Asymptotic Analysis - Volume 95, issue 1-2

Authors: Yang, Yunyun | Estrada, Ricardo

Article Type: Research Article

Abstract: We give a theory of asymptotic expansions of thick distributions of rapid decay at infinity. We show that the moment asymptotic expansion of standard distributions of rapid decay follows by projection of our result. We also study in which spaces of thick test functions the asymptotic Taylor approximation is valid. We employ our formulas to obtain the asymptotic expansion of several multidimensional integrals that are divergent but are regularized by using the Hadamard method.

Keywords: thick points, delta functions, distributions, generalized functions, asymptotic expansions

DOI: 10.3233/ASY-151310

Citation: Asymptotic Analysis, vol. 95, no. 1-2, pp. 1-19, 2015

Authors: Ammari, Kaïs | Nicaise, Serge | Pignotti, Cristina

Article Type: Research Article

Abstract: In this paper we consider a stabilization problem for an abstract wave equation with delay and a Kelvin–Voigt damping. We prove an exponential stability result for appropriate damping coefficients. The proof of the main result is based on a frequency-domain approach.

Keywords: internal stabilization, Kelvin–Voigt damping, abstract wave equation with delay

DOI: 10.3233/ASY-151317

Citation: Asymptotic Analysis, vol. 95, no. 1-2, pp. 21-38, 2015

Authors: Salort, Ariel | Terra, Joana | Wolanski, Noemi

Article Type: Research Article

Abstract: In this paper we continue our study of the large time behavior of the bounded solution to the nonlocal diffusion equation with absorption u t = L u − u p in  R N × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ( x ) in  R N , where p > 1 , u 0 …⩾ 0 and bounded and L u ( x , t ) = ∫ J ( x − y ) ( u ( y , t ) − u ( x , t ) ) d y with J ∈ C 0 ∞ ( B d ) , radially symmetric, J > 0 in B d , with ∫ J = 1 . Our assumption on the initial datum is that 0 ⩽ u 0 ∈ L ∞ ( R N ) and | x | α u 0 ( x ) → A > 0 as  | x | → ∞ . This problem was studied in [Proc. Amer. Math. Soc. 139 (4) (2011), 1421–1432; Discrete Cont. Dyn. Syst. A , 31 (2) (2011), 581–605] in the supercritical and critical cases p ⩾ 1 + 2 / α . In the present paper we study the subcritical case 1 < p < 1 + 2 / α . More generally, we consider bounded nonnegative initial data such that | x | 2 / ( p − 1 ) u 0 ( x ) → ∞ as  | x | → ∞ and prove that t 1 / ( p − 1 ) u ( x , t ) → ( 1 p − 1 ) 1 / ( p − 1 ) as  t → ∞ uniformly in | x | ⩽ k t for every k > 0 . Of independent interest is our study of the positive eigenfunction of the operator L in the ball B R in the L ∞ setting that we include in Section 3. Show more

Keywords: nonlocal diffusion, large time behavior

DOI: 10.3233/ASY-151320

Citation: Asymptotic Analysis, vol. 95, no. 1-2, pp. 39-57, 2015

Authors: Zakrevskiy, Timofey

Article Type: Research Article

Abstract: We are interested in the connection between kinetic models with Fermi–Dirac statistics and fluid dynamics. We establish that moments and parameters of Fermi–Dirac distributions are related by a diffeomorphism. We obtain the macroscopic limits when the fluid is dense enough that particles undergo many collisions per unit of time. This situation is described via a small parameter ε , called the Knudsen number, that represents the ratio of mean free path of particles between collisions to some characteristic length of the flow. We give the conditions that allow us to formally derive the generalized Euler equations from the Boltzmann equation …by adopting the formalism proposed in [Advances in Kinetic Theory and Continuum Mechanics , Springer, Berlin, 1991, pp. 57–71]. These conditions are related to the H -theorem and assume a formally consistent convergence for fluid dynamical moments and entropy of the kinetic equation. We also discuss the well-posedness of the obtained Euler equations by using Godunov’s criterion of hyperbolicity. Show more

Keywords: kinetic model, Boltzmann equation, Euler equations, entropy

DOI: 10.3233/ASY-151323

Citation: Asymptotic Analysis, vol. 95, no. 1-2, pp. 59-77, 2015

Authors: Mosconi, Sunra | Shioji, Naoki | Squassina, Marco

Article Type: Research Article

Abstract: We prove the existence of a positive solution for nonlocal problems involving the fractional Laplacian and a critical growth power nonlinearity when the equation is set in a suitable contractible domain.

Keywords: fractional equation, critical embedding, contractible domains, existence

DOI: 10.3233/ASY-151324

Citation: Asymptotic Analysis, vol. 95, no. 1-2, pp. 79-100, 2015

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

Article Type: Research Article

Abstract: In this article we consider the implicit Euler scheme for a homogeneous two-phase flow model in a two-dimensional domain and with the aid of the discrete Gronwall lemma and of the discrete uniform Gronwall lemma we prove that the global attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero.

Keywords: semi-implicit scheme, long-time stability, incompressible two-phase flow, discrete attractors

DOI: 10.3233/ASY-151325

Citation: Asymptotic Analysis, vol. 95, no. 1-2, pp. 101-127, 2015

Authors: Aouadi, Moncef | Miranville, Alain

Article Type: Research Article

Abstract: A nonlinear system of sixth-order evolution equations which takes into account the hereditary effects via Gurtin–Pipkin’s model and the rotational inertia is considered. The system describes the behavior of thermoelastic diffusion thin plates, recently derived by Aouadi [Applied Mathematics and Mechanics (English Edition) 36 (2015), 619–632], where the heat and diffusion fluxes depend on the past history of the temperature and diffusion gradients through memory kernels, respectively. We prove the existence and uniqueness of global solutions as well as the exponential stability of the linear system at a rate proportional to the rotational inertia parameter. The existence of …a global attractor whose fractal dimension is finite is proved. Finally, a smoothness property of the attractor is established with respect to the rotational inertia parameter. Show more

Keywords: thermoelastic diffusion plate, Gurtin–Pipkin’s model, exponential stability, global attractor

DOI: 10.3233/ASY-151330

Citation: Asymptotic Analysis, vol. 95, no. 1-2, pp. 129-160, 2015

Authors: Jiang, Song | Li, Fucai

Article Type: Research Article

Abstract: The full compressible magnetohydrodynamic equations can be derived formally from the complete electromagnetic fluid system in some sense as the dielectric constant tends to zero. This process is usually referred as magnetohydrodynamic approximation in physical books. In this paper we justify this singular limit rigorously in the framework of smooth solutions for well-prepared initial data.

Keywords: complete electromagnetic fluid system, full compressible magnetohydrodynamic equations, zero dielectric constant limit

DOI: 10.3233/ASY-151321

Citation: Asymptotic Analysis, vol. 95, no. 1-2, pp. 161-185, 2015