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 $L_t^{\infty }(L_x^p)$ 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. Such estimates for the solution of the continuity equation are derived on non Cartesian grids and without the need to assume a CFL condition.

Corrected Remark 12. In the previous version it was incorrectly stated that the method needed a CFL condition to be satisfied. The lack of a CFL condition is also remarked in the abstract and in the introduction