Dissipative Hyperbolic Geometric Flow

TL;DR

This paper introduces a new dissipative hyperbolic geometric flow, explores its properties, solutions, and stability, and establishes foundational theoretical results for this novel mathematical construct.

Contribution

It defines the dissipative hyperbolic geometric flow, introduces hyperbolic Ricci solitons, and proves short-time existence, uniqueness, and stability results.

Findings

Existence and uniqueness of solutions established

Hyperbolic Ricci solitons introduced and characterized

Flow stability proven on Euclidean spaces of dimension > 2

Abstract

In this paper we introduce and study a new kind of hyperbolic geometric flows --dissipative hyperbolic geometric flow. This kind of flow is defined by a system of quasilinear wave equations with dissipative terms. Some interesting exact solutions are given, in particular, a new concept-- hyperbolic Ricci soliton is introduced and some of its geometric properties are described. We also establish the short-time existence and uniqueness theorem for the dissipative hyperbolic geometric flow, and prove the nonlinear stability of the flow defined on the Euclidean space of dimension larger than 2. Wave character of the evolving metrics and curvatures is illustrated and the nonlinear wave equations satisfied by the curvatures are derived.