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

By Luigi Ambrosio, Dario Trevisan. 2014

110 citations recorded by the source corpus.

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 ${\sf RCD}(K,\infty)$ 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7c1e2be024199f74614f6e873479ad71f0fb1c02

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Well-posedness of Multidimensional Diffusion Processes with Weakly Differentiable Coefficients (2015, methodology): “Let us point out that such a technique was first developed in [6] to deal with an analogue problem for derivations in metric measure spaces.”
Second order differentiation formula on RCD*$(K,N)$ spaces (2018, methodology): “We shall use the following formulation of their result (weaker than the one provided in [10]):”
Nonsmooth differential geometry - An approach tailored for spaces with Ricci curvature bounded from below (2014, background): “3, one can then verify that the theory developed in [15] can be fully read in terms of the language proposed here.”
Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows (2018, methodology): “Under these assumptions, the theory developed in [14] guarantees existence and uniqueness of the regular Lagrangian flow .”
Superposition and mimicking theorems for conditional McKean–Vlasov equations (2020, background): “Superposition principles in infinite-dimensional spaces, specifically in R and general metric measure spaces, have been established recently in [65] (see also [5, 62] for deterministic counterparts).”