Dynamical systems generated by Sobolev class vector fields in finite and infinite dimensions
type: Item
instance of: scholarly article
title: Dynamical systems generated by Sobolev class vector fields in finite and infinite dimensions
zbMATH Open document ID: 0938.37006
DOI: 10.1007/BF02365019
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-10-12T00:00:00Z
review text: The authors consider the fundamental problems of dynamical systems theory on existence and uniqueness of a global flow \(\{U_t\}\) generated by a vector field \(A\). The authors introduce the following definition: If \(X\) is a topological vector space equipped with a positive Radon measure \(\mu\) on its Borel \(\sigma\)-field and a measurable vector field \(A: X\to X\), a mapping \((t,x)\in \mathbb{R}\times X\to U^A_t(x)\in X\) is said to be a solution to the equation \[ U_t(x)=x+ \int^t_0 A\bigl(U_s(x)\bigr) ds\tag{*} \] if (a) for \(\mu\)-almost every \(x\), (*) holds with \(U=U^A\) for all \(t\in \mathbb{R}\) (its right-hand side being implicitly well defined), and (b) for every \(t\in\mathbb{R}\), the measure \(\mu\circ U_t^{-1}\) is absolutely continuous with respect to \(\mu\). The Radon-Nikodym derivative \(d(\mu\circ U^{-1}_t)/d\mu\) will always be denoted by \(r_t\). Moreover, \(U=U^A\) is a flow solution if, in addition, \(\mu\) -- a.e., \(\bigwedge_{s,t\in\mathbb{R}} U_{t+s} =U_tU_s\). The authors prove the following main theorem: Theorem 1. Let \(\mu\) be a measure on \(\mathbb{R}^n\) possessing a density \(\rho\in W^{1,1}_{\text{loc}} (\mathbb{R}^n)\), which is locally uniformly positive, and let \(A\in W^{1,1}_{\text{loc}} (\mathbb{R}^n, \mathbb{R}^n)\) be a vector field such that either \(e^{\|\nabla A\|}\in \bigcap_{p>1} L^p(\mu)\) or \(e^{\|\nabla A\|}\in \bigcap_{p>1} L^p_{\text{loc}} (\mu)\) and \(\|A\|\in L^{1+\varepsilon} (\mathbb{R}^n)\), and such that \(e^{|\delta_\mu A|} \in\bigcap_{p>1} L^p(\mu)\), where \(\delta_\mu A=\text{div} A+ (A,{\nabla\rho \over\rho})\). Then equation (*) possesses a flow solution. Assume that the measure \(\mu\) on a locally convex space \(X\) is such that there is a separable Hilbert space \(H\) continuously embedded into \(X\). This gives rise to the embedding \(j:X^*\to H\). The measure \(\mu\) is assumed to be Fomin differentiable along all vectors in \(j(X^*)\) such that the corresponding logarithmic derivati…
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 37A05
Mathematics Subject Classification ID: 60H07
zbMATH DE Number: 1348492
zbMATH Keywords: global flow
zbMATH Keywords: vector field
zbMATH Keywords: Radon measure
zbMATH Keywords: Radon Nikodym derivative
zbMATH Keywords: Radon-Nikodym process
author: Vladimir I. Bogachev
author: Eddy Mayer-Wolf
reviewed by: Stanisław Wedrychowicz
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02365019
OpenAlex ID: W1991837558
description: scientific article; zbMATH DE number 1348492