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Thin-film limit of the parabolic $p$-Laplace equation in a moving thin domain

Tatsu-Hiko Miura

TL;DR

The paper proves a rigorous thin-film limit for the parabolic $p$-Laplace equation in moving thin domains when $p>2$, showing that the weighted normal average of the solution converges to a pair $(v, zeta)$ on the moving surface that solves a nonlinear diffusion system coupled to an algebraic constraint. The limit problem on the moving hypersurface consists of a nonlinear tangential diffusion term with flux $w=|_{v, zeta}|^{p-2}_{v, zeta}$ and a normal-velocity constraint that yields a local mass conservation law on $igl\Gamma_tigr)$ through $ zeta$, the normal component of the flux. The analysis extends prior linear ($p=2$) thin-film results to the nonlinear regime, employing the evolving Bochner-space framework, a weighted-thin-direction average, and a careful characterization of the limit flux to overcome the nonlinearity and moving-domain geometry. The findings provide a rigorous reduction of dimension that preserves mass balance on moving surfaces and offer a foundation for approximating nonlinear diffusion on evolving manifolds in thin-geometry settings.

Abstract

We consider the parabolic $p$-Laplace equation with $p>2$ in a moving thin domain under a Neumann type boundary condition corresponding to the total mass conservation. When the moving thin domain shrinks to a given closed moving hypersurface as its thickness tends to zero, we rigorously derive a limit problem by showing the weak convergence of the weighted average of a weak solution to the thin-domain problem and characterizing the limit function as a unique weak solution to the limit problem. The limit problem obtained in this paper is a system of a nonlinear partial differential equation and an algebraic equation on the moving hypersurface. This seems to be somewhat strange, but we also find that the limit problem can be seen as a new kind of local mass conservation law on the moving hypersurface with a normal flux.

Thin-film limit of the parabolic $p$-Laplace equation in a moving thin domain

TL;DR

The paper proves a rigorous thin-film limit for the parabolic -Laplace equation in moving thin domains when , showing that the weighted normal average of the solution converges to a pair on the moving surface that solves a nonlinear diffusion system coupled to an algebraic constraint. The limit problem on the moving hypersurface consists of a nonlinear tangential diffusion term with flux and a normal-velocity constraint that yields a local mass conservation law on through , the normal component of the flux. The analysis extends prior linear () thin-film results to the nonlinear regime, employing the evolving Bochner-space framework, a weighted-thin-direction average, and a careful characterization of the limit flux to overcome the nonlinearity and moving-domain geometry. The findings provide a rigorous reduction of dimension that preserves mass balance on moving surfaces and offer a foundation for approximating nonlinear diffusion on evolving manifolds in thin-geometry settings.

Abstract

We consider the parabolic -Laplace equation with in a moving thin domain under a Neumann type boundary condition corresponding to the total mass conservation. When the moving thin domain shrinks to a given closed moving hypersurface as its thickness tends to zero, we rigorously derive a limit problem by showing the weak convergence of the weighted average of a weak solution to the thin-domain problem and characterizing the limit function as a unique weak solution to the limit problem. The limit problem obtained in this paper is a system of a nonlinear partial differential equation and an algebraic equation on the moving hypersurface. This seems to be somewhat strange, but we also find that the limit problem can be seen as a new kind of local mass conservation law on the moving hypersurface with a normal flux.
Paper Structure (32 sections, 47 theorems, 458 equations)

This paper contains 32 sections, 47 theorems, 458 equations.

Table of Contents

  1. Introduction
  2. Problem settings and main result
  3. Physical interpretation of the limit problem
  4. Literature overview
  5. Organization of the paper
  6. Outline of Proof
  7. Basic outline
  8. Difference from the linear case
  9. Characterization of $\mathbf{w}$
  10. Uniqueness of a weak solution
  11. Average of the material derivative
  12. Preliminaries
  13. Preliminaries
  14. Vectors and matrices
  15. Moving surface
  16. ...and 17 more sections

Key Result

Theorem 1.1

For $u_0^\varepsilon\in L^2(\Omega_0^\varepsilon)$ and $f^\varepsilon\in L_{[W^{1,p}]^\ast}^{p'}(Q_T^\varepsilon)$, let $u^\varepsilon\in\mathbb{W}^{p,p'}(Q_T^\varepsilon)$ be a unique weak solution to E:pLap_MTD. Suppose that the following conditions are satisfied: Then, there exist functions $v\in\mathbb{W}^{p,p'}(S_T)$ and $\zeta\in L_{L^p}^p(S_T)$ such that and $(v,\zeta)$ is a unique weak s

Theorems & Definitions (96)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Lemma 3.8
  • proof
  • ...and 86 more