Geometric estimates for the logarithmic fast diffusion equation
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.
scientific article; zbMATH DE number 2133084