On smooth dependence on initial conditions for dissipative PDEs, an ODE-type approach.
type: Item
instance of: scholarly article
title: On smooth dependence on initial conditions for dissipative PDEs, an ODE-type approach.
zbMATH Open document ID: 1043.35034
author: Piotr Zgliczyński
published in: Journal of Differential Equations
publication date: 2004-01-26T00:00:00Z
review text: The goal of the paper is to show using relatively elementary ODE methods that for a class of PDEs for which there exist the compact trapping region and fast decay of Fourier coefficients, the partial derivatives with respect to initial conditions for Galerkin projections converge to the partial derivatives with respect to initial conditions of the semiflow induced by PDE. This also proves smooth dependence on initial conditions for full PDE on a trapping region. An abstract part of present paper depends only on the content of Sections 4 and 5 in [\textit{P. Zgliczynski}, Trapping regions and an ODE-type proof of an existence and uniqueness for Navier-Stokes equations with periodic boundary conditions on the plane, Zesz. Nauk. Uniw. Jagiell. 1269, Univ. Iagell. Acta Math. 41, 89--113 (2003; Zbl 1067.35068)]. An application of the main result -- Theorem 2 -- to the planar Navier-Stokes equations in Section 3 depends on the trapping regions, which are given in the paper cited above. It should be pointed out that in the literature there exist much more general results about regularity of solutions of dissipative PDEs. For example, it is shown that solutions of Navier-Stokes equations in dimension two depend analytically on initial conditions. But still the author believes that the simplicity of the tools used in the paper under review will make this paper interesting for researches with a dynamical system theory and ODE background, especially in the context of a computer-assisted program of rigorous investigation of dynamics of dissipative PDE. It is rather obvious that the presented method should be easily applicable to any PDE for which there exists a compact attractor and Gevrey class regularity results have been established. Here is a list of a few examples for which Theorem 2 holds: -- Navier-Stokes equations with periodic boundary conditions in three dimensions with small initial data and small force; -- Kuramoto-Sivashinsky equations in 1D with periodic boundary …
reviewed by: Leonid B. Chubarov
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 34A45
Mathematics Subject Classification ID: 37L05
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 76D05
zbMATH DE Number: 2032990
zbMATH Keywords: compact trapping region
zbMATH Keywords: Galerkin projections
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: compact attractor
MaRDI profile type: Publication
full work available at URL: j.jde.2003.07.009
OpenAlex ID: W1982884543
cites work: Low-dimensional behaviour in the complex Ginzburg-Landau equation
cites work: Recent results on mathematical and statistical hydrodynamics
cites work: Gevrey class regularity for the solutions of the Navier-Stokes equations
cites work: Q3779682
cites work: Geometric theory of semilinear parabolic equations
cites work: Q4735543
cites work: AN ELEMENTARY PROOF OF THE EXISTENCE AND UNIQUENESS THEOREM FOR THE NAVIER–STOKES EQUATIONS
cites work: Q4831749
cites work: Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation
DOI: 10.1016/J.JDE.2003.07.009
description: scientific article; zbMATH DE number 2032990