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The journal Asymptotic Analysis fulfills a twofold function. It aims at publishing original mathematical results in the asymptotic theory of problems affected by the presence of small or large parameters on the one hand, and at giving specific indications of their possible applications to different fields of natural sciences on the other hand.
Asymptotic Analysis thus provides mathematicians with a concentrated source of newly acquired information which they may need in the analysis of asymptotic problems.
Authors: Liu, Jitao | Wang, Shasha | Xu, Wen-Qing
Article Type: Research Article
Abstract: Recently, Niche [J. Differential Equations, 260 (2016), 4440–4453] established upper bounds on the decay rates of solutions to the 3D incompressible Navier–Stokes–Voigt equations in terms of the decay character r ∗ of the initial data in H 1 ( R 3 ) . Motivated by this work, we focus on characterizing the large-time behavior of all space-time derivatives of the solutions for the 2D case and establish upper bounds and lower bounds on their decay rates by making use of the decay character and Fourier splitting …methods. In particular, for the case − n 2 < r ∗ ⩽ 1 , we verify the optimality of the upper bounds, which is new to the best of our knowledge. Similar improved decay results are also true for the 3D case. Show more
Keywords: Incompressible Navier–Stokes–Voigt equations, decay characterization, Fourier splitting, large-time behavior
DOI: 10.3233/ASY-241900
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-27, 2024
Authors: Huo, Wenwen | Teng, Kaimin | Zhang, Chao
Article Type: Research Article
Abstract: We consider the Cauchy problem for the 3-D incompressible Navier–Stokes–Allen–Cahn system, which can effectively describe large deformations or topological deformations. Under the assumptions of small initial data, we study the global well-posedness and time-decay of solutions to such system by means of pure energy method and Fourier-splitting technique.
Keywords: Navier–Stokes–Allen–Cahn, global well-posedness, time-decay, Fourier-splitting
DOI: 10.3233/ASY-241901
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-21, 2024
Authors: Chakrabortty, Amartya | Griso, Georges | Orlik, Julia
Article Type: Research Article
Abstract: This paper focuses on the simultaneous homogenization and dimension reduction of periodic composite plates within the framework of non-linear elasticity. The composite plate in its reference (undeformed) configuration consists of a periodic perforated plate made of stiff material with holes filled by a soft matrix material. The structure is clamped on a cylindrical part. Two cases of asymptotic analysis are considered: one without pre-strain and the other with matrix pre-strain. In both cases, the total elastic energy is in the von-Kármán (vK) regime (ε 5 ). A new splitting of the displacements is introduced to …analyze the asymptotic behavior. The displacements are decomposed using the Kirchhoff–Love (KL) plate displacement decomposition. The use of a re-scaling unfolding operator allows for deriving the asymptotic behavior of the Green St. Venant’s strain tensor in terms of displacements. The limit homogenized energy is shown to be of vK type with linear elastic cell problems, established using the Γ-convergence. Additionally, it is shown that for isotropic homogenized material, our limit vK plate is orthotropic. The derived results have practical applications in the design and analysis of composite structures. Show more
Keywords: Homogenization, dimension reduction, unfolding operators, Γ-convergence, non-linear elasticity, von-Kármán plate, pre-strain
DOI: 10.3233/ASY-241896
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-56, 2024
Authors: Ganguly, Debdip | Gupta, Diksha | Sreenadh, K.
Article Type: Research Article
Abstract: We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space − Δ B N u − λ u = a ( x ) u p − 1 + ε u 2 ∗ − 1 in B N , u ∈ H 1 ( B N ) , where B N …denotes the hyperbolic space, 2 < p < 2 ∗ : = 2 N N − 2 , if N ⩾ 3 ; 2 < p < + ∞ , if N = 2 , λ < ( N − 1 ) 2 4 , and 0 < a ∈ L ∞ ( B N ) . We first prove the existence of a positive radially symmetric ground-state solution for a ( x ) ≡ 1 . Next, we prove that for a ( x ) ⩾ 1 , there exists a ground-state solution for ε small. For proof, we employ “conformal change of metric” which allows us to transform the original equation into a singular equation in a ball in R N . Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case a ( x ) ⩽ 1 is considered where we first show that there is no ground-state solution, and prove the existence of a bound-state solution (high energy solution) for ε small. We employ variational arguments in the spirit of Bahri–Li to prove the existence of high energy-bound-state solutions in the hyperbolic space. Show more
Keywords: Hyperbolic space, hyperbolic bubbles, Palais–Smale decomposition, semilinear elliptic problem
DOI: 10.3233/ASY-241895
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-29, 2024
Authors: Pelinovsky, Dmitry E. | Sobieszek, Szymon
Article Type: Research Article
Abstract: Ground state of the energy-critical Gross–Pitaevskii equation with a harmonic potential can be constructed variationally. It exists in a finite interval of the eigenvalue parameter. The supremum norm of the ground state vanishes at one end of this interval and diverges to infinity at the other end. We explore the shooting method in the limit of large norm to prove that the ground state is pointwise close to the Aubin–Talenti solution of the energy-critical wave equation in near field and to the confluent hypergeometric function in far field. The shooting method gives the precise dependence of the eigenvalue parameter versus …the supremum norm. Show more
Keywords: Gross–Pitaevskii equation, ground state, energy-critical case, shooting method
DOI: 10.3233/ASY-241897
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-29, 2024
Authors: Kassan, Mouna | Carbou, Gilles | Jazar, Mustapha
Article Type: Research Article
Abstract: In this paper, we establish the existence of global-in-time weak solutions for the Landau–Lifschitz–Gilbert equation with magnetostriction in the case of mixed boundary conditions. From this model, we derive by asymptotic method a two-dimensional model for thin ferromagnetic plates taking into account magnetostrictive effects.
Keywords: Ferromagnetism, magnetostriction, weak solutions, thin plates
DOI: 10.3233/ASY-241899
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-29, 2024
Authors: Deng, Ting | Squassina, Marco | Zhang, Jianjun | Zhong, Xuexiu
Article Type: Research Article
Abstract: We are concerned with solutions of the following quasilinear Schrödinger equations − div ( φ 2 ( u ) ∇ u ) + φ ( u ) φ ′ ( u ) | ∇ u | 2 + λ u = f ( u ) , x ∈ R N with prescribed mass ∫ R N u 2 d x = c , …where N ⩾ 3 , c > 0 , λ ∈ R appears as the Lagrange multiplier and φ ∈ C 1 ( R , R + ) . The nonlinearity f ∈ C ( R , R ) is allowed to be mass-subcritical, mass-critical and mass-supercritical at origin and infinity. Via a dual approach, the fixed point index and a global branch approach, we establish the existence of normalized solutions to the problem above. The results extend previous results by L. Jeanjean, J. J. Zhang and X.X. Zhong to the quasilinear case. Show more
Keywords: Quasilinear Schrödinger equations, normalized solutions, mass critical exponent
DOI: 10.3233/ASY-241908
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-20, 2024
Authors: Chipot, Michel
Article Type: Research Article
Abstract: The goal of this paper is to explore the asymptotic behaviour of anisotropic problems governed by operators of the pseudo p -Laplacian type when the size of the domain goes to infinity in different directions.
Keywords: Anisotropic operators, nonlinear elliptic operators, pseudo p-Laplacian, asymptotic behaviour, cylinder like domains
DOI: 10.3233/ASY-241906
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-27, 2024
Authors: Rawat, Rama | Roy, Haripada | Roy, Prosenjit
Article Type: Research Article
Abstract: The aim of this work is to characterize the asymptotic behaviour of the first eigenfunction of the generalised p -Laplace operator with mixed (Dirichlet and Neumann) boundary conditions in cylindrical domains when the length of the cylindrical domains tends to infinity. This generalises an earlier work of Chipot et al. (Asymptot. Anal. 85 (3–4) (2013) 199–227) where the linear case p = 2 is studied. Asymptotic behavior of all the higher eigenvalues of the linear case and the second eigenvalues of general case (using topological degree) for such problems is also studied.
Keywords: p-Laplacian, uniform elliptcity, Poincaré inequality, Krasnoselskii’s genus
DOI: 10.3233/ASY-241907
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-33, 2024
Authors: Kundu, A. | Mahato, H.S.
Article Type: Research Article
Abstract: We present an optimal control problem associated to a chemical transportation phenomena in a periodic porous medium. Posing controls on the porous part of the medium (distributed control), we set up a convex minimization problem. The main objective of this article is to characterize an arbitrary control to be an optimal control. We establish a relation between the optimal control and the corresponding adjoint state. At first, we analyse the microscopic description of the controlled system, then we homogenised the system by rigorous two-scale convergence method and periodic unfolding method.
Keywords: Diffusion–reaction–precipitation equations, optimal control problem, homogenisation, periodic porous medium, existence of solution, asymptotic expansion, two-scale convergence
DOI: 10.3233/ASY-241905
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-33, 2024
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