The Free Boundary Value Problem of \(\alpha \)-Harmonic Maps Flow

We study the \(\alpha \) $\alpha$ -harmonic map flow with free boundary for the Sacks–Uhlenbeck functional on a compact Riemann surface with smooth boundary and prove that this flow exists globally for smooth initial data and converges to an \(\alpha \) $\alpha$ -harmonic map with free boundary as \(t\rightarrow \infty \) $t\to \infty$ .