Gradient flow of the Dirichlet energy for the curvature of plane curves

This thesis studies curvature flows of planar curves with Neumann boundary condition and flows of closed planar curves without boundary. We describe the local existence for them. For the global existence results for curvature flows of planar curves, we consider the sixth and higher order curvature flows of planar curves with suitable associated generalised Neumann boundary condition. The conclusion is that the solution of each flow problem exists for all time and converges to a unique line segment exponentially. Moreover, we study the curve diffusion flow and ideal curve flow of planar curves with constrained length, as well as the ideal curve flow of planar curves with preserved area. Closed curve satisfying one of these flows exists for all time and converges to a unique round circle exponentially.