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11. On the Lagrangian branched transport model and the equivalence with its Eulerian formulation

  • Paul Pegon
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Topological Optimization and Optimal Transport
This chapter is in the book Topological Optimization and Optimal Transport

Abstract

First,we present two classicalmodels of branched transport: the Lagrangian model introduced by Bernot et al. and Maddalena et al. [3, 7], and the Eulerianmodel introduced by Xia [13]. An emphasis is put on the Lagrangian model, forwhichwe give a complete proof of existence of minimizers in a - hopefully - simplified manner.We also treat in detail some σ-finiteness and rectifiability issues to yield rigorously the energy formula connecting the irrigation cost Iα to the Gilbert energy Eα. Our main purpose is to use this energy formula and exploit a Smirnov decomposition of vector flows, which was proved via the Dacorogna-Moser approach in [9], to establish the equivalence between the Lagrangian and Eulerian models, as stated in Theorem 4.7.

Abstract

First,we present two classicalmodels of branched transport: the Lagrangian model introduced by Bernot et al. and Maddalena et al. [3, 7], and the Eulerianmodel introduced by Xia [13]. An emphasis is put on the Lagrangian model, forwhichwe give a complete proof of existence of minimizers in a - hopefully - simplified manner.We also treat in detail some σ-finiteness and rectifiability issues to yield rigorously the energy formula connecting the irrigation cost Iα to the Gilbert energy Eα. Our main purpose is to use this energy formula and exploit a Smirnov decomposition of vector flows, which was proved via the Dacorogna-Moser approach in [9], to establish the equivalence between the Lagrangian and Eulerian models, as stated in Theorem 4.7.

Chapters in this book

  1. Frontmatter I
  2. Contents V
  3. Part I
  4. 1. Geometric issues in PDE problems related to the infinity Laplace operator 3
  5. 2. Solution of free boundary problems in the presence of geometric uncertainties 20
  6. 3. Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies 40
  7. 4. High-order topological expansions for Helmholtz problems in 2D 64
  8. 5. On a new phase field model for the approximation of interfacial energies of multiphase systems 123
  9. 6. Optimization of eigenvalues and eigenmodes by using the adjoint method 142
  10. 7. Discrete varifolds and surface approximation 159
  11. Part II
  12. Preface 173
  13. 8. Weak Monge–Ampère solutions of the semi-discrete optimal transportation problem 175
  14. 9. Optimal transportation theory with repulsive costs 204
  15. 10. Wardrop equilibria: long-term variant, degenerate anisotropic PDEs and numerical approximations 257
  16. 11. On the Lagrangian branched transport model and the equivalence with its Eulerian formulation 281
  17. 12. On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows 304
  18. 13. Pressureless Euler equations with maximal density constraint: a time-splitting scheme 333
  19. 14. Convergence of a fully discrete variational scheme for a thin-film equation 356
  20. 15. Interpretation of finite volume discretization schemes for the Fokker–Planck equation as gradient flows for the discrete Wasserstein distance 400
  21. Index 417
https://doi.org/10.1515/9783110430417-012

Chapters in this book

  1. Frontmatter I
  2. Contents V
  3. Part I
  4. 1. Geometric issues in PDE problems related to the infinity Laplace operator 3
  5. 2. Solution of free boundary problems in the presence of geometric uncertainties 20
  6. 3. Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies 40
  7. 4. High-order topological expansions for Helmholtz problems in 2D 64
  8. 5. On a new phase field model for the approximation of interfacial energies of multiphase systems 123
  9. 6. Optimization of eigenvalues and eigenmodes by using the adjoint method 142
  10. 7. Discrete varifolds and surface approximation 159
  11. Part II
  12. Preface 173
  13. 8. Weak Monge–Ampère solutions of the semi-discrete optimal transportation problem 175
  14. 9. Optimal transportation theory with repulsive costs 204
  15. 10. Wardrop equilibria: long-term variant, degenerate anisotropic PDEs and numerical approximations 257
  16. 11. On the Lagrangian branched transport model and the equivalence with its Eulerian formulation 281
  17. 12. On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows 304
  18. 13. Pressureless Euler equations with maximal density constraint: a time-splitting scheme 333
  19. 14. Convergence of a fully discrete variational scheme for a thin-film equation 356
  20. 15. Interpretation of finite volume discretization schemes for the Fokker–Planck equation as gradient flows for the discrete Wasserstein distance 400
  21. Index 417
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