An explicit Euler method for Sobolev vector fields with applications to the continuity equation on non cartesian grids

Abstract: We prove a novel stability estimate in $L^\infty _t (L^p _x)$ between the regular Lagrangian flow of a Sobolev vector field and a piecewise affine approximation of such flow. This approximation of the flow is obtained by a (sort of) explicit Euler method, and it is the crucial tool to prove approximation results for the solution of the continuity equation by using the representation of the solution as the push-forward via the regular Lagrangian flow of the initial datum. We approximate the solution in two ways, one probabilistic and one deterministic, using different approximations for both the flow and the initial datum.