On an anisotropic fractional Stefan-type problem with Dirichlet boundary conditions

Catharine W. K. Lo; José Francisco Rodrigues; Catharine W. K. Lo; José Francisco Rodrigues

doi:10.3934/mine.2023047

Mathematics in Engineering

2023, Volume 5, Issue 3: 1-38. doi: 10.3934/mine.2023047

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Research article

On an anisotropic fractional Stefan-type problem with Dirichlet boundary conditions

CMAFcIO – Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa P-1749-016 Lisboa, Portugal

Received: 08 March 2022 Revised: 19 July 2022 Accepted: 19 July 2022 Published: 08 August 2022

In this work, we consider the fractional Stefan-type problem in a Lipschitz bounded domain $ \Omega\subset\mathbb{R}^d $ with time-dependent Dirichlet boundary condition for the temperature $ \vartheta = \vartheta(x, t) $, $ \vartheta = g $ on $ \Omega^c\times]0, T[$, and initial condition $ \eta_0 $ for the enthalpy $ \eta = \eta(x, t) $, given in $ \Omega\times]0, T[$ by

$ \frac{\partial \eta}{\partial t} +\mathcal{L}_A^s \vartheta = f\quad\text{ with }\eta\in \beta(\vartheta), $

where $ \mathcal{L}_A^s $ is an anisotropic fractional operator defined in the distributional sense by

$ \langle\mathcal{L}_A^su, v\rangle = \int_{\mathbb{R}^d}AD^su\cdot D^sv\, dx, $

$ \beta $ is a maximal monotone graph, $ A(x) $ is a symmetric, strictly elliptic and uniformly bounded matrix, and $ D^s $ is the distributional Riesz fractional gradient for $ 0 < s < 1 $. We show the existence of a unique weak solution with its corresponding weak regularity. We also consider the convergence as $ s\nearrow 1 $ towards the classical local problem, the asymptotic behaviour as $ t\to\infty $, and the convergence of the two-phase Stefan-type problem to the one-phase Stefan-type problem by varying the maximal monotone graph $ \beta $.
- Stefan problem,
- fractional derivatives,
- boundary value problem,
- nonlocal diffusion,
- phase transitions,
- subdifferential,
- nonlinear,
- fractional evolution equation
Citation: Catharine W. K. Lo, José Francisco Rodrigues. On an anisotropic fractional Stefan-type problem with Dirichlet boundary conditions[J]. Mathematics in Engineering, 2023, 5(3): 1-38. doi: 10.3934/mine.2023047

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Abstract

In this work, we consider the fractional Stefan-type problem in a Lipschitz bounded domain $ \Omega\subset\mathbb{R}^d $ with time-dependent Dirichlet boundary condition for the temperature $ \vartheta = \vartheta(x, t) $, $ \vartheta = g $ on $ \Omega^c\times]0, T[$, and initial condition $ \eta_0 $ for the enthalpy $ \eta = \eta(x, t) $, given in $ \Omega\times]0, T[$ by

$ \frac{\partial \eta}{\partial t} +\mathcal{L}_A^s \vartheta = f\quad\text{ with }\eta\in \beta(\vartheta), $

where $ \mathcal{L}_A^s $ is an anisotropic fractional operator defined in the distributional sense by

$ \langle\mathcal{L}_A^su, v\rangle = \int_{\mathbb{R}^d}AD^su\cdot D^sv\, dx, $

$ \beta $ is a maximal monotone graph, $ A(x) $ is a symmetric, strictly elliptic and uniformly bounded matrix, and $ D^s $ is the distributional Riesz fractional gradient for $ 0 < s < 1 $. We show the existence of a unique weak solution with its corresponding weak regularity. We also consider the convergence as $ s\nearrow 1 $ towards the classical local problem, the asymptotic behaviour as $ t\to\infty $, and the convergence of the two-phase Stefan-type problem to the one-phase Stefan-type problem by varying the maximal monotone graph $ \beta $.

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