eBook ISBN:  9781470420284 
Product Code:  MEMO/234/1101.E 
List Price:  $70.00 
MAA Member Price:  $63.00 
AMS Member Price:  $42.00 
eBook ISBN:  9781470420284 
Product Code:  MEMO/234/1101.E 
List Price:  $70.00 
MAA Member Price:  $63.00 
AMS Member Price:  $42.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 234; 2014; 81 ppMSC: Primary 35; Secondary 33; 37; 58; 76
This paper quantifies the speed of convergence and higherorder asymptotics of fast diffusion dynamics on \(\mathbf{R}^n\) to the Barenblatt (self similar) solution. Degeneracies in the parabolicity of this equation are cured by reexpressing 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 LaplaceBeltrami operator plus a transport term (which can be suppressed by introducing appropriate weights into the function space norm), plus a finitedepth 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 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.

Table of Contents

Chapters

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 Hölder spaces

6. Schauder estimates for the heat equation

7. Quantitative global wellposedness of the linear and nonlinear equations in Hölder spaces

8. The spectrum of the linearized equation

9. Proof of Theorem

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 and its corollaries.

A. Pedestrian derivation of all Schauder Estimates


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This paper quantifies the speed of convergence and higherorder asymptotics of fast diffusion dynamics on \(\mathbf{R}^n\) to the Barenblatt (self similar) solution. Degeneracies in the parabolicity of this equation are cured by reexpressing 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 LaplaceBeltrami operator plus a transport term (which can be suppressed by introducing appropriate weights into the function space norm), plus a finitedepth 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 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.

Chapters

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 Hölder spaces

6. Schauder estimates for the heat equation

7. Quantitative global wellposedness of the linear and nonlinear equations in Hölder spaces

8. The spectrum of the linearized equation

9. Proof of Theorem

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 and its corollaries.

A. Pedestrian derivation of all Schauder Estimates