Backward uniqueness for parabolic equations
type: Item
instance of: scholarly article
title: Backward uniqueness for parabolic equations
zbMATH Open document ID: 1039.35052
author: L. Escauriaza
author: G. Seregin
author: Vladimír Šverák
published in: Archive for Rational Mechanics and Analysis
publication date: 2004-01-05T00:00:00Z
review text: The authors show that any solution of the parabolic equation \[ u_ t +\Delta u + b\cdot Du +cu =0 \] in the unbounded domain \((\mathbb R^ n\setminus B_ R)\times (0,T)\) must vanish identically if it vanishes at \(t=0\) and if it grows no faster than \(M\exp(M| x| ^2)\) for some positive constant \(M\) as \(| x| \to\infty\). The only assumptions on the coefficients \(b\) and \(c\) are that they are bounded. The basic idea of the proof is to show that two Carleman inequalities are satisfied; these inequalities give weighted \(L^2\) estimates on \(u\) and its gradient in terms of corresponding weighted estimates of \(u_ t +\Delta u\). Several applications are mentioned, with references for details: first, the uniqueness result implies smoothness of weak solutions to the Navier-Stokes equations in a borderline case; and, second, it is related (in a negative way) to the controllability of such problems.
reviewed by: Gary M. Lieberman
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35R45
zbMATH DE Number: 2021066
zbMATH Keywords: exterior domain
zbMATH Keywords: two Carleman inequalities
zbMATH Keywords: controllability
MaRDI profile type: Publication
full work available at URL: s00205-003-0263-8
OpenAlex ID: W2051926666
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DOI: 10.1007/S00205-003-0263-8
description: scientific article; zbMATH DE number 2021066