Local and global solvability for advection-diffusion equation on an evolving surface with a boundary

Koba, Hajime

doi:10.2140/apde.2022.15.1617

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Local and global solvability for advection-diffusion equation on an evolving surface with a boundary

Hajime Koba

Vol. 15 (2022), No. 7, 1617–1654

DOI: 10.2140/apde.2022.15.1617

Abstract

This paper considers the existence of local and global-in-time strong solutions to the advection-diffusion equation with variable coefficients on an evolving surface with a boundary. We apply both the maximal $L^{p}$ -in-time regularity for Hilbert space-valued functions and semigroup theory to construct local and global-in-time strong solutions to an evolution equation. Using the approach and our function spaces on the evolving surface, we show the existence of local and global-in-time strong solutions to the advection-diffusion equation. Moreover, we derive the asymptotic stability of the global-in-time strong solution.

Keywords

advection-diffusion equation with variable coefficients, time-dependent Laplace–Beltrami operator, function spaces on evolving surfaces, maximal $L^p$-regularity, asymptotic stability

Mathematical Subject Classification 2010

Primary: 35A01

Secondary: 35R01, 35R37, 47D06

Milestones

Received: 3 May 2019

Revised: 2 February 2021

Accepted: 19 March 2021

Published: 5 December 2022

Authors

Hajime Koba
Graduate School of Engineering Science Osaka University Osaka Japan

Vol. 15, No. 7, 2022