Dissipative hyperbolic geometric flow. (English) Zbl 1165.58013

Dai, Wen-Rong; Kong, De-Xing; Liu, Kefeng

Asian J. Math. 12, No. 3, 345-364 (2008).

The authors 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. Short-time existence and uniqueness theorem is also established for the dissipative hyperbolic geometric flow, and nonlinear stability of the flow defined on the Euclidean space of dimension larger than \(2\) is proved. Wave character of the evolving metrics and curvatures is illustrated and the nonlinear wave equations satisfied by the curvatures are derived.

Reviewer: Dian K. Palagachev (Bari)

MSC:

58J45	Hyperbolic equations on manifolds
58J47	Propagation of singularities; initial value problems on manifolds

Keywords:

dissipative hyperbolic geometric flow; quasilinear wave equation; hyperbolic Ricci soliton; short-time existence; nonlinear stability