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Nonlocal to local convergence of phase field systems with inertial term

Pierluigi Colli, Shunsuke Kurima, Luca Scarpa

Abstract

This paper deals with a nonlocal model for a hyperbolic phase field system coupling the standard energy balance equation for temperature with a dynamic for the phase variable: the latter includes an inertial term and a nonlocal convolution-type operator where the family of kernels depends on a small parameter. We rigorously study the asymptotic convergence of the system as the approximating parameter tends to zero and we obtain at the limit the local system with the elliptic laplacian operator acting on the phase variable. Our analysis is based on some asymptotic properties on nonlocal-to-local convergence that have been recently and successfully applied to families of Cahn--Hilliard models.

Nonlocal to local convergence of phase field systems with inertial term

Abstract

This paper deals with a nonlocal model for a hyperbolic phase field system coupling the standard energy balance equation for temperature with a dynamic for the phase variable: the latter includes an inertial term and a nonlocal convolution-type operator where the family of kernels depends on a small parameter. We rigorously study the asymptotic convergence of the system as the approximating parameter tends to zero and we obtain at the limit the local system with the elliptic laplacian operator acting on the phase variable. Our analysis is based on some asymptotic properties on nonlocal-to-local convergence that have been recently and successfully applied to families of Cahn--Hilliard models.
Paper Structure (6 sections, 10 theorems, 56 equations)

This paper contains 6 sections, 10 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Preliminaries
  4. Proof of Theorem \ref{['maintheorem2']}
  5. Uniform estimates for \ref{['Pep']}
  6. Passage to the limit

Key Result

Theorem 2.1

Assume A1- A5. Then, for all $\varepsilon \in (0, 1)$ there exists a unique solution $(\theta_{\varepsilon}, \varphi_{\varepsilon})$ of Pep.

Theorems & Definitions (20)

  • Remark 2.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.2
  • Theorem 2.1: K7
  • Theorem 2.2
  • Lemma 3.1: DST_JDE2021
  • Lemma 3.2: DST_JDE2021
  • Lemma 3.3: DST_JDE2021
  • Remark 3.1
  • ...and 10 more