Compact traveling waves for anisotropic curvature flows with driving force

To study the dynamics of an anisotropic curvature flow with external driving force depending only on the normal vector, we focus on traveling waves composed of Jordan curves in R². Here we call them compact traveling waves. The objective of this study is to investigate thoroughly the condition of the driving force for the existence of compact traveling waves to the anisotropic curvature flow. It is shown that all traveling waves are strictly convex and unstable, and that a compact traveling wave is unique, if they exist. To determine the existence of compact traveling waves, three cases are considered: if the driving force is positive, there exists a compact traveling wave; if it is negative, there is no traveling wave; if it is sign-changing, a positive answer is obtained under the assumption called “admissible condition”. We also obtain a necessary and sufficient condition for the existence of axisymmetric compact traveling waves. Lastly, we make reference to the inverse problem and non-convex compact traveling waves.