Lipschitz regularity of harmonic map heat flows into $CAT(0)$ spaces

In 1964, Eells and Sampson proved the celebrated long-time existence and convergence for the harmonic map heat flow into non-positively curved Riemannian manifolds. In 1992, Gromov and Schoen initiated the study of harmonic maps into $CAT(0) $ metric spaces. It naturally motivates the study of the harmonic map heat flow into singular metric spaces. In the 1990s, Mayer and Jost independently studied convex functionals on $CAT(0)$ spaces and extended Crandall-Liggett's theory of gradient flows from Banach spaces to $CAT(0) $ spaces to obtain the weak solutions for the harmonic map heat flow into $CAT(0)$ spaces. The weak solutions enjoy the favorable long-time existence, uniqueness and well-established long-time behaviors. It is a long-standing open question to ask if the weak solutions possess the Lipschitz regularity. Very recently, by using elliptic approximation method, Lin, Segatti, Sire, and Wang proved the weak solutions are Lipschitz in space and $1\over 2$-Hölder continuous in time, for a wide class of $CAT(0)$ spaces. In the present paper, we give a complete answer to the question. We show that every weak solution of the harmonic map heat flow into $CAT(0)$ spaces is Lipschitz continuous in both space and time. We also establish an Eells-Sampson-type Bochner inequality.