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On $α$-entropy solutions of a nonlocal thin film equation: existence and finite speed of propagation

Antonio Segatti, Roman Taranets

TL;DR

This work analyzes a nonlocal thin film model driven by a spectral Neumann fractional Laplacian, establishing an $\alpha$-entropy framework that yields global a priori bounds and enables the proof of finite propagation speed and waiting-time phenomena. The authors develop a careful regularization scheme for mobility and entropy, obtain uniform entropy/energy estimates, and then take limits to construct $\alpha$-entropy solutions in an extended parameter range for the exponent $n$. A localized entropy analysis combined with Stampacchia-type arguments yields explicit finite-speed propagation, while a boundary-entropy-based argument provides a lower bound on waiting times. Overall, the paper deepens the understanding of how nonlocal diffusion interacts with thin-film dynamics and broadens the known parameter regime for existence, finite propagation, and waiting-time behavior in multi-dimensional settings.

Abstract

We consider an initial-boundary value problem for a class of nonlocal thin film equations governed by the spectral fractional Laplacian with homogeneous Neumann boundary conditions. We were the first to establish an $α$-entropy estimate for nonlocal thin film equations, which yields essential a priori bounds for the regularity and long-time behavior of weak solutions. By developing a localized version of this estimate, we prove finite speed of propagation, showing that the support of nonnegative solutions remains compact for positive times. Furthermore, we find a sufficient condition for a waiting time phenomenon, whereby the solution remains identically zero in a region for a nontrivial time interval. These results highlight new features in the interaction between nonlocal effects and classical thin film dynamics.

On $α$-entropy solutions of a nonlocal thin film equation: existence and finite speed of propagation

TL;DR

This work analyzes a nonlocal thin film model driven by a spectral Neumann fractional Laplacian, establishing an -entropy framework that yields global a priori bounds and enables the proof of finite propagation speed and waiting-time phenomena. The authors develop a careful regularization scheme for mobility and entropy, obtain uniform entropy/energy estimates, and then take limits to construct -entropy solutions in an extended parameter range for the exponent . A localized entropy analysis combined with Stampacchia-type arguments yields explicit finite-speed propagation, while a boundary-entropy-based argument provides a lower bound on waiting times. Overall, the paper deepens the understanding of how nonlocal diffusion interacts with thin-film dynamics and broadens the known parameter regime for existence, finite propagation, and waiting-time behavior in multi-dimensional settings.

Abstract

We consider an initial-boundary value problem for a class of nonlocal thin film equations governed by the spectral fractional Laplacian with homogeneous Neumann boundary conditions. We were the first to establish an -entropy estimate for nonlocal thin film equations, which yields essential a priori bounds for the regularity and long-time behavior of weak solutions. By developing a localized version of this estimate, we prove finite speed of propagation, showing that the support of nonnegative solutions remains compact for positive times. Furthermore, we find a sufficient condition for a waiting time phenomenon, whereby the solution remains identically zero in a region for a nontrivial time interval. These results highlight new features in the interaction between nonlocal effects and classical thin film dynamics.
Paper Structure (14 sections, 21 theorems, 259 equations)

This paper contains 14 sections, 21 theorems, 259 equations.

Table of Contents

  1. Introduction
  2. Fractional Sobolev spaces and fractional laplacian
  3. Main results: existence and upper bound on interface propagation speed
  4. Existence of $\alpha$-entropy solutions
  5. A regularized problem
  6. Approximated $\alpha$-entropy
  7. Proof of the Theorem \ref{['th:ex']}: Existence of $\alpha$-entropy solutions.
  8. Limits $\varepsilon,\delta \to 0$ in the $\alpha$-entropy and energy estimate
  9. Finite Speed of Propagation
  10. Local $\alpha$-entropy estimate
  11. Proof of FSP for $n \in (\tfrac{2(s+1)}{ 4(s+1)-d },2)$
  12. Lower-bound on the waiting time
  13. Auxiliary Lemmas
  14. Proof of Lemma \ref{['lem-GG']}

Key Result

Proposition 2.1

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

Theorems & Definitions (30)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 3.1: Weak solution
  • Theorem 1: existence of an $\alpha$-entropy solution
  • Remark 3.1
  • Theorem 2: Upper bounds on interface propagation speed
  • Theorem 3: Lower bounds on waiting times
  • Lemma 3.1: Nonlocal Chain Rule
  • ...and 20 more