Seminar Analysis and Mathematical Physics: Tommaso Cortopassi (Scuola Normale Superiore di Pisa)

An explicit Euler method for Sobolev vector fields with applications to the continuity equation

We introduce a new stability estimate for comparing the regular Lagrangian flow of a Sobolev vector field to a piecewise affine approximation generated by an explicit Euler-like method, in the spirit of Crippa-De Lellis's estimates. We use this estimate to prove approximation results for solutions of the continuity equation, which can be represented as the push forward of the initial datum via the regular Lagrangian flow. We give two examples: a probabilistic one using Dirac deltas to approximate the initial datum and a deterministic one using a diffuse approximation instead. In both cases, we assume no regularity on the mesh partitioning the spatial domain.

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