Quasi-differentiable Banach manifold and phase-diagram of invariant parabolic differential equation in such manifold

TL;DR

This paper introduces a new class of quasi-differentiable Banach manifolds allowing differential calculus and studies the phase diagram of invariant parabolic differential equations within these manifolds, with applications to fluid dynamics.

Contribution

It defines quasi-differentiable Banach manifolds and analyzes the structure of phase diagrams near center manifolds under group actions, extending the understanding of invariant differential equations.

Findings

Quasi-differentiable Banach manifolds enable differential calculus in non-traditionally differentiable settings.

The phase diagram near the center manifold forms a homogeneous fibre bundle under certain conditions.

Application to Hele-Shaw problem demonstrates the practical relevance of the theoretical results.

Abstract

The purpose of this paper is twofold. First we study a class of Banach manifolds which are not differentiable in traditional sense but they are quasi-differentiable in the sense that a such Banach manifold has an embedded submanifold such that all points in that submanifold are differentiable and tangent spaces at those points can be defined. It follows that differential calculus can be performed in that submanifold and, consequently, differential equations in a such Banach manifold can be considered. Next we study the structure of phase diagram near center manifold of a parabolic differential equation in Banach manifold which is invariant or quasi-invariant under a finite number of mutually quasi-commutative Lie group actions. We prove that under certain conditions, near the center manifold $\mathcal{M}_c$ the underline manifold is a homogeneous fibre bundle over $\mathcal{M}_c$, with fibres being stable manifolds of the differential equation. As an application, asymptotic behavior of the solution of a two-free-surface Hele-Shaw problem is also studied.