Analyticity of solutions to fully nonlinear parabolic evolution equations on symmetric spaces
type: Item
instance of: scholarly article
title: Analyticity of solutions to fully nonlinear parabolic evolution equations on symmetric spaces
zbMATH Open document ID: 1064.58011
author: Joachim Escher
author: Gieri Simonett
published in: Journal of Evolution Equations
publication date: 2004-09-22T00:00:00Z
review text: The authors consider the following initial value problem: \(\{{{du}\over{dt}}+P(u)=0, u(0)=u_0\}\), where \(u\in C([0,t^+),B)\cap C^1([0,t^+),E_0)\), where \(t^+\equiv t^+(u_0)>0\) stands for the maximal existence time of \(u\), \(E_0\) is a Banach space, \(B\subset E_1\) is open in the Banach space \(E_1\) and \(P\in C^\omega(B,E_0)\). It is assumed that \(E_1\) is continuously injected and dense in \(E_0\). Furthermore, it is assumed that \(\forall u\in{\mathcal L}(E_1,E_0)\) posses the property of maximal regularity in the sense of Da Prato-Grisvard. In particular the spaces \(E_j\), \(j=0,1\), are built by means of numerical functions, of a suitable class, on \([0,t^+)\times M\), with \(M\) a closed symmetric Riemannian manifold. Then, it is proven that if \(P\) is compatible with the underlying Lie structure, above system has real analytic solutions \(u\in C^\omega([0,t^+)\times M)\). The authors emphasize that this theorem is a special case of another more general one proved in the same paper. Some applications of this general result are also considered. More precisely, Bellman's equation and a class of mean curvature flows are studied in this framework. Reviewer's remark: The above results can be directly gained in the framework of the geometric theory of PDEs. In fact, we can more generally consider the following differential equation: \(u_t+P(u)=0\), on the fiber bundle \(\pi:W\equiv{\mathbb R}\times M\times{\mathbb R}\to {\mathbb R}\times M\), \((t,x^\alpha,u)\mapsto(t,x^\alpha)\), \(1\leq\alpha\leq n\), where \(M\) is a smooth \(n\)-dimensional manifold. Let us assume, for simplicity, that \(P\) is defined at the zero order, i.e., does not depend on derivatives of \(u\). Then we get a first order differential equation \(E_1\subset J{\mathcal D}^1(W)\), locally defined by the function \(F\equiv u_t+P(u)\) on \(J{\mathcal D}^1(W)\). One can prove that this equation is involutive formally integrable, therefore in the analytic case it is also completely integ…
reviewed by: Agostino Prástaro
Mathematics Subject Classification ID: 58D25
Mathematics Subject Classification ID: 53C44
zbMATH DE Number: 2101651
zbMATH Keywords: evolution equation
MaRDI profile type: Publication
full work available at URL: s00028-003-0093-z
OpenAlex ID: W4237722738
DOI: 10.1007/S00028-003-0093-Z
description: scientific article; zbMATH DE number 2101651