Rotationally Symmetric 1-Harmonic Flows From $D^2$ to $S^2$: Local Well-Posedness and Finite Time Blowup

Abstract

The 1-harmonic flow from the disk to the sphere with constant Dirichlet boundary conditions is analyzed in the case of rotational symmetry. Sufficient conditions on the initial datum are given, such that a unique classical solution exists for short times. Also, a sharp criterion on the boundary condition is identified, such that any classical solution will blow up in finite time. Finally, nongeneric examples of finite time blowup are exhibited for any boundary condition.