Well posedness of Lagrangian flows and continuity equations in metric measure spaces

Abstract

We establish, in a rather general setting, an analogue of DiPerna-Lions theory on well-posedness of flows of ODE's associated to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into ${\mathbb{R}}^{\infty }$. When specialized to the setting of Euclidean or infinite dimensional (e.g. Gaussian) spaces, large parts of previously known results are recovered at once. Moreover, the class of $\mathsf{RCD}(K,\infty )$ metric measure spaces object of extensive recent research fits into our framework. Therefore we provide, for the first time, well-posedness results for ODE's under low regularity assumptions on the velocity and in a nonsmooth context.

Slightly expanded some remarks on the technical assumption (7.11); Journal reference inserted

References in corpus (1)

• Derivations and Alberti representations

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• Parallel transport on non-collapsed $\mathsf{RCD}(K,N)$ spaces
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• Rigidity of some functional inequalities on RCD spaces
• On the uniqueness of solutions to continuity equations
• Convergence to the Product of the Standard Spheres and Eigenvalues of the Laplacian