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

2019/01/16 by Shangbin Cui, Cui, Shangbin
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1901.05443

openalex publication_date 2019/01/16 · openalex created_date 2019/03/22 · openalex updated_date 2026/07/28

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 M_c the underline manifold is a homogeneous fibre bundle over 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.

Citations

• The principle of linearized stability for quasilinear parabolic evolution equations
• Asymptotic exponential stability in quasilinear parabolic equations
• On the Banach manifold of simple domains in the Euclidean space and applications to free boundary problems

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