\(L^ 2\) flow of curve straightening in the plane (Q1320610)

For a plane curve considered as an elastic wire, the time evolution is modeled by decreasing total squared curvature. The resulting model is a partial differential equation \(u_ t = u_{ss} + f(u)\) with boundary conditions. Long time existence of solutions is established. Moreover, it is shown that convex curves stay convex and that a curve with rotation number \(\eta \neq 0\) approaches an \(\eta\)-fold circle.