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Fractional fast diffusion with initial data a Radon measure

Jorge Ruiz-Cases

TL;DR

The paper addresses the fractional fast diffusion equation $\partial_t u + (-\Delta)^{\sigma/2}u^m =0$ with $m_c<m<1$ and $N>\sigma$, seeking a Widder-type theory that links measure-valued initial data to nonnegative very weak solutions. It introduces weighted very weak solutions with controlled growth at infinity and proves that any such solution has a unique initial trace $\mu$ in a growth class $\mathfrak{M}_\sigma$, while establishing existence of solutions from any $\mu\in\mathfrak{M}_\sigma$ for $m>m_c$ and uniqueness of these solutions. The uniqueness proof hinges on a backward-in-time problem with a coefficient $\alpha=(u^m-v^m)/(u-v)$ and Riesz-potential techniques, along with compactness and capacity arguments to handle measure data in the nonlocal setting. The work extends the classical Widder theory to the nonlocal fractional porous medium regime, providing a rigorous framework for measure-valued initial data and clarifying the role of the growth-at-infinity condition. Open directions include lowering dimensional restrictions, handling non-weighted initial data, and extending to more general operators or nonlinearities and smoothing effects.

Abstract

We establish a complete Widder Theory for the fractional fast diffusion equation. Our work focuses on nonnegative solutions satisfying a certain integral size condition at infinity. We prove that these solutions possess a Radon measure as initial trace, and prove the existence and uniqueness of solutions originating from such initial data. The uniqueness result is the main issue. Most of its difficulty comes from the singular character of the nonlinearity.

Fractional fast diffusion with initial data a Radon measure

TL;DR

The paper addresses the fractional fast diffusion equation with and , seeking a Widder-type theory that links measure-valued initial data to nonnegative very weak solutions. It introduces weighted very weak solutions with controlled growth at infinity and proves that any such solution has a unique initial trace in a growth class , while establishing existence of solutions from any for and uniqueness of these solutions. The uniqueness proof hinges on a backward-in-time problem with a coefficient and Riesz-potential techniques, along with compactness and capacity arguments to handle measure data in the nonlocal setting. The work extends the classical Widder theory to the nonlocal fractional porous medium regime, providing a rigorous framework for measure-valued initial data and clarifying the role of the growth-at-infinity condition. Open directions include lowering dimensional restrictions, handling non-weighted initial data, and extending to more general operators or nonlinearities and smoothing effects.

Abstract

We establish a complete Widder Theory for the fractional fast diffusion equation. Our work focuses on nonnegative solutions satisfying a certain integral size condition at infinity. We prove that these solutions possess a Radon measure as initial trace, and prove the existence and uniqueness of solutions originating from such initial data. The uniqueness result is the main issue. Most of its difficulty comes from the singular character of the nonlinearity.

Paper Structure

This paper contains 18 sections, 20 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Goal
  3. Precedents
  4. Main results
  5. Outline of the paper
  6. Preliminaries
  7. On the fractional Laplacian and fractional Sobolev spaces
  8. On the concepts of solution
  9. On the Radon measures
  10. Potential theory and capacity
  11. Banach-Saks: On weak and pointwise convergence
  12. Solutions growing at infinity
  13. Existence of initial traces
  14. Existence of solutions with initial data a measure
  15. Uniqueness
  16. ...and 3 more sections

Key Result

Theorem 2.3

Let $m>m_c$. Assume $u$, a weak solution of equation, verifies $\lim_{t\to 0}u(t) = u_0$ in $L^1(\mathbb{R}^N)$. Then, with $\alpha = N /(N(m-1)+\sigma)$ and $\gamma = \sigma \alpha / N$, where the constant $C$ depends only on $m, N$, and $\sigma$.

Theorems & Definitions (43)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Smoothing effect
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Lemma 2.10
  • ...and 33 more