Asymptotic Analysis - Volume Pre-press, issue Pre-press

Authors: Tachim Medjo, T. | Tone, C. | Tone, F.

Article Type: Research Article

Abstract: In this article we study the long-time dynamics of the dynamical system generated by the Ericksen–Leslie model. More precisely, we discretize the Ericksen–Leslie equations in time using the implicit Euler scheme, and with the aid of the discrete Gronwall lemmas we prove that the scheme is uniformly bounded. Moreover, using the theory of the multi-valued attractors we prove in a particular case the convergence of the global attractors generated by the numerical scheme to the global attractor of the continuous system as the time-step approaches zero.

Keywords: Ericksen–Leslie equations, implicit Euler scheme, long-time stability, attractors

DOI: 10.3233/ASY-241941

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-29, 2024

Authors: Belhachmi, Zakaria | Jacumin, Thomas

Article Type: Research Article

Abstract: We consider a shape optimization based method for finding the best interpolation data in the compression of images with noise. The aim is to reconstruct missing regions by means of minimizing a data fitting term in an L p -norm, for 1 ⩽ p < + ∞ , between original images and their reconstructed counterparts using linear diffusion PDE-based inpainting. Reformulating the problem as a constrained optimization over sets (shapes), we derive the topological asymptotic expansion of the considered shape functionals with respect to the insertion of small ball (a single pixel) using …the adjoint method. Based on the achieved distributed topological shape derivatives, we propose a numerical approach to determine the optimal set and present numerical experiments showing the efficiency of our method. Numerical computations are presented that confirm the usefulness of our theoretical findings for PDE-based image compression. Show more

Keywords: Image compression, shape optimization, adjoint method, image interpolation, inpainting, PDEs, image denoising

DOI: 10.3233/ASY-241944

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-28, 2024

Authors: Hajaiej, Hichem | Kumar, Rohit | Mukherjee, Tuhina | Song, Linjie

Article Type: Research Article

Abstract: This article focuses on the existence and non-existence of solutions for the following system of local and nonlocal type − ∂ x x u + ( − Δ ) y s 1 u + u − u 2 s 1 − 1 = κ α h ( x , y ) u α − 1 v β in  R 2 …, − ∂ x x v + ( − Δ ) y s 2 v + v − v 2 s 2 − 1 = κ β h ( x , y ) u α v β − 1 in  R 2 , u , v ⩾ 0 in  R 2 , where s 1 , s 2 ∈ ( 0 , 1 ) , α , β > 1 , α + β ⩽ min { 2 s 1 , 2 s 2 } , and 2 s i = 2 ( 1 + s i ) 1 − s i , i = 1 , 2 . The existence of a ground state solution entirely depends on the behaviour of the parameter κ > 0 and on the function h . In this article, we prove that a ground state solution exists in the subcritical case if κ is large enough and h satisfies (H ). Further, if κ becomes very small, then there is no solution to our system. The study of the critical case, i.e., s 1 = s 2 = s , α + β = 2 s , is more complex, and the solution exists only for large κ and radial h satisfying (H1 ). Finally, we establish a Pohozaev identity which enables us to prove the non-existence results under some smooth assumptions on h . Show more

Keywords: Mixed Schrödinger operator, system of PDEs in plane, variational methods, concentration-compactness, non-existence

DOI: 10.3233/ASY-241922

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-36, 2024

Authors: Ding, Jiajia | Estrada, Ricardo | Yang, Yunyun

Article Type: Research Article

Abstract: In this article we continue our research in (Yang and Estrada in Asymptot. Anal. 95 (1–2) (2015 ) 1–19), about the asymptotic expansion of thick distributions. We compute more examples of asymptotic expansion of integral transforms using the techniques developed in (Yang and Estrada in Asymptot. Anal. 95 (1–2) (2015 ) 1–19). Besides, we derive a new “Laplace Formula” for the situation in which a point singularity is allowed.

Keywords: Distributions, asymptotic expansion, singular integral, Laplace formula, thick distributions

DOI: 10.3233/ASY-241924

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-11, 2024

Authors: Schäffner, M. | Schweizer, B.

Article Type: Research Article

Abstract: The wave equation with stochastic rapidly oscillating coefficients can be classically homogenized on bounded time intervals; solutions converge in the homogenization limit to solutions of a wave equation with constant coefficients. This is no longer true on large time scales: Even in the periodic case with periodicity ε , classical homogenization fails for times of the order ε − 2 . We consider the one-dimensional wave equation with random rapidly oscillating coefficients on scale ε and are interested in the critical time scale ε − β from where …on classical homogenization fails. In the general setting, we derive upper and lower bounds for β in terms of the growth rate of correctors. In the specific setting of i.i.d. coefficients with matched impedance, we show that the critical time scale is ε − 1 . Show more

Keywords: Stochastic homogenization, wave equation

DOI: 10.3233/ASY-241923

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-39, 2024

Authors: Kinra, Kush | Mohan, Manil T.

Article Type: Research Article

Abstract: In this work, we consider the two-dimensional Oldroyd model for the non-Newtonian fluid flows (viscoelastic fluid) in Poincaré domains (bounded or unbounded) and study their asymptotic behavior. We establish the existence of a global attractor in Poincaré domains using asymptotic compactness property. Since the high regularity of solutions is not easy to establish, we prove the asymptotic compactness of the solution operator by applying Kuratowski’s measure of noncompactness, which relies on uniform-tail estimates and the flattening property of the solution. Finally, the estimates for the Hausdorff as well as fractal dimensions of global attractors are also obtained.

Keywords: Oldroyd fluids, Global attractors, Kuratowski’s measure of noncompactness, uniform-tail estimates, flattening property, fractal and Hausdorff dimensions

DOI: 10.3233/ASY-241932

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-27, 2024

Authors: Giacomoni, J. | Nidhi, Nidhi | Sreenadh, K.

Article Type: Research Article

Abstract: In this paper we study the existence and regularity results of normalized solutions to the following critical growth Choquard equation with mixed diffusion type operators: − Δ u + ( − Δ ) s u = λ u + g ( u ) + ( I α ∗ | u | 2 α ∗ ) | u | 2 α ∗ − 2 u in  R N , ∫ R …N | u | 2 d x = τ 2 , where N ⩾ 3 , τ > 0 , I α is the Riesz potential of order α ∈ ( 0 , N ) , ( − Δ ) s is the fractional laplacian operator, 2 α ∗ = N + α N − 2 is the critical exponent with respect to the Hardy Littlewood Sobolev inequality, λ appears as a Lagrange multiplier and g is a real valued function satisfying some L 2 -supercritical conditions. Show more

Keywords: Normalized solutions, Choquard equation, critical growth, local and nonlocal operator, L2-supercritical growth, existence results, Sobolev regularity

DOI: 10.3233/ASY-241933

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-34, 2024

Authors: Hernández-Santamaría, Víctor | Peña-García, Alberto

Article Type: Research Article

Abstract: The shadow limit is a versatile tool used to study the reduction of reaction-diffusion systems into simpler PDE-ODE models by letting one of the diffusion coefficients tend to infinity. This reduction has been used to understand different qualitative properties and their interplay between the original model and its reduced version. The aim of this work is to extend previous results about the controllability of linear reaction-diffusion equations and how this property is inherited by the corresponding shadow model. Defining a suitable class of nonlinearities and improving some uniform Carleman estimates, we extend the results to the semilinear case and prove …that the original model is null-controllable and that the shadow limit preserves this important feature. Show more

Keywords: Shadow limit, semilinear reaction-diffusion equations, uniform null-controllability, Carleman estimates

DOI: 10.3233/ASY-241930

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-39, 2024

Authors: Coclite, Giuseppe Maria | di Ruvo, Lorenzo

Article Type: Research Article

Abstract: The wave propagation in dilatant granular materials is described by a nonlinear evolution equation of the fifth order deduced by Giovine–Oliveri in (Meccanica 30 (4) (1995 ) 341–357). In this paper, we study the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.

Keywords: Existence, Uniqueness, Stability, Wave propagation, Dilatant granular materials, Cauchy problem

DOI: 10.3233/ASY-241920

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-28, 2024

Authors: Shen, Liejun | Squassina, Marco

Article Type: Research Article

Abstract: We consider the existence of ground state solutions for a class of zero-mass Chern–Simons–Schrödinger systems − Δ u + A 0 u + ∑ j = 1 2 A j 2 u = f ( u ) − a ( x ) | u | p − 2 u , ∂ 1 A 2 − ∂ 2 A 1 = − 1 2 | u …| 2 , ∂ 1 A 1 + ∂ 2 A 2 = 0 , ∂ 1 A 0 = A 2 | u | 2 , ∂ 2 A 0 = − A 1 | u | 2 , where a : R 2 → R + is an external potential, p ∈ ( 1 , 2 ) and f ∈ C ( R ) denotes the nonlinearity that fulfills the critical exponential growth in the Trudinger–Moser sense at infinity. By introducing an improvement of the version of Trudinger–Moser inequality approached in (J. Differential Equations 393 (2024 ) 204–237), we are able to investigate the existence of positive ground state solutions for the given system using variational method. Show more

Keywords: Zero-mass, Chern–Simons–Schrödinger system, Trudinger–Moser inequality, Critical exponential growth, Ground state solution, Variational method

DOI: 10.3233/ASY-241921

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-25, 2024