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

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. Such estimates for the solution of the continuity equation are derived on non Cartesian grids and without the need to assume a CFL condition.

💡 Research Summary

The paper addresses the numerical approximation of continuity equations driven by Sobolev‑regular velocity fields on arbitrary, possibly non‑Cartesian meshes. The authors start from the DiPerna–Lions theory, which guarantees existence and uniqueness of a regular Lagrangian flow (RLF) for velocity fields (\mathbf v) belonging to (L^1(0,T;W^{1,\alpha}(\mathbb T^d))) with (\alpha>1) and bounded in space‑time. A key novelty is the introduction of an explicit Euler‑type scheme that directly approximates the RLF itself, rather than approximating the solution of the PDE in an Eulerian fashion.

Construction of the Euler approximation.
The velocity field is first mollified in space with a standard kernel (\rho_\varepsilon) to obtain (\mathbf v^\varepsilon = \rho_\varepsilon * \mathbf v). For a uniform time step (\Delta t) and a partition (0=t_0<t_1<\dots<t_N=T), the approximate flow (\Phi^{E}) is defined recursively by
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