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Let $(X,\omega)$ be a compact connected K\"ahler manifold and denote by$(\mathcal E^p,d_p)$ the metric completion of the space of K\"ahler potentials$\mathcal H_\omega$ with respect to the $L^p$-type path length metric $d_p$.First, we show that the natural analytic extension of the (twisted) MabuchiK-energy to $\mathcal E^p$ is a $d_p$-lsc functional that is convex alongfinite energy geodesics. Second, following the program of J. Streets, we usethis to study the asymptotics of the weak (twisted) Calabi flow inside theCAT(0) metric space $(\mathcal E^2,d_2)$. This flow exists for all times andcoincides with the usual smooth (twisted) Calabi flow whenever the latterexists. We show that the weak (twisted) Calabi flow either diverges withrespect to the $d_2$-metric or it $d_1$-converges to some minimizer of theK-energy inside $\mathcal E^2$. This gives the first concrete result about thelong time convergence of this flow on general K\"ahler manifolds, partiallyconfirming a conjecture of Donaldson. Finally, we investigate the possibilityof constructing destabilizing geodesic rays asymptotic to diverging weak(twisted) Calabi trajectories, and give a result in the case when the twistingform is K\"ahler. If the twisting form is only smooth, we reduce this problemto a conjecture on the regularity of minimizers of the K-energy on $\mathcalE^1$, known to hold in case of Fano manifolds.

Convexity of the extended K-energy and the large time behaviour of the weak Calabi flow