Abstract or Additional Information
The heat flow of harmonic maps can be viewed as the gradient flow of Dirichlet energy functional of maps between manifolds,
which has been extensively studied thanks to its important roles in Calculus of Variations and Geometric Evolution Equations.
In this talk, I will introduce an extension into the nonlocal setting, namely, the heat flow of fractional harmonic maps into spheres (or general manifolds N)
that takes the geometric form: $(\partial_t-\Delta)^s u\perp T_u N$, $0<s<1$. I will describe a recent joint work, together with Du, Huang, and Schikorra,
that establishes the partial regularity for their suitable weak solutions which satisfy both energy monotonicity and local energy inequality.
The ideas are based on the parabolic Cafferalli-Silvestre extension, the div-curl structure of nonlinearity of the system in extended variable,
and the weighted Hardy-BMO duality with A_2 weights.