On the analyticity of the flow map for the AHT equations

TL;DR

This paper proves that the flow map for the AHT equations in a bounded domain with an analytic boundary is analytic in time, extending understanding of solution regularity in optimal transport models.

Contribution

It demonstrates the analyticity of the flow map for the AHT equations in bounded domains with analytic boundaries, a novel regularity result.

Findings

Flow map is analytic in time for the AHT equations.

Analytic boundary conditions ensure solution regularity.

Extends regularity theory for nonlocal transport equations.

Abstract

The AHT equation is a non linear and non local vectorial transport equation which was introduced in 2003 by Angenent, Haker and Tannenbaum in optimal transport theory. For this equation, classical solutions are known to exist at least locally in time, and a flow map can thus be uniquely associated with these solutions. In this paper we consider the case where the equation is set in a bounded domain with an analytic boundary and we prove that the flow map is analytic with respect to time.