Log-Lipschitz regularity and uniqueness of the flow for a field in \((W_{\text{loc}}^{n/p+1,p}(\mathbb{R}^n))^n\)
type: Item
instance of: scholarly article
title: Log-Lipschitz regularity and uniqueness of the flow for a field in \((W_{\text{loc}}^{n/p+1,p}(\mathbb{R}^n))^n\)
zbMATH Open document ID: 1022.34002
DOI: 10.1016/S1631-073X(02)02426-3
author: Enrique Zuazua
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2003-10-30T00:00:00Z
review text: The author is interested in the uniqueness property for solutions to the initial value problem \(\dot x= b(t,x)\), \(t> 0\), \(x(0)= x_0\), with \(x= x(t)\in\mathbb{R}^n\). There are two well-known results. The theorem of Peano yields the existence of a solution, the theorem of Picard-Lindelöf implies besides existence a uniqueness result, too. One possibility to weaken the Lipschitz condition on \(b\) with respect to \(x\) is to consider \(b\) in Sobolev spaces. The author explains the situation for \(b\in (W^{s,p}_{\text{loc}}(\mathbb{R}^n))^n\) and even for \(b\in L^1(0, T;(W^{s,p}_{\text{loc}}(\mathbb{R}^n))^n)\). The critical case for uniqueness is \(s={n\over p}+1\), \(p\geq 1\). A sketch of the proof of uniqueness is given, the cases \(p=2\) and \(p=n\), \(n\geq 3\) are proved.
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 1777357
zbMATH Keywords: log-Lipschitz regularity
zbMATH Keywords: Picard-Lindelöf
zbMATH Keywords: uniqueness of the flow
zbMATH Keywords: Sobolev spaces
MaRDI profile type: Publication
cites work: Q4073917
cites work: Flows of non-Lipschitzian vector fields and Navier-Stokes equations
cites work: Q4831566
cites work: Linear transport equations with initial values in Sobolev spaces and application to the Navier-Stokes equations
cites work: Ordinary differential equations, transport theory and Sobolev spaces
cites work: Q4399897
cites work: Sobolev imbedding theorems in borderline cases
cites work: Elliptic partial differential equations of second order
cites work: Sur les équations différentielles ordinaires et les équations de transport
cites work: On the uniqueness theorem for generalized solutions of initial-boundary problems for the Maguerre-Vlasov vibrations of shallow shells with clamped boundary conditions
cites work: Q5607484
full work available at URL: s1631-073x(02)02426-3
OpenAlex ID: W1599696556
description: scientific article; zbMATH DE number 1777357