Asymptotic Analysis - Volume 124, issue 3-4

Authors: Zhu, Yichun

Article Type: Research Article

Abstract: In this paper, we generalize the classical Freidlin-Wentzell’s theorem for random perturbations of Hamiltonian systems. In (Probability Theory and Related Fields 128 (2004 ) 441–466), M.Freidlin and M.Weber generalized the original result in the sense that the coefficient for the noise term is no longer the identity matrix but a state-dependent matrix and taking the drift term into consideration. In this paper, We generalize the result by adding a state-dependent matrix that converges uniformly to 0 on any compact sets as ϵ tends to 0 to a state-dependent noise and considering the drift term which contains two …parts, the state-dependent mapping and a state-dependent mapping that converges uniformly to 0 on any compact sets as ϵ tends to 0. In the proof, we adapt a new way to prove the weak convergence inside the edge by constructing an auxiliary process and modify the proof in (Probability Theory and Related Fields 128 (2004 ) 441–466) when proving gluing condition. Show more

Keywords: Freidlin–Wentcell theorem, Hamiltonian system, averaging principle

DOI: 10.3233/ASY-201641

Citation: Asymptotic Analysis, vol. 124, no. 3-4, pp. 199-233, 2021

Authors: Alouini, Brahim

Article Type: Research Article

Abstract: In the current issue, we consider a system of N-coupled weakly dissipative fractional Schrödinger equations with cubic nonlinearities. We will prove that the asymptotic dynamics of the solutions will be described by the existence of a regular compact global attractor with finite fractal dimension.

Keywords: Schrödinger equation, asymptotic behavior, global attractor, fractal dimension

DOI: 10.3233/ASY-201643

Citation: Asymptotic Analysis, vol. 124, no. 3-4, pp. 235-258, 2021

Authors: Jaffal-Mourtada, Basma

Article Type: Research Article

Abstract: We consider the equations of a rotating incompressible non-Newtonian fluid flow of grade two in a three dimensional torus. We prove two different results of global existence of strong solutions. In the first case, we consider that the elasticity coefficient α is arbitrary and we suppose that the third components of the vertical average of the initial data and of the forcing term are small compared to the horizontal components. In the second case, we consider a forcing term and initial data of arbitrary size but we restrict the size of α . In both cases, we show that …the limit system is composed of a linear system and a second grade fluid system with two variables and three components. Show more

Keywords: Non-newtonian fluid, second grade fluid, global existence, rotating fluid, limit system

DOI: 10.3233/ASY-201644

Citation: Asymptotic Analysis, vol. 124, no. 3-4, pp. 259-290, 2021

Authors: Liu, Zhenhai | Papageorgiou, Nikolaos S.

Article Type: Research Article

Abstract: We consider a Dirichlet double phase problem with unbalanced growth. In the reaction we have the combined effects of a critical term and of a locally defined Carathéodory perturbation. Using cut-off functions and truncation techniques we bypass the critical term and deal with a coercive problem. Using this auxillary problem, we show that the original Dirichlet equation has a whole sequence of nodal (sign-changing) solutions which converge to zero in the Musielak–Orlice–Sobolev space and in L ∞ .

Keywords: Double phase integrand, unbalanced growth, Musielak–Orlice spaces critical term, nodal solutions, cut-off function

DOI: 10.3233/ASY-201645

Citation: Asymptotic Analysis, vol. 124, no. 3-4, pp. 291-302, 2021

Authors: Fukao, Takeshi | Wu, Hao

Article Type: Research Article

Abstract: We consider a class of Cahn–Hilliard equation that models phase separation process of binary mixtures involving nontrivial boundary interactions in a bounded domain with non-permeable wall. The system is characterized by certain dynamic type boundary conditions and the total mass, in the bulk and on the boundary, is conserved for all time. For the case with physically relevant singular (e.g., logarithmic) potential, global regularity of weak solutions is established. In particular, when the spatial dimension is two, we show the instantaneous strict separation property such that for arbitrary positive time any weak solution stays away from the pure phases ± …1 , while in the three dimensional case, an eventual separation property for large time is obtained. As a consequence, we prove that every global weak solution converges to a single equilibrium as t → ∞ , by the usage of an extended Łojasiewicz–Simon inequality. Show more

Keywords: Cahn–Hilliard equation, dynamic boundary condition, singular potential, separation from pure states, convergence to equilibrium

DOI: 10.3233/ASY-201646

Citation: Asymptotic Analysis, vol. 124, no. 3-4, pp. 303-341, 2021

Authors: Debyaoui, Mohamed Ali | Ersoy, Mehmet

Article Type: Research Article

Abstract: In this paper, we present a new non-linear dispersive model for open channel and river flows. These equations are the second-order shallow water approximation of the section-averaged (three-dimensional) incompressible and irrotational Euler system. This new asymptotic model generalises the well-known one-dimensional Serre–Green–Naghdi (SGN) equations for rectangular section on uneven bottom to arbitrary channel/river section.

Keywords: Open channel flow, river flow, Euler equations, asymptotic approximation, Serre–Green–Naghdi equations, free surface shallow water equations, non-hydrostatic pressure, dispersive

DOI: 10.3233/ASY-201647

Citation: Asymptotic Analysis, vol. 124, no. 3-4, pp. 343-369, 2021

Authors: Isernia, Teresa | Repovš, Dušan D.

Article Type: Research Article

Abstract: We consider the following ( p , q ) -Laplacian Kirchhoff type problem − ( a + b ∫ R 3 | ∇ u | p d x ) Δ p u − ( c + d ∫ R 3 | ∇ u | q d x ) Δ q u + V ( x ) ( | u | p − …2 u + | u | q − 2 u ) = K ( x ) f ( u ) in  R 3 , where a , b , c , d > 0 are constants, 3 2 < p < q < 3 , V : R 3 → R and K : R 3 → R are positive continuous functions allowed for vanishing behavior at infinity, and f is a continuous function with quasicritical growth. Using a minimization argument and a quantitative deformation lemma we establish the existence of nodal solutions. Show more

Keywords: (p, q)-Kirchhoff, nodal solutions, vanishing potentials, Nehari manifold

DOI: 10.3233/ASY-201648

Citation: Asymptotic Analysis, vol. 124, no. 3-4, pp. 371-396, 2021