Asymptotic Analysis - Volume 60, issue 1-2

Authors: de Andrade, E.X.L. | Bracciali, C.F. | Sri Ranga, A.

Article Type: Research Article

Abstract: Inner products of the type 〈f, g〉S =〈f, g〉ψ0 +〈f′ , g′ 〉ψ1 , where one of the measures ψ0 or ψ1 is the measure associated with the Gegenbauer polynomials, are usually referred to as Gegenbauer–Sobolev inner products. This paper deals with some asymptotic relations for the orthogonal polynomials with respect to a class of Gegenbauer–Sobolev inner products. The inner products are such that the associated pairs of symmetric measures (ψ0 , ψ1 ) are not within the concept of symmetrically coherent pairs of measures.

Keywords: orthogonal polynomials, Sobolev orthogonal polynomials, asymptotics

DOI: 10.3233/ASY-2008-0891

Citation: Asymptotic Analysis, vol. 60, no. 1-2, pp. 1-14, 2008

Authors: Moulin, Simon

Article Type: Research Article

Abstract: We prove dispersive estimates at low frequency in dimensions n≥4 for the wave equation for a very large class of real-valued potentials, provided zero is neither an eigenvalue nor a resonance. This class includes potentials V∈L∞ (Rn ) satisfying V(x)=O(〈x〉−(n+1)/2−ε ), ε>0.

Keywords: dispersive estimates, wave equation

DOI: 10.3233/ASY-2008-0896

Citation: Asymptotic Analysis, vol. 60, no. 1-2, pp. 15-27, 2008

Authors: Dirr, Nicolas | Lucia, Marcello | Novaga, Matteo

Article Type: Research Article

Abstract: We consider the Γ-limit of a family of functionals which model the interaction of a material that undergoes phase transition with a rapidly oscillating conservative vector field. These functionals consist of a gradient term, a double-well potential and a vector field. The scaling is such that all three terms scale in the same way and the frequency of the vector field is equal to the interface thickness. Difficulties arise from the fact that the two global minimizers of the functionals are nonconstant and converge only in the weak L2 -topology.

Keywords: phase transitions, homogenization, Γ-convergence

DOI: 10.3233/ASY-2008-0897

Citation: Asymptotic Analysis, vol. 60, no. 1-2, pp. 29-59, 2008

Authors: Glass, O. | Guerrero, S.

Article Type: Research Article

Abstract: In this paper, we deal with controllability properties of linear and nonlinear Korteweg–de Vries equations in a bounded interval. The main part of this paper is a result of uniform controllability of a linear KdV equation in the limit of zero-dispersion. Moreover, we establish a result of null controllability for the linear equation via the left Dirichlet boundary condition, and of exact controllability via both Dirichlet boundary conditions. As a consequence, we obtain some local exact controllability results for the nonlinear KdV equation.

Keywords: controllability, KdV equation, zero dispersion limit

DOI: 10.3233/ASY-2008-0900

Citation: Asymptotic Analysis, vol. 60, no. 1-2, pp. 61-100, 2008

Authors: Guo, Zhenhua | Jiang, Song | Xie, Feng

Article Type: Research Article

Abstract: This paper is concerned with the existence of global weak solutions to the 1D compressible Navier–Stokes equations with density-dependent viscosity and initial density that is connected to vacuum with discontinuities. When the viscosity coefficient is proportional to ρθ with 0<θ<max{3−γ, 3/2} where ρ is the density, we prove a global existence theorem, improving thus the result in Meth. Appl. Anal. 12 (2005), 239–252, where 0<θ<1 is required. Moreover, we show that the domain occupied by the fluid expands into vacuum at an algebraic rate as time grows up due to the dispersion effect of the total pressure. It is …worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., θ=1, γ=2). Show more

Keywords: compressible Navier–Stokes equations, vacuum, existence, asymptotic behavior

DOI: 10.3233/ASY-2008-0902

Citation: Asymptotic Analysis, vol. 60, no. 1-2, pp. 101-123, 2008