2000/05/24 by Piotr Zgliczyński, Zgliczynski, P., Konstantin Mischaikow +1
Computer Science · Engineering · Mathematics · #35Q35 #37B30 #37L65 #65M60 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Biology Tumor Growth #Nonlinear Dynamics and Pattern Formation #Numerical Analysis (math.NA) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.math/0005247
openalex publication_date 2000/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a new topological method for the study of the dynamics of dissipative PDE's. The method is based on the concept of the self-consistent apriori bounds, which allows to justify rigorously the Galerkin projection. As a result we obtain a low-dimensional system of ODE's subject to rigorously controlled small perturbation from the neglected modes. To this ODE's we apply the Conley index to obtain information about the dynamics of the PDE under consideration. As an application we present a computer assisted proof of the existence of fixed points for the Kuramoto-Sivashinsky equation.