[Paper Review] Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods

Matteo Bonforte, Jean Dolbeault+2 more|arXiv (Cornell University)|Feb 26, 2016

Nonlinear Partial Differential Equations12 references3 citations

TL;DR

This paper establishes sharp asymptotic rates of convergence to self-similar solutions for weighted fast diffusion equations using entropy methods and Caffarelli-Kohn-Nirenberg inequalities. It rigorously proves exponential convergence in relative error via spectral gap estimates and Harnack inequalities, distinguishing the weighted case from the non-weighted one by identifying symmetry-breaking regimes and optimal convergence rates beyond the supercritical exponent range.

ABSTRACT

This paper is the second part of the study. In Part~I, self-similar solutions of a weighted fast diffusion equation (WFD) were related to optimal functions in a family of subcritical Caffarelli-Kohn-Nirenberg inequalities (CKN) applied to radially symmetric functions. For these inequalities, the linear instability (symmetry breaking) of the optimal radial solutions relies on the spectral properties of the linearized evolution operator. Symmetry breaking in (CKN) was also related to large-time asymptotics of (WFD), at formal level. A first purpose of Part~II is to give a rigorous justification of this point, that is, to determine the asymptotic rates of convergence of the solutions to (WFD) in the symmetry range of (CKN) as well as in the symmetry breaking range, and even in regimes beyond the supercritical exponent in (CKN). Global rates of convergence with respect to a free energy (or entropy) functional are also investigated, as well as uniform convergence to self-similar solutions in the strong sense of the relative error. Differences with large-time asymptotics of fast diffusion equations without weights will be emphasized.

Motivation & Objective

• To rigorously justify the large-time asymptotics of weighted fast diffusion equations (WFD) in both symmetry and symmetry-breaking regimes of the associated Caffarelli-Kohn-Nirenberg (CKN) inequalities.
• To determine sharp rates of convergence in relative error for solutions to WFD, extending beyond the subcritical and supercritical regimes of the CKN inequalities.
• To establish global convergence rates using free energy (entropy) functionals and relate them to the relative Fisher information via a spectral gap estimate.
• To analyze the role of weights in altering the asymptotic behavior compared to the non-weighted fast diffusion equation, particularly in terms of convergence speed and symmetry properties.
• To prove Hölder regularity and Harnack-type inequalities for solutions in the weighted framework, enabling refined convergence analysis.

Proposed method

• Transform the WFD into a Fokker-Planck-type equation using self-similar variables, converting time-dependent solutions into stationary profiles.
• Define a free energy functional $ \mathcal{F}[v] $ measuring relative entropy to the Barenblatt-type stationary solution $ \mathfrak{B} $, and derive its time evolution via $ \frac{d}{dt}\mathcal{F}[v] = -\frac{m}{1-m}\mathcal{I}[v] $, where $ \mathcal{I}[v] $ is the relative Fisher information.
• Establish a connection between the decay of $ \mathcal{F}[v] $ and $ \mathcal{I}[v] $ using a family of weighted Caffarelli-Kohn-Nirenberg inequalities with optimal constants.
• Apply spectral gap estimates derived from the linearization of the evolution operator around the self-similar solution to quantify the rate of decay of the free energy.
• Use a Harnack inequality for nonnegative solutions of the transformed equation to deduce Hölder continuity of solutions, enabling uniform convergence estimates.
• Perform a change of variables and iterate Harnack estimates to derive intrinsic scaling-invariant regularity and convergence results in the weighted parabolic setting.

Experimental results

Research questions

• RQ1 What are the sharp asymptotic rates of convergence in relative error for solutions to the weighted fast diffusion equation across all parameter regimes, including symmetry-breaking and supercritical exponents?
• RQ2 How does the presence of weights in the diffusion equation alter the convergence dynamics compared to the non-weighted fast diffusion equation?
• RQ3 Can the entropy method be rigorously applied to establish exponential convergence to self-similar solutions in the weighted case, and what is the role of the spectral gap in this process?
• RQ4 To what extent do Caffarelli-Kohn-Nirenberg inequalities with optimal constants govern the convergence rates in the weighted setting?
• RQ5 What regularity properties, such as Hölder continuity, can be derived for solutions of the weighted fast diffusion equation using Harnack-type estimates?

Key findings

• The paper establishes exponential convergence of solutions to the self-similar Barenblatt profile in the weighted fast diffusion equation, with sharp rates determined by the spectral gap of the linearized operator.
• Convergence in relative error is proven to be exponential and uniform, with the rate governed by the optimal constant in the Caffarelli-Kohn-Nirenberg inequality for the given parameters.
• The symmetry-breaking transition in the Caffarelli-Kohn-Nirenberg inequalities corresponds precisely to a change in the asymptotic behavior of the WFD, with different convergence rates in the symmetric and symmetry-breaking regimes.
• The method identifies a basin of attraction for the self-similar solution that persists for all $ m \in (0,1) $, extending beyond the classical fast diffusion range.
• Harnack and Hölder regularity estimates are derived for the transformed equation, ensuring that solutions are locally Hölder continuous and enabling uniform convergence in relative error.
• The analysis reveals that the weighted case exhibits fundamentally different asymptotic dynamics than the non-weighted case, particularly in the supercritical regime where convergence rates are slower and symmetry breaking occurs.