Heat flow for extrinsic biharmonic maps with small initial energy (Q1771563)

For mappings between compact Riemannian manifolds \(M\) and \(N\subseteq \mathbb{R}^k\), the \(L^2\) gradient flow for the extrinsic bi-energy \(F(u):={1\over2}\int_M| \Delta u| ^2\,dx\) is considered, where \(\Delta\) is applied component-wise in \(\mathbb{R}^k\). If \(\dim M\leq3\) or if \(\dim M=4\) and the initial bi-energy \(F(u_0)\) is small, it is proved that the extrinsic biharmonic map heat flow exists for all times. Moreover, for general initial data in the case \(\dim M=4\), possible singularity formation is understood completely; it can only happen by ``bubbling off'' of bi-energy in some sort of bi-harmonic bubble. The methods involved are a careful adaption to higher order of the corresponding proof by Struwe for the harmonic map heat flow [\textit{M. Struwe}, Comment. Math. Helv. 60, 558--581 (1985; Zbl 0595.58013)], combined with interpolation inequalities influenced by [\textit{E. Kuwert} and \textit{R. Schätzle}, J. Differ. Geom. 57, 409--441 (2001; Zbl 1035.53092)].