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Higher-order time asymptotics of fast diffusion in Euclidean space : a dynamical systems methods / Jochen Denzler, Herbert Koch and Robert J. McCann.

By: Contributor(s): Material type: TextTextSeries: Memoirs of the American Mathematical Society ; v 234, no 1101.Publication details: Providence : American Mathematical Society, 2015.Description: v, 81 p. : illustrations ; 26 cmISBN:
  • 9781470414085 (pbk. : acidfree paper)
Subject(s): DDC classification:
  • 510 23 Am512
Contents:
1. Introduction -- 2. Overview of obstructions and strategies, and notation -- 3. The nonlinear and linear equations in cigar coordinates -- 4. The cigar as a Riemannian manifold -- 5. Uniform manifolds and Holder spaces -- 6. Schauder estimates for the heat equation -- 7. Quantitative global well-posedness of the linear and nonlinear equations in Holder spaces -- 8. The spectrum of the linearized equation -- 9. Proof of Theorem 1.1 -- 10. Asymptotic estimates in weighted spaces: The case $m< \frac{n}{n+2}$ -- 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 -- Bibliography.
Summary: This paper quantifies the speed of convergence and higher-order asymptotics of fast diffusion dynamics on R [superscript]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 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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Holdings
Item type Current library Call number Status Date due Barcode Item holds
Books ISI Library, Kolkata 510 Am512 (Browse shelf(Opens below)) Available 136690
Total holds: 0

Includes bibliographical references.

1. Introduction --
2. Overview of obstructions and strategies, and notation --
3. The nonlinear and linear equations in cigar coordinates --
4. The cigar as a Riemannian manifold --
5. Uniform manifolds and Holder spaces --
6. Schauder estimates for the heat equation --
7. Quantitative global well-posedness of the linear and nonlinear equations in Holder spaces --
8. The spectrum of the linearized equation --
9. Proof of Theorem 1.1 --
10. Asymptotic estimates in weighted spaces: The case $m< \frac{n}{n+2}$ --
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 --
Bibliography.

This paper quantifies the speed of convergence and higher-order asymptotics of fast diffusion dynamics on R [superscript]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 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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