A hyperbolic-type affine invariant curve flow

A new hyperbolic version of affine geometric flow is proposed, which is a family of plane curve flows whose acceleration is constant along affine normal direction. The equations satisfied by the graph and support function of the curve under this flow give rise to fully nonlinear hyperbolic equations. By reducing the flow to a single nonlinear hyperbolic equation, we obtain the existence for local solutions of the flow. Global existence is established by a method of LeFloch-Smoczyk for studying the hyperbolic mean curvature flow. The equations for both perimeter and area of closed curves under this flow are also obtained. Based on this, we show that for a closed curve, the solution of Cauchy problem of this flow blows up in finite time. Furthermore, some group-invariant solutions to this flow are discussed.

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Published 13 May 2014