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Existence and Uniqueness of Maximal Regular Flows for Non-smooth Vector Fields

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Abstract

In this paper we provide a complete analogy between the Cauchy–Lipschitz and the DiPerna–Lions theories for ODE’s, by developing a local version of the DiPerna–Lions theory. More precisely, we prove the existence and uniqueness of a maximal regular flow for the DiPerna–Lions theory using only local regularity and summability assumptions on the vector field, in analogy with the classical theory, which uses only local regularity assumptions. We also study the behaviour of the ODE trajectories before the maximal existence time. Unlike the Cauchy–Lipschitz theory, this behaviour crucially depends on the nature of the bounds imposed on the spatial divergence of the vector field. In particular, a global assumption on the divergence is needed to obtain a proper blow-up of the trajectories.

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Authors and Affiliations

  1. Scuola Normale Superiore, Pisa, Italy

    Luigi Ambrosio & Maria Colombo

  2. University of Texas at Austin, Austin, USA

    Alessio Figalli

Authors
  1. Luigi Ambrosio
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  2. Maria Colombo
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  3. Alessio Figalli
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Correspondence to Luigi Ambrosio.

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Communicated by A. Bressan

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Ambrosio, L., Colombo, M. & Figalli, A. Existence and Uniqueness of Maximal Regular Flows for Non-smooth Vector Fields. Arch Rational Mech Anal 218, 1043–1081 (2015). https://doi.org/10.1007/s00205-015-0875-9

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  • DOI: https://doi.org/10.1007/s00205-015-0875-9

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