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We prove local Calabi and higher order estimates for solutions to thecontinuity equation introduced by La Nave-Tian and extended to Hermitianmetrics by Sherman-Weinkove. We apply the estimates to show that on a compactcomplex manifold the Chern scalar curvature of a solution must blow up at afinite-time singularity. Additionally, starting from certain classes of initialdata on Oeljeklaus-Toma manifolds we prove Gromov-Hausdorff and smoothconvergence of the metric to a particular non-negative $(1,1)$-form as$t\to\infty$.

The continuity equation for Hermitian metrics: Calabi estimates, Chern scalar curvature and Oeljeklaus-Toma manifolds