Asymptotic Analysis - Volume 32, issue 2

Authors: Hayashi, Nakao | Kaikina, Elena I. | Naumkin, Pavel I.

Article Type: Research Article

Abstract: We obtain the large time asymptotic expansion of small solutions to the Landau–Ginzburg type equations \[\cases{u_{t}-\alpha u_{xx}+\beta |u|^{2}u=0,\quad x\in {\mathbf{R}},\ t>0,\cr \noalign{\vskip3pt}u(0,x)=\phi (x),\quad x\in {\mathbf{R}},\cr}\] where α,β∈C,Rα>0, R(β/$\sqrt{2|\alpha |^{2}+\alpha ^{2}}$ )≥0, under the condition that the initial data ϕ are sufficiently small in a suitable weighted norm and the mean value ∫ϕ(x) dx≠0.

Keywords: dissipative nonlinear evolution equation, asymptotic expansions, Landau–Ginzburg equation

Citation: Asymptotic Analysis, vol. 32, no. 2, pp. 91-106, 2002

Authors: Hamouda, Makram

Article Type: Research Article

Abstract: In this article, we derive the first term in the asymptotic expansion of the regularised minimal surface solution in the radially symmetric case when the domain is a pair of concentric circles in R2 . General domains and time‐dependent problems will be considered elsewhere.

Keywords: minimal surfaces, singular perturbations, asymptotic analysis, boundary layer, small viscosity

Citation: Asymptotic Analysis, vol. 32, no. 2, pp. 107-130, 2002

Authors: Buscaglia, G. | Ciuperca, I. | Jai, M.

Article Type: Research Article

Abstract: State‐of‐the‐art magnetic storage devices have head‐to‐disk distances of about 300 Angstrom, for which compressibility, slip‐flow and roughness effects are significant. Since the head and the disk are in relative motion, the air‐gap thickness when both surfaces are rough varies rapidly in both space and time. A rigorous homogenization of the transient compressible Reynolds equation appropriate for such situation is presented in this article. If ε is the roughness length and pε the pressure field for that roughness, the existence of p0 ∈L2 (0,T,H1 (Ω)) such that pε →p0 strongly in L2 (Ω×]0,T[) when ε→0 is proved. A …homogenized problem for p0 is introduced together with a uniqueness result under remarkably weak assumptions, i.e., p0 ∈L2 (0,T,H1 (Ω)) and ∂p0 /∂t∈L2 (0,T,H−1 (Ω)). Interestingly, no time derivatives appear in the auxiliary local problems, which are thus computed as in the steady state case. The role of the time variable is to parameterize the relative positions of the roughness shapes, and the homogenized coefficients result from averaging all such positions. To our knowledge, this is the first rigorous treatment of lubrication problems accounting for roughness on both surfaces. Show more

Citation: Asymptotic Analysis, vol. 32, no. 2, pp. 131-152, 2002

Authors: Bouchitté, Guy | Bellieud, Michel

Article Type: Research Article

Abstract: We study the effective properties of an elastic composite medium under the assumption of small deformations. This composite is made of a periodic possibly disconnected subset filled up with a strong material surrounded by another material whose elastic coefficients are very small. The effective macroscopic behaviour obtained by homogenization turns out to be nonlocal and depends highly on the geometry of the strong component. In particular, second order derivative of the displacement appear in the limit energy when disconnected fibers are considered. These results were announced in [8].

Keywords: homogenization, linear elasticity, fiber reinforced structures, two‐scale convergence, Γ‐convergence

Citation: Asymptotic Analysis, vol. 32, no. 2, pp. 153-183, 2002