Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations |
| |
Authors: | Ciprian Foias George R Sell Edriss S Titi |
| |
Institution: | (1) Department of Mathematics, Indiana University, 47405 Bloomington, Indiana;(2) Institute for Mathematics and Its Applications, University of Minnesota, 55455 Minneapolis, Minnesota;(3) Department of Mathematics, University of Chicago, 60637 Chicago, Illinois |
| |
Abstract: | In this paper, we study the long-time behavior of a class of nonlinear dissipative partial differential equations. By means of the Lyapunov-Perron method, we show that the equation has an inertial manifold, provided that certain gap condition in the spectrum of the linear part of the equation is satisfied. We verify that the constructed inertial manifold has the property of exponential tracking (i.e., stability with asymptotic phase, or asymptotic completeness), which makes it a faithful representative to the relevant long-time dynamics of the equation. The second feature of this paper is the introduction of a modified Galerkin approximation for analyzing the original PDE. In an illustrative example (which we believe to be typical), we show that this modified Galerkin approximation yields a smaller error than the standard Galerkin approximation. |
| |
Keywords: | Dissipation exponential tracking inertial manifolds nonlinear equations |
本文献已被 SpringerLink 等数据库收录! |
|