An improvement of regularity result for pseudo Calabi flow

In this paper, we observe that if the initial data of pseudo Calabi flow has volume form $C^0$ close to a smooth one, then the flow is immediately smooth for $t>0$. As an application, we show that if the initial data has volume form $C^0$ close to that of a cscK metric, then the pseudo Calabi flow exists for $t\in (0,+\mathrm{\infty })$. We also prove similar improvement of regularity and long time existence result for pseudo Calabi flow on a Fano manifold when the volume form is bounded and the class is close to $c_1(M)$.