A priori error estimates for the $\theta$-method for the flow of nonsmooth velocity fields

Velocity fields with low regularity (below the Lipschitz threshold) naturally arise in many models from mathematical physics, such as the inhomogeneous incompressible Navier-Stokes equations, and play a fundamental role in the analysis of nonlinear PDEs. The DiPerna-Lions theory ensures existence and uniqueness of the flow associated with a divergence-free velocity field with Sobolev regularity. In this paper, we establish a priori error estimates showing a logarithmic rate of convergence of numerical solutions, constructed via the $\theta$-method, towards the exact (analytic) flow for a velocity field with Sobolev regularity. In addition, we derive analogous a priori error estimates for Lagrangian solutions of the associated transport equation, exhibiting the same logarithmic rate of convergence. Our theoretical results are supported by numerical experiments, which confirm the predicted logarithmic behavior.