Higher-Order Time Asymptotics of Fast Diffusion in Euclidean Space: A Dynamical Systems Approach

ISBN-13: 9781470414085 / Angielski

Jochen Denzler;Herbert Koch;Robert John McCann

Opis

This paper quantifies the speed of convergence and higher-order asymptotics of fast diffusion dynamics on $mathbf^n$ to the Barenblatt (self similar) solution. Degeneracies in the parabolicity of this equation are cured by re-expressing the dynamics on a manifold with a cylindrical end, called the cigar. The nonlinear evolution becomes differentiable in Holder spaces on the cigar. The linearization of the dynamics is given by the Laplace-Beltrami operator plus a transport term (which can be suppressed by introducing appropriate weights into the function space norm), plus a finite-depth potential well with a universal profile. In the limiting case of the (linear) heat equation, the depth diverges, the number of eigenstates increases without bound, and the continuous spectrum recedes to infinity. The authors provide a detailed study of the linear and nonlinear problems in Holder spaces on the cigar, including a sharp boundedness estimate for the semigroup, and use this as a tool to obtain sharp convergence results toward the Barenblatt solution, and higher order asymptotics. In finer convergence results (after modding out symmetries of the problem), a subtle interplay between convergence rates and tail behavior is revealed. The difficulties involved in choosing the right functional spaces in which to carry out the analysis can be interpreted as genuine features of the equation rather than mere annoying technicalities.

Spis treści

• Introduction
• Overview of obstructions and strategies, and notation
• The nonlinear and linear equations in cigar coordinates
• The cigar as a Riemannian manifold
• Uniform manifolds and Holder spaces
• Schauder estimates for the heat equation
• Quantitative global well-posedness of the linear and nonlinear equations in Holder spaces
• The spectrum of the linearized equation
• Proof of Theorem 1.1
• Asymptotic estimates in weighted spaces: The case $m< \frac{n+2}$
• Higher asymptotics in weighted spaces: The case $m> \frac{n+2}$.
• Proof of Theorem 1.2 and its corollaries
• Appendix A. Pedestrian derivation of all Schauder estimates
• Bibliography