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Existence, uniqueness and homogenization of nonlinear parabolic problems with dynamical boundary conditions in perforated media

María Anguiano

TL;DR

The paper analyzes a nonlinear parabolic equation with nonlinear dynamical boundary conditions on periodically distributed holes and proves existence/uniqueness of solutions in perforated domains. It develops an energy-based variational framework and ε-independent a priori estimates, then employs an extension argument and compactness (à la Vanninathan) to pass to the homogenization limit. The homogenized model on the intact domain Ω features an effective diffusion tensor Q and extra nonlinear boundary contributions turning into volumetric terms via cell problems. The work provides a rigorous derivation of the macroscopic limit and shows the homogenized problem has a unique weak solution, clarifying how boundary nonlinearities influence the bulk dynamics.

Abstract

We consider a nonlinear parabolic problem with nonlinear dynamical boundary conditions of pure-reactive type in a media perforated by periodically distributed holes of size $\varepsilon$. The novelty of our work is to consider a nonlinear model where the nonlinearity also appears in the boundary. The existence and uniqueness of solution is analyzed. Moreover, passing to the limit when $\varepsilon$ goes to zero, a new nonlinear parabolic problem defined on a unified domain without holes with zero Dirichlet boundary condition and with extra-terms coming from the influence of the nonlinear dynamical boundary conditions is rigorously derived.

Existence, uniqueness and homogenization of nonlinear parabolic problems with dynamical boundary conditions in perforated media

TL;DR

The paper analyzes a nonlinear parabolic equation with nonlinear dynamical boundary conditions on periodically distributed holes and proves existence/uniqueness of solutions in perforated domains. It develops an energy-based variational framework and ε-independent a priori estimates, then employs an extension argument and compactness (à la Vanninathan) to pass to the homogenization limit. The homogenized model on the intact domain Ω features an effective diffusion tensor Q and extra nonlinear boundary contributions turning into volumetric terms via cell problems. The work provides a rigorous derivation of the macroscopic limit and shows the homogenized problem has a unique weak solution, clarifying how boundary nonlinearities influence the bulk dynamics.

Abstract

We consider a nonlinear parabolic problem with nonlinear dynamical boundary conditions of pure-reactive type in a media perforated by periodically distributed holes of size . The novelty of our work is to consider a nonlinear model where the nonlinearity also appears in the boundary. The existence and uniqueness of solution is analyzed. Moreover, passing to the limit when goes to zero, a new nonlinear parabolic problem defined on a unified domain without holes with zero Dirichlet boundary condition and with extra-terms coming from the influence of the nonlinear dynamical boundary conditions is rigorously derived.

Paper Structure

This paper contains 7 sections, 10 theorems, 131 equations.

Table of Contents

  1. Introduction and setting of the problem
  2. Some notations
  3. Existence and uniqueness of solution
  4. A priori estimates
  5. The extension of $u_\varepsilon$ to the whole $\Omega \times (0,T)$
  6. A compactness result
  7. Homogenized model

Key Result

Theorem 3.3

Under the assumptions (hyp 0)--(hip_4), there exists a unique solution $(u_\varepsilon,\psi_\varepsilon)$ of the problem (PDE). Moreover, this solution satisfies the energy equality

Theorems & Definitions (25)

  • Remark 1.1
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 15 more