we take an optimal control for each α , then there exists a sequence of optimal controls converging to an optimal control of the same optimal control problem for the Navier–Stokes system with the Dirichlet boundary condition. We also show the convergence of the corresponding direct and adjoint states.

Keywords: Optimal control, Navier–Stokes system, Navier slip boundary condition

DOI: 10.3233/ASY-211685

Citation: Asymptotic Analysis, vol. 126, no. 3-4, pp. 379-399, 2022