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


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2015-11

AUTHORS

Luigi Ambrosio, Maria Colombo, Alessio Figalli

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. More... »

PAGES

1043-1081

References to SciGraph publications

  • 2004-03. Renormalized solutions of some transport equations with partially W1,1 velocities and applications in ANNALI DI MATEMATICA PURA ED APPLICATA (1923 -)
  • 2001-03. Renormalized Solutions to the Vlasov Equation with Coefficients of Bounded Variation in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2008. Transport Equation and Cauchy Problem for Non-Smooth Vector Fields in CALCULUS OF VARIATIONS AND NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
  • 1989-10. Ordinary differential equations, transport theory and Sobolev spaces in INVENTIONES MATHEMATICAE
  • 2004-11. Transport equation and Cauchy problem for BV vector fields in INVENTIONES MATHEMATICAE
  • 2008. Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields in TRANSPORT EQUATIONS AND MULTI-D HYPERBOLIC CONSERVATION LAWS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00205-015-0875-9

    DOI

    http://dx.doi.org/10.1007/s00205-015-0875-9

    DIMENSIONS

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