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Existence and finite speed of propagation of solutions for a multi-dimensional fractional thin-film equation

Nicola De Nitti, Stefano Lisini, Antonio Segatti, Roman Taranets

TL;DR

The article develops a rigorous existence theory for a multi-dimensional fractional thin-film equation with spectral Neumann Laplacian, proving nonnegative weak solutions under precise parameter ranges and then establishing two key interface properties: finite speed of propagation and a waiting-time phenomenon. The authors tackle the nonlocal, degenerate nature via a nested regularization scheme and Faedo–Galerkin approximations, obtaining uniform entropy and energy estimates that enable compactness and limit-passage to a global weak solution. A local entropy estimate feeds into Stampacchia-type iteration to derive finite propagation speed, while a detailed waiting-time analysis yields explicit time lower bounds dependent on initial entropy tails. Overall, the work extends known one-dimensional results to higher dimensions and bounded domains, illuminating how nonlocal diffusion and slip regimes govern interface evolution in fractional thin-film dynamics.

Abstract

In this paper, we discuss existence and finite speed of propagation for the solutions to an initial-boundary value problem for a family of fractional thin-film equations in a bounded domain in $\mathbb{R}^d$. The nonlocal operator we consider is the spectral fractional Laplacian with Neumann boundary conditions. In the case of a ``strong slippage'' regime with ``complete wetting'' interfacial conditions, we prove local entropy estimates that entail finite speed of propagation of the support and a lower bound for the waiting time phenomenon.

Existence and finite speed of propagation of solutions for a multi-dimensional fractional thin-film equation

TL;DR

The article develops a rigorous existence theory for a multi-dimensional fractional thin-film equation with spectral Neumann Laplacian, proving nonnegative weak solutions under precise parameter ranges and then establishing two key interface properties: finite speed of propagation and a waiting-time phenomenon. The authors tackle the nonlocal, degenerate nature via a nested regularization scheme and Faedo–Galerkin approximations, obtaining uniform entropy and energy estimates that enable compactness and limit-passage to a global weak solution. A local entropy estimate feeds into Stampacchia-type iteration to derive finite propagation speed, while a detailed waiting-time analysis yields explicit time lower bounds dependent on initial entropy tails. Overall, the work extends known one-dimensional results to higher dimensions and bounded domains, illuminating how nonlocal diffusion and slip regimes govern interface evolution in fractional thin-film dynamics.

Abstract

In this paper, we discuss existence and finite speed of propagation for the solutions to an initial-boundary value problem for a family of fractional thin-film equations in a bounded domain in . The nonlocal operator we consider is the spectral fractional Laplacian with Neumann boundary conditions. In the case of a ``strong slippage'' regime with ``complete wetting'' interfacial conditions, we prove local entropy estimates that entail finite speed of propagation of the support and a lower bound for the waiting time phenomenon.
Paper Structure (22 sections, 15 theorems, 301 equations)

This paper contains 22 sections, 15 theorems, 301 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Fractional Sobolev norms and the fractional Laplacian
  4. Main results
  5. Proof of the existence result
  6. A regularized problem
  7. The Faedo--Galerkin scheme
  8. Limit for gamma tending to zero
  9. Entropy estimate and its consequences
  10. Limit for gamma tending to zero in the weak formulation
  11. Limit for gamma tending to zero in the energy identity and identification of the flux $J_{\varepsilon,\delta}$
  12. Limits for epsilon and delta tending to zero
  13. A priori estimates and compactness
  14. Limits for epsilon and delta tending to zero in the weak formulation
  15. Limits for epsilon and delta tending to zero in the entropy and energy estimates
  16. ...and 7 more sections

Key Result

Proposition 2.1

Let $r\in [0,+\infty)$.

Theorems & Definitions (25)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Theorem 3.1: Existence of weak solutions
  • Theorem 3.2: Upper bounds on interface propagation speed
  • Theorem 3.3: Lower bounds on waiting times
  • Lemma 5.1
  • ...and 15 more