Higher Order Time Asymptotics Of Fast Diffusion In Euclidean Space A Dynamical Systems Approach
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Higher-Order Time Asymptotics of Fast Diffusion in Euclidean Space: A Dynamical Systems Approach
Author: Jochen Denzler
language: en
Publisher: American Mathematical Soc.
Release Date: 2015-02-06
This paper quantifies the speed of convergence and higher-order asymptotics of fast diffusion dynamics on Rn 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. 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.
Stability in Gagliardo–Nirenberg–Sobolev Inequalities: Flows, Regularity and the Entropy Method
Author: Matteo Bonforte
language: en
Publisher: American Mathematical Society
Release Date: 2025-05-29
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Higher Moments of Banach Space Valued Random Variables
Author: Svante Janson
language: en
Publisher: American Mathematical Soc.
Release Date: 2015-10-27
The authors define the :th moment of a Banach space valued random variable as the expectation of its :th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. The authors study both the projective and injective tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations: Bochner integrals, Pettis integrals and Dunford integrals.