Differential and Integral Equations

Global existence and uniqueness of solutions of the Ricci flow equation

Shu-Yu Hsu

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Abstract

In this paper we will prove the global existence and uniqueness of solutions of the Ricci flow equation on $R^2$ $u_t=\Delta \text{ log }u$, $u>0$, in $R^2\times (0,\infty)$, $u(x,0) =u_0(x)$ for $x\in R^2$, satisfying the inequality $u_t\le u/t$ in $R^2\times (0,\infty)$ and the condition $\liminf_{r\to\infty}$ log $u(x,t)/\text{log }r\ge -2$ uniformly on any compact subset of $(0,\infty)$ as $r=|x|\to\infty$ for any $u_0\not\in L^1(R^2)$, $u_0\ge 0$, satisfying $u_0\in L_{loc}^p(R^2)$ for some $p>1$.

Article information

Source
Differential Integral Equations, Volume 14, Number 3 (2001), 305-320.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356123330

Mathematical Reviews number (MathSciNet)
MR1799897

Zentralblatt MATH identifier
1011.35085

Subjects
Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Secondary: 35A05 35B25: Singular perturbations 35Kxx: Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35]

Citation

Hsu, Shu-Yu. Global existence and uniqueness of solutions of the Ricci flow equation. Differential Integral Equations 14 (2001), no. 3, 305--320. https://projecteuclid.org/euclid.die/1356123330


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