Asymptotic Analysis - Volume 43, issue 4

Authors: Rodríguez, José M. | Taboada-Vázquez, Raquel

Article Type: Research Article

Abstract: In this paper, we study the Navier–Stokes equations in a domain with small depth. With this aim, we introduce a small adimensional parameter ε related to the depth. First we make a change of variable to a domain independent of ε and then we use asymptotic analysis to study what happens when ε becomes small. In this way we obtain a model for ε small that, after coming back to the original domain and without making a priori assumptions about velocity or pressure behavior, gives us a shallow water model including a new diffusion term.

Keywords: asymptotic analysis, shallow waters with viscosity

Citation: Asymptotic Analysis, vol. 43, no. 4, pp. 267-285, 2005

Authors: Demengel, Françoise

Article Type: Research Article

Abstract: In this paper we study necessary and sufficient conditions on f and the first eigenfunctions for the 1-Laplacian, for equations of the form \[\cases{-\mathrm{div}(\sigma)-\lambda =fu^{q-1},\quad u\geqslant 0\ \hbox{in}\ {\varOmega},\cr \noalign{\vskip3pt}\sigma\cdot \nabla u=|\nabla u|,\quad |\sigma|_{L^{\infty}({\varOmega})}\leqslant 1,\ u\in W_{0}^{1,1}({\varOmega})\cr}\] when q≤N/(N−1) and λ≥λ1 , λ1 is the first eigenvalue for the 1-Laplacian.

Citation: Asymptotic Analysis, vol. 43, no. 4, pp. 287-322, 2005

Authors: Hillairet, M.

Article Type: Research Article

Abstract: We consider the Burgers–Hopf equation on \[$\mathbb {R}$ outside a finite set of (moving) points coupled with transmission conditions in these points prescribing the dynamics of these points. We address the problem of asymptotic collisions. We give new proof of result in J.L. Vàzquez and E. Zuazua preprint, May 2004, and complementary results.

Keywords: fluid-solid interactions, long-time behavior, lack of collision

Citation: Asymptotic Analysis, vol. 43, no. 4, pp. 323-338, 2005

Authors: Leung, Anthony W.

Article Type: Research Article

Abstract: This article considers the dynamics of a coupled system of incompressible Navier–Stokes equations with second-order wave equations. The system may be used to approximate the interaction of ionized plasma particles with an electromagnetic field. Under appropriate assumptions and provided that the viscosity is sufficiently large, we prove the existence of an invariant manifold. Moreover, the manifold is attractive as t→+∞ for all close neighboring solutions.

Keywords: coupled parabolic–hyperbolic system, invariant manifold, Navier–Stokes equations, asymptotic stability, electro- magnetic waves

Citation: Asymptotic Analysis, vol. 43, no. 4, pp. 339-357, 2005

Article Type: Other

Citation: Asymptotic Analysis, vol. 43, no. 4, pp. 359-360, 2005