Dissipative Hyperbolic Geometric Flow

In this paper we introduce 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.

Keywords

Dissipative hyperbolic geometric flow, quasilinear wave equation, hyperbolic Ricci soliton, short-time existence, nonlinear stability

2010 Mathematics Subject Classification

58J45, 58J47

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Published 1 January 2008