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Flows of Non-Smooth Vector Fields and Degenerate Elliptic Equations

With Applications to the Vlasov-Poisson and Semigeostrophic Systems

  • Book
  • © 2017

Overview

Authors:
  1. Maria Colombo

    1. Institute for Theoretical Studies, ETH Zürich, Zürich, Switzerland

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  • Provides a mathematical tool to describe the motion of large-scale oceanic/atmospheric flow
  • Focuses on interactions between transport equations and PDEs
  • The Theses was defended at Scuola Normale Superiore, Pisa

Part of the book series: Publications of the Scuola Normale Superiore (PSNS, volume 22)

Part of the book sub series: Theses (Scuola Normale Superiore) (TSNS)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xxiii
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  2. An overview on flows of vector fields and on optimal transport

    • Maria Colombo
    Pages 1-25

  3. Maximal regular flows for non-smooth vector fields

    • Maria Colombo
    Pages 27-44

  4. Main properties of maximal regular flows and analysis of blow-up

    • Maria Colombo
    Pages 45-70

  5. Lagrangian structure of transport equations

    • Maria Colombo
    Pages 71-97

  6. The continuity equation with an integrable damping term

    • Maria Colombo
    Pages 99-117

  7. Regularity results for very degenerate elliptic equations

    • Maria Colombo
    Pages 119-157

  8. An excess-decay result for a class of degenerate elliptic equations

    • Maria Colombo
    Pages 159-188

  9. The Vlasov-Poisson system

    • Maria Colombo
    Pages 189-230

  10. The semigeostrophic system

    • Maria Colombo
    Pages 231-253

  11. Back Matter

    Pages 255-269
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Keywords

About this book

The first part of the book is devoted to the transport equation for a given vector field, exploiting the lagrangian structure of solutions. It also treats the regularity of solutions of some degenerate elliptic equations, which appear in  the eulerian counterpart of some transport models with congestion. The second part of the book deals with the lagrangian structure of solutions of the Vlasov-Poisson system, which describes the evolution of a system of particles under the self-induced gravitational/electrostatic field, and the existence of solutions of the semigeostrophic system, used in meteorology to describe the motion of large-scale oceanic/atmospheric flows.​

Authors and Affiliations

  • Institute for Theoretical Studies, ETH Zürich, Zürich, Switzerland

    Maria Colombo

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