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Characterisation of homogenisation for nonlocal diffusion by local topologies

Andreas Buchinger, Krešimir Burazin, Ivana Crnjac, Marko Erceg, Maja Jolić, Marcus Waurick

TL;DR

The paper develops a comprehensive homogenisation theory for nonlocal diffusion with local oscillatory coefficients by introducing $H^s$-convergence, linking it to classical $H$-convergence on the domain and weak-$*$ limits outside the domain. It builds a robust analytical framework using fractional Sobolev spaces, fractional compensated compactness, and a fractional div-curl machinery to prove compactness, uniqueness, and energy convergence. A central result is the precise characterization of $H^s$-convergence in terms of local $H$-convergence and exterior weak-$*$ convergence, with connections to Schur topologies and two Schur-topology descriptions that yield the same metric structure. The work also extends the theory to the one-dimensional case, to coefficients defined on all of ${\mathbb R}^d$, and to symmetric settings via $G^s$-convergence, culminating in a homogenisation result for a fractional heat equation and demonstrating the practical impact for nonlocal models with heterogeneous media.

Abstract

We consider fractional variants of divergence form problems with highly oscillatory local coefficients. We characterise the convergence of these coefficients by means of classical $H$-convergence covering the local behaviour of the fractional divergence form problem and weak-$\ast$ convergence on the complement caused by the nonlocality of the differential operators. The results are further described in the light of nonlocal $H$-convergence as introduced in [Waurick, Calc Var PDEs, 57, 2018] and certain Schur topologies. Applications to symmetric coefficients and a homogenisation problem for a fractional heat type equation are provided.

Characterisation of homogenisation for nonlocal diffusion by local topologies

TL;DR

The paper develops a comprehensive homogenisation theory for nonlocal diffusion with local oscillatory coefficients by introducing -convergence, linking it to classical -convergence on the domain and weak- limits outside the domain. It builds a robust analytical framework using fractional Sobolev spaces, fractional compensated compactness, and a fractional div-curl machinery to prove compactness, uniqueness, and energy convergence. A central result is the precise characterization of -convergence in terms of local -convergence and exterior weak- convergence, with connections to Schur topologies and two Schur-topology descriptions that yield the same metric structure. The work also extends the theory to the one-dimensional case, to coefficients defined on all of , and to symmetric settings via -convergence, culminating in a homogenisation result for a fractional heat equation and demonstrating the practical impact for nonlocal models with heterogeneous media.

Abstract

We consider fractional variants of divergence form problems with highly oscillatory local coefficients. We characterise the convergence of these coefficients by means of classical -convergence covering the local behaviour of the fractional divergence form problem and weak- convergence on the complement caused by the nonlocality of the differential operators. The results are further described in the light of nonlocal -convergence as introduced in [Waurick, Calc Var PDEs, 57, 2018] and certain Schur topologies. Applications to symmetric coefficients and a homogenisation problem for a fractional heat type equation are provided.
Paper Structure (32 sections, 31 theorems, 184 equations, 1 figure)

This paper contains 32 sections, 31 theorems, 184 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fractional operators
  4. Fractional Sobolev spaces
  5. Proof.
  6. Proof.
  7. Fractional compensated compactness
  8. Proof.
  9. Proof.
  10. Fractional diffusion: well-posedness
  11. $H^s$-convergence on ${\cal M}(\alpha,\beta;\Omega, {\bf A}_0)$
  12. Proof of \ref{['th:ccm1']}.
  13. Proof.
  14. Proof.
  15. Proof of Theorem \ref{['thm:metr']}.
  16. ...and 17 more sections

Key Result

Theorem 1.3

Let $\Omega$ have a boundary of measure zero, $d\geq 1$, $s\in \langle 0,1\rangle$. Let $({\bf A}_n)_n$, ${\bf A}$ in ${\cal M}(\alpha,\beta;{{\mathbb R}^{d}})$. Then where $\xrightharpoonup{*}$ denotes the weak-* convergence in ${\rm L}^\infty({{\mathbb R}^{d}}\setminus{\sf Cl}\Omega;{\mathbb R}^{d\times d})$.

Figures (1)

  • Figure 1: Characterisation of $H^s$-convergence for the classes ${\cal M}(\alpha,\beta;\Omega, {\bf A}_0)$ and ${\cal M}(\alpha,\beta;{{\mathbb R}^{d}})$. The former is a subset of the latter, consisting of operators with coefficients fixed outside $\Omega$. Here, weak-$\ast$ convergence is understood in the sense of the ${\rm L}^\infty(\mathbb{R}^d\setminus{\sf Cl}\Omega)^{d\times d}$ space (middle box). In a broader sense, $H^s$-convergence can be interpreted within the framework of nonlocal $H$-convergence (largest box).

Theorems & Definitions (48)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • ...and 38 more