Geometric estimates for the logarithmic fast diffusion equation
type: Item
instance of: scholarly article
title: Geometric estimates for the logarithmic fast diffusion equation
zbMATH Open document ID: 1070.53041
published in: Communications in Analysis and Geometry
publication date: 2005-02-09T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.cag/1088454730
review text: The authors consider the logarithmic fast diffusion equation \[ {{\partial u}\over{\partial t}} = \Delta \log u \leqno(*) \] on the plane \({\mathbb R}^2\) with initial data \(f\geq 0\) of finite total mass. S.~Angenent [see \textit{L.~Wu}, Bull. Am. Math. Soc. 28, No. 1, 90--94 (1993; Zbl 0780.58009); Commun. Anal. Geom. 1, No. 3, 439--472 (1993; Zbl 0854.58011)] has observed that \((*)\) describes the evolution of the conformal metric \(g=u(dx^2+dy^2)\) under the Ricci flow. \((*)\) also arises in some fluid flow models. Here the authors consider complete finite area solutions with bounded curvature. In this case \(A(t)\), the area of \(({\mathbb R}^2,g)\) at time \(t\), and \(T\), the maximal time of existence, are related by \(A(t)=4\pi(T-t)\), and \(T\) is determined by the initial data via \(T={1\over{4\pi}}\int_{{\mathbb R}^2} f\). The authors' main result states that there exist positive constants \(\gamma\) and \(C\) such that the geometric width \(w\) and the maximum curvature \(R_{max}\) of the solution satisfy \[ \gamma(T-t) \leq w(t) \leq C(T-t) \] and \[ {\gamma\over{(T-t)^2}} \leq R_{max}(t) \leq {C\over{(T-t)^2}} \] for all \(t\in(0,T)\). Precise pointwise bounds on \(u\) near its vanishing time were obtained earlier by \textit{J. R.~King} [Philos. Trans. R. Soc. Lond., Ser. A 343, No. 1668, 337--375 (1993; Zbl 0797.35097)] in the rotationally symmetric case.
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 2133084
zbMATH Keywords: logarithmic fast diffusion equation
zbMATH Keywords: Ricci flow
zbMATH Keywords: geometric width
author: Panagiota Daskalopoulos
author: Richard S. Hamilton
reviewed by: John Urbas
MaRDI profile type: Publication
OpenAlex ID: W1966687081
DOI: 10.4310/CAG.2004.V12.N1.A8
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description: scientific article; zbMATH DE number 2133084