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Weak solutions to the parabolic $p$-Laplace equation in a moving domain under a Neumann type boundary condition

Tatsu-Hiko Miura

TL;DR

This work proves the existence and uniqueness of weak solutions for the parabolic $p$-Laplace equation with $p>2$ in a moving domain under a mass-conserving Neumann-type boundary condition. The authors develop a Galerkin scheme with time-dependent basis functions in evolving Bochner spaces and overcome a key limit passage obstacle from the boundary term by deriving a uniform-in-time Friedrichs-type inequality, enabling strong convergence of approximate solutions. They establish the strong identification of the nonlinear term and, under higher data regularity, gain time-derivative regularity in $L^2$, while also extending the results to Leray–Lions type operators. The approach provides a robust framework for nonlinear parabolic equations on moving domains and has potential applications to moving-thin domain limits and surface PDEs on evolving manifolds.

Abstract

This paper studies the parabolic $p$-Laplace equation with $p>2$ in a moving domain under a Neumann type boundary condition corresponding to the total mass conservation. We establish the existence and uniqueness of a weak solution by the Galerkin method in evolving Bochner spaces and a monotonicity argument. The main difficulty is in characterizing the weak limit of the nonlinear gradient term, where we need to deal with a term which comes from the boundary condition and cannot be absorbed into a monotone operator. To overcome this difficulty, we prove a uniform-in-time Friedrichs type inequality on a moving domain with time-dependent basis functions and make use of it to get the strong convergence of approximate solutions. We also show that the time derivative exists in the $L^2$ sense when given data have a better regularity, and discuss extension of the existence and uniqueness results to a Leray-Lions type operator.

Weak solutions to the parabolic $p$-Laplace equation in a moving domain under a Neumann type boundary condition

TL;DR

This work proves the existence and uniqueness of weak solutions for the parabolic -Laplace equation with in a moving domain under a mass-conserving Neumann-type boundary condition. The authors develop a Galerkin scheme with time-dependent basis functions in evolving Bochner spaces and overcome a key limit passage obstacle from the boundary term by deriving a uniform-in-time Friedrichs-type inequality, enabling strong convergence of approximate solutions. They establish the strong identification of the nonlinear term and, under higher data regularity, gain time-derivative regularity in , while also extending the results to Leray–Lions type operators. The approach provides a robust framework for nonlinear parabolic equations on moving domains and has potential applications to moving-thin domain limits and surface PDEs on evolving manifolds.

Abstract

This paper studies the parabolic -Laplace equation with in a moving domain under a Neumann type boundary condition corresponding to the total mass conservation. We establish the existence and uniqueness of a weak solution by the Galerkin method in evolving Bochner spaces and a monotonicity argument. The main difficulty is in characterizing the weak limit of the nonlinear gradient term, where we need to deal with a term which comes from the boundary condition and cannot be absorbed into a monotone operator. To overcome this difficulty, we prove a uniform-in-time Friedrichs type inequality on a moving domain with time-dependent basis functions and make use of it to get the strong convergence of approximate solutions. We also show that the time derivative exists in the sense when given data have a better regularity, and discuss extension of the existence and uniqueness results to a Leray-Lions type operator.
Paper Structure (22 sections, 27 theorems, 296 equations)

This paper contains 22 sections, 27 theorems, 296 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Moving domain
  4. Function spaces
  5. Other basic results
  6. Definition and uniqueness of a weak solution
  7. Existence of a weak solution
  8. Basis functions
  9. Approximate solutions
  10. Energy estimate and weak convergence
  11. Limit weak forms
  12. Strong convergence
  13. Characterization of the nonlinear term
  14. Regularity of the time derivative
  15. Another approximate solution
  16. ...and 7 more sections

Key Result

Lemma 2.2

The mappings $\Phi_{(\cdot)}^{-1}$ and $\nabla\Phi_{(\cdot)}^{-1}$ are of class $C^1$ on $\overline{Q_T}$.

Theorems & Definitions (57)

  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 47 more