# Local and Global Solvability for Advection-Diffusion Equation on an Evolving Surface with a Boundary

## TL;DR

This paper establishes the existence and stability of local and global strong solutions to the advection-diffusion equation on evolving surfaces with boundaries, using advanced functional analysis techniques.

## Contribution

It introduces a novel approach combining maximal $L^p$-regularity and semigroup theory to solve the advection-diffusion equation on evolving surfaces with boundaries.

## Key findings

- Existence of local strong solutions on evolving surfaces.
- Existence of global-in-time strong solutions.
- Proven asymptotic stability of the global solutions.

## 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 the 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.