Asymptotic Analysis - Volume 21, issue 2

Authors: Iftimie, Dragoş

Article Type: Research Article

Abstract: In this paper we show that the quasigeostrophic system is well approximated by the primitive systems. More precisely, we prove that if the initial data are weakly well‐prepared then the maximal time existence of the regular solution of the primitive system goes to infinity and the regular solution goes to the solution of the quasigeostrophic system, strongly on an arbitrary time interval. By weakly well‐prepared initial data we mean that the initial data of the primitive systems is converging to an initial data with zero oscillating part, without any assumptions on the speed.

Citation: Asymptotic Analysis, vol. 21, no. 2, pp. 89-97, 1999

Authors: Cerqueti, Katiuscia

Article Type: Research Article

Abstract: We consider the problem \cases{ - \Delta u = N(N-2) u^p +\varepsilon u & \mbox{in} $\varOmega$,\cr u >0 & \mbox{in} $\varOmega$,\cr u =0 & \mbox{on} $\curpartial\varOmega$,\cr} where \varOmega is a bounded smooth domain of \mathbf{R}^N \ (N \geq 5) which is symmetric with respect to the coordinate hyperplanes \{x_i = 0\} , i = 1, \ldots, N , and it is convex in the x_i ‐directions for i = 1, \ldots, N ; here 0 < \varepsilon < \lambda_1 (\lambda_1 …being the first eigenvalue of the Laplace operator in H_0^1 (\varOmega) ) and p = ({N+2})/({N-2}) . For \varepsilon small, we prove uniqueness and nondegeneracy of the solution u_\varepsilon with the property that \lim_{\varepsilon \rightarrow0} \frac{\int_\varOmega|\nabla u_\varepsilon|^2 \,\mathrm{d} x }{ (\int_\varOmega|u_\varepsilon|^{p+1}\,\mathrm{d} x)^{2/p+1} } = S_N, where S_N is the best Sobolev constant in \mathbf{R}^N . Show more

Citation: Asymptotic Analysis, vol. 21, no. 2, pp. 99-115, 1999

Authors: Bayada, Guy | Chambat, Michèle | Ciuperca, Ionel

Article Type: Research Article

Abstract: We consider a fluid flow in a thin domain with moving boundary. Using a rescaling of the domain in time and in height, we obtain the generalized Reynolds equation for the limit pressure.

Citation: Asymptotic Analysis, vol. 21, no. 2, pp. 117-132, 1999

Authors: Hayashi, Nakao | Miao, Changxing | Naumkin, Pavel I.

Article Type: Research Article

Abstract: We study the global in time existence of small solutions to the generalized derivative nonlinear Schrödinger equations of the form \begin{equation} \cases {\mathrm{i}\curpartial_t u + (1/2)\Delta u = \mathcal{N}(u,\nabla u,\overline u,\nabla \overline u), &$(t,x) \in \mathbf{R}\times {\mathbf{R}}^n$,\cr \noalign{\vskip4pt} u(0,x) = u_0 (x),\quad x\in\mathbf{R}^n,\cr} \end{equation} where the space dimension n \geqslant 3 , the initial data u_0 are sufficiently small, \bar u is the complex conjugate of u and the nonlinear term \mathcal{N} is a smooth complex valued function \mathbf{C}\times \mathbf{C}^n\times \mathbf{C}\times …\mathbf{C}^n\rightarrow \mathbf{C} . We assume that {\mathcal N} is a quadratic function in the neighborhood of the origin and always includes at least one derivative, that is, |{\mathcal N}(u,w,\bar u,\bar w)| \leqslant C|w|(|u|+|w|) , for small u and w in the case of space dimensions n = 3, 4 . As a typical example we consider the case of the polynomial type nonlinearity of the form {\mathcal N}(u,w,\bar u,\bar w) = \sum_{\tiny{\matrix{2 \leqslant |\alpha| + |\beta| + |\gamma| \leqslant l\cr \noalign{\vskip2pt} m \leqslant |\beta| + |\gamma| \leqslant l}}} \normalsize\lambda_{\alpha\beta\gamma} u^{\alpha_1}\bar u^{\alpha_2} w^{\beta}\bar w^{\gamma} with w = (w_j)_{1\leqslant j\leqslant n} , \lambda_{\alpha\beta\gamma} \in \mathbf{C} , l\geqslant 2, m \geqslant 1 for n=3,4 , and m \geqslant 0 for n\geqslant 5 . We prove the global existence of solutions to the Cauchy problem (A) under the condition that the initial data u_0 \in \mathbf{H}^{[n/2]+5,0} \cap \mathbf{H}^{[n/2]+3,2} , where \mathbf{H}^{m,s} = \{ \phi \in \mathbf{L}^2;\ \|\phi\|_{m,s} = \|(1+x^2)^{s/2} (1-\Delta)^{m/2}\phi\|_{\mathbf{L}^2} <\infty\} is the weighted Sobolev space. We also show the existence of the usual scattering states. Our result for n=3,4 is an improvement of Hayashi and Hirata, Nonlinear Anal. 31 (1998), 671–685. Show more

Keywords: Derivative nonlinear Schrödinger equations, global small solutions, general space dimensions

Citation: Asymptotic Analysis, vol. 21, no. 2, pp. 133-147, 1999

Authors: Bambusi, Dario | Graffi, Sandro | Paul, Thierry

Article Type: Research Article

Abstract: Let \mathcal{H} be a holomorphic Hamiltonian of quadratic growth on \mathbb{R}^{2n} , b a holomorphic exponentially localized observable, H,B the corresponding operators on L^2(\mathbb{R}^n) generated by Weyl quantization, and U(t)=\exp{\mathrm{i}Ht/\hbar} . It is proved that the L^2 norm of the difference between the Heisenberg observable B_t=U(t)BU(-t) and its semiclassical approximation of order N-1 is majorized by K^N N^{(6n+1)N}\hbar^{-4/9}(-\hbar\log\hbar)^N for t\in [0,T_n(\hbar)] , where T_n(\hbar):=-2\log\hbar/[\alpha(6n+3)(N-1)] and \alpha:=\Vert\mathrm{Hess}_{(x,\xi)}\,\mathcal{H}\Vert . Choosing a suitable N(\hbar) the error is majorized by …C\hbar^{\log\vert\log\hbar\vert} , 0\leq t\leq \vert\log\hbar\vert/\log\vert\log\hbar\vert (here K and C are explicit constants independent of N,\hbar ). Show more

Citation: Asymptotic Analysis, vol. 21, no. 2, pp. 149-160, 1999

Authors: El Hajji, Mohamed | Donato, Patrizia

Article Type: Research Article

Abstract: In Nonlinear Partial Differential Equations and Their Applications, Vol. 13 (to appear), Briane, Damlamian and Donato extend to the case of perforated domains the H ‐covergence introduced by Murat and Tartar (Sém. Anal. Fonct. Numér., 1977/1978; Topics in the Mathematical Modelling of Composite Materials, 1997) for the diffusion equation. In this paper, we define in the same spirit a notion of H^0 ‐convergence for the linearized elasticity system in perforated domains. This definition allows to obtain compactness, locality and corrector results.

Keywords: Homogenization, elasticity, H‐convergence, H^0‐convergence

Citation: Asymptotic Analysis, vol. 21, no. 2, pp. 161-186, 1999