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Higher-order time asymptotics of fast diffusion in Euclidean space: A dynamical systems approach


About this Title

Jochen Denzler, Herbert Koch and Robert J. McCann

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 234, Number 1101
ISBNs: 978-1-4704-1408-5 (print); 978-1-4704-2028-4 (online)
DOI: http://dx.doi.org/10.1090/memo/1101
Published electronically: June 4, 2014

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Overview of Obstructions and Strategies, and Notation
  • Chapter 3. The nonlinear and linear equations in cigar coordinates
  • Chapter 4. The cigar as a Riemannian manifold
  • Chapter 5. Uniform manifolds and Hölder spaces
  • Chapter 6. Schauder estimates for the heat equation
  • Chapter 7. Quantitative global well-posedness of the linear and nonlinear equations in Hölder spaces
  • Chapter 8. The spectrum of the linearized equation
  • Chapter 9. Proof of Theorem 1.1
  • Chapter 10. Asymptotic estimates in weighted spaces: The case $m< \frac {n}{n+2}$
  • Chapter 11. Higher asymptotics in weighted spaces: The case $m> \frac {n}{n+2}$. Proof of Theorem 1.2 and its corollaries.
  • Appendix A. Pedestrian derivation of all Schauder Estimates

Abstract


This paper quantifies the speed of convergence and higher-order asymptotics of fast diffusion dynamics on 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 Hölder 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. We provide a detailed study of the linear and nonlinear problems in Hölder 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.

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