The transverse Chern-Ricci flow

We introduce transverse Chern-Ricci flow for transversely Hermitian foliations, which is analogous to the Chern-Ricci flow. We show that when $\mathcal{F}$ is homologically orientable and the basic first Bott-Chern class is zero, starting at any transversely Hermitian metric the flow exists for all time and as $t\to \mathrm{\infty }$ converges smoothly to a transversely Hermitian metric ${\omega }_{\mathrm{\infty }}$ with the transverse Chern-Ricci form ${\rho }^T({\omega }_{\mathrm{\infty }})=0$. We also determine the maximal existence time of the flow in the general case. These are foliated version of results of Gill and Tosatti-Weinkove, and also extend recent work of Bedulli-He-Vezzoni.