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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: 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: 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: Poblete, Felipe | Silva, Clessius | Viana, Arlúcio
Article Type: Research Article
Abstract: This paper studies the existence of local and global self-similar solutions for a Boussinesq system with fractional memory and fractional diffusions u t + u · ∇ u + ∇ p + ν ( − Δ ) β u = θ f , x ∈ R n , t > 0 , θ t + u · ∇ θ + g α ∗ ( − Δ ) γ θ …= 0 , x ∈ R n , t > 0 , div u = 0 , x ∈ R n , t > 0 , u ( x , 0 ) = u 0 , θ ( x , 0 ) = θ 0 , x ∈ R n , where g α ( t ) = t α − 1 Γ ( α ) . The existence results are obtained in the framework of pseudo-measure spaces. We find that the existence and self-similarity of global solutions is strongly influenced by the relationship among the memory and the fractional diffusions. Indeed, we obtain the existence and self-similarity of global solutions only when γ = ( α + 1 ) β . Moreover, we prove a stability result for the global solutions and the existence of asymptotically self-similar solutions. Show more
Keywords: Nonlocal Navier–Stokes, Boussinesq system, PDEs in connection with fluid mechanics, fractional memory, self-similarity
DOI: 10.3233/ASY-241904
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-25, 2024
Authors: Liu, Chungen | Zhong, Yuyou | Zuo, Jiabin
Article Type: Research Article
Abstract: In this paper, we study a fractional Schrödinger–Poisson system with p -Laplacian. By using some scaling transformation and cut-off technique, the boundedness of the Palais–Smale sequences at the mountain pass level is gotten. As a result, the existence of non-trivial solutions for the system is obtained.
Keywords: Fractional Schrödinger–Poisson system, p-Laplacian, mountain pass lemma
DOI: 10.3233/ASY-241903
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-17, 2024
Authors: Nika, Grigor | Muntean, Adrian
Article Type: Research Article
Abstract: We derive effective models for a heterogeneous second-gradient elastic material taking into account chiral scale-size effects. Our classification of the effective equations depends on the hierarchy of four characteristic lengths: The size of the heterogeneities ℓ , the intrinsic lengths of the constituents ℓ SG and ℓ chiral , and the overall characteristic length of the domain L. Depending on the different scale interactions between ℓ SG , ℓ chiral , ℓ , and L we obtain either an …effective Cauchy continuum or an effective second-gradient continuum. The working technique combines scaling arguments with the periodic homogenization asymptotic procedure. Both the passage to the homogenization limit and the unveiling of the correctors’ structure rely on a suitable use of the periodic unfolding operator. Show more
Keywords: Second-gradient elasticity, scale-size effects, partial scale separation, chirality, multi-continuum homogenization
DOI: 10.3233/ASY-241902
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-27, 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: Da Silva, João Pablo P.
Article Type: Research Article
Abstract: In this work, we consider a functional I : W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) → R of the form I ( u , v ) = 1 p ∫ Ω ( | ∇ u | p + | ∇ v | p ) d x − ∫ Ω H …( x , u ( x ) , v ( x ) ) d x where Ω ⊂ R N is a smooth bounded domain, max { | ∂ s H ( x , s , t ) | , | ∂ t H ( x , s , t ) | } ⩽ C ( 1 + | s | σ 1 − 1 + | t | σ 2 − 1 ) a.e. x ∈ Ω , for some C > 0 , ∀ t , s ∈ R , p < σ i ⩽ p ∗ : = N p / ( N − p ) , i = 1 , 2 , and 1 < p < N . We prove that a local minimum in the topology of C 0 1 ( Ω ) × C 0 1 ( Ω ) is a local minimum in the topology of W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) . An important application of this result is related to the question of multiplicity of solutions for a class of systems with concave-convex type nonlinearities. Show more
Keywords: Local minimization, p-Laplacian system, Sub-super solution method
DOI: 10.3233/ASY-241911
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-18, 2024
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