Abstract
In this paper we devote our attention to a class of weighted ultrafast diffusion equations arising from the problem of quantisation for probability measures. These equations have a natural gradient flow structure in the space of probability measures endowed with the quadratic Wasserstein distance. Exploiting this structure, in particular through the so-called JKO scheme, we introduce a notion of weak solutions, prove existence, uniqueness, BV and H1 estimates, L1 weighted contractivity, Harnack inequalities, and exponential convergence to a steady state.
Highlights
In this work we investigate the well-posedness and the long-time behaviour of solutions u = u(t, x) of the nonlinear diffusion equation
We are interested in the case α < 0, i.e., when (0.1) takes the name of ultrafast diffusion equation
This class of equations has completely different properties from those found in the case α 1, which corresponds to the porous medium and heat framework
Summary
In [16], the authors introduced a new approach to the quantisation problem based on gradient flows; their idea was to study the evolution of the points of N when they follow the steepest descent curves of the functional (0.2) (which is nothing but a continuous-time version of the well-known Lloyd’s algorithm for the optimal quantisation; see [40], or [12,42] for more recent accounts and related topics), and to compare it to the gradient flow of a continuous functional This analysis was performed in detail in the one-dimensional case in [16], and in the two-dimensional case in [17] when ρ ≡ 1. Each time we choose which approach to favour depending on the easiest one to adopt
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