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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
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