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A new transportation distance with bulk/interface interactions and flux penalization

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Abstract

We introduce and study a new optimal transport problem on a bounded domain \({{\bar{\Omega }}}\subset {\mathbb {R}}^d\), defined via a dynamical Benamou–Brenier formulation. The model handles differently the motion in the interior and on the boundary, and penalizes the transfer of mass between the two. The resulting distance interpolates between classical optimal transport on \({{\bar{\Omega }}}\) on the one hand, and on the other hand between two independent optimal transport problems set on \(\Omega \) and \({\partial \Omega }\).

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References

  1. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Springer, Berlin (2008)

    MATH  Google Scholar 

  2. Benamou, J.-D., Brenier, Y.: A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numerische Mathematik 84(3), 375–393 (2000)

    Article  MathSciNet  Google Scholar 

  3. Bouchitté, G., Buttazzo, G.: New lower semicontinuity results for nonconvex functionals defined on measures. Nonlinear Anal. Theory Methods Appl. 15(7), 679–692 (1990)

    Article  MathSciNet  Google Scholar 

  4. Brenier, Y: Extended Monge–Kantorovich theory. In: Optimal Transportation and Applications, pp. 91–121. Springer (2003)

  5. Caffarelli, L.A., McCann, R.J.: Free boundaries in optimal transport and Monge–Ampère obstacle problems. Ann. Math. 171, 673–730 (2010)

    Article  MathSciNet  Google Scholar 

  6. Chizat, L., Peyré, G., Schmitzer, B., Vialard, F.-X.: An interpolating distance between optimal transport and Fisher–Rao metrics. Found. Comput. Math. 18(1), 1–44 (2018)

    Article  MathSciNet  Google Scholar 

  7. Chizat, L., Peyré, G., Schmitzer, B., Vialard, F.-X.: Scaling algorithms for unbalanced optimal transport problems. Math. Comput. 87(314), 2563–2609 (2018)

    Article  MathSciNet  Google Scholar 

  8. Chizat, L., Peyré, G., Schmitzer, B., Vialard, F.-X.: Unbalanced optimal transport: dynamic and Kantorovich formulations. J. Funct. Anal. 274(11), 3090–3123 (2018)

    Article  MathSciNet  Google Scholar 

  9. Dolbeault, J., Nazaret, B., Savaré, G.: A new class of transport distances between measures. Calc. Var. Partial Differ. Equ. 34(2), 193–231 (2009)

    Article  MathSciNet  Google Scholar 

  10. Dudley, R.M.: Real Analysis and Probability. Chapman and Hall/CRC, Cambridge (2018)

    Google Scholar 

  11. Figalli, A.: The optimal partial transport problem. Arch. Ration. Mech. Anal. 195(2), 533–560 (2010)

    Article  MathSciNet  Google Scholar 

  12. Figalli, A., Gigli, N.: A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions. J. Math. Pures Appl. 94(2), 107–130 (2010)

    Article  MathSciNet  Google Scholar 

  13. Gallouët, T., Laborde, M., Monsaingeon, L.: An unbalanced optimal transport splitting scheme for general advection–reaction–diffusion problems. ESAIM Control Optim. Calc. Var. 25, 8 (2019)

    Article  MathSciNet  Google Scholar 

  14. Gallouët, T., Monsaingeon, L.: A JKO splitting scheme for Kantorovich–Fisher–Rao gradient flows. SIAM J. Math. Anal. 49(2), 1100–1130 (2017)

    Article  MathSciNet  Google Scholar 

  15. Gallouët, T., Natale, A., Vialard, F.-X.: Generalized compressible flows and solutions of the \({H}\)(div) geodesic problem. Arch. Ration. Mech. Anal. 235, 1–56 (2019)

    MathSciNet  Google Scholar 

  16. Gallouët, T., Vialard, F.-X.: The Camassa–Holm equation as an incompressible Euler equation: a geometric point of view. J. Differ. Equ. 264(7), 4199–4234 (2018)

    Article  MathSciNet  Google Scholar 

  17. Gangbo, W., Li, W., Osher, S., Puthawala, M.: Unnormalized optimal transport. J. Comput. Phys. 399, 108940 (2019)

    Article  MathSciNet  Google Scholar 

  18. Glitzky, A., Mielke, A.: A gradient structure for systems coupling reaction–diffusion effects in bulk and interfaces. Zeitschrift für angewandte Mathematik und Physik 64(1), 29–52 (2013)

    Article  MathSciNet  Google Scholar 

  19. Kantorovich, L.V.: On the translocation of masses. Dokl. Akad. Nauk. USSR (NS) 37, 199–201 (1942)

    MathSciNet  MATH  Google Scholar 

  20. Kondratyev, S., Monsaingeon, L., Vorotnikov, D.: A fitness-driven cross-diffusion system from population dynamics as a gradient flow. J. Differ. Equ. 261(5), 2784–2808 (2016)

    Article  MathSciNet  Google Scholar 

  21. Kondratyev, S., Monsaingeon, L., Vorotnikov, D.: A new optimal transport distance on the space of finite Radon measures. Adv. Differ. Equ. 21(11/12), 1117–1164 (2016)

    MathSciNet  MATH  Google Scholar 

  22. Kondratyev, S., Monsaingeon, L., Vorotnikov, D.: A new multicomponent Poincaré–Beckner inequality. J. Funct. Anal. 272(8), 3281–3310 (2017)

    Article  MathSciNet  Google Scholar 

  23. Kondratyev, S., Vorotnikov, D.: Convex Sobolev inequalities related to unbalanced optimal transport. J. Differ. Equ. 268(7), 3705–3724 (2020)

  24. Kondratyev, S., Vorotnikov, D.: Spherical Hellinger–Kantorovich gradient flows. SIAM J. Math. Anal. 51(3), 2053–2084 (2019)

    Article  MathSciNet  Google Scholar 

  25. Kondratyev, S., Vorotnikov, D.: Nonlinear Fokker–Planck equations with reaction as gradient flows of the free energy. J. Funct. Anal. 278(2), 108310 (2020)

    Article  MathSciNet  Google Scholar 

  26. Laschos, V., Mielke, A.: Geometric properties of cones with applications on the Hellinger–Kantorovich space, and a new distance on the space of probability measures. J. Funct. Anal. 276(11), 3529–3576 (2019)

    Article  MathSciNet  Google Scholar 

  27. Lavenant, H.: Unconditional convergence for discretizations of dynamical optimal transport. arXiv preprint arXiv:1909.08790 (2019)

  28. Liero, M., Mielke, A.: Gradient structures and geodesic convexity for reaction–diffusion systems. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 371(2005), 20120346 (2013)

    Article  MathSciNet  Google Scholar 

  29. Liero, M., Mielke, A., Savaré, G.: Optimal transport in competition with reaction: the Hellinger–Kantorovich distance and geodesic curves. SIAM J. Math. Anal. 48(4), 2869–2911 (2016)

    Article  MathSciNet  Google Scholar 

  30. Liero, M., Mielke, A., Savaré, G.: Optimal entropy-transport problems and a new Hellinger–Kantorovich distance between positive measures. Inventiones mathematicae 211(3), 969–1117 (2018)

    Article  MathSciNet  Google Scholar 

  31. Mielke, A.: Thermomechanical modeling of energy–reaction–diffusion systems, including bulk-interface interactions. Disc. Continuous Dyn. Syst-S, 6(2), 479 (2013)

  32. Monge, G.: Mémoire sur la théorie des déblais et des remblais. Histoire de l’Académie Royale des Sciences de Paris (1781)

  33. Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial. Differ. Equ. 26(1–2), 101–174 (2001)

  34. Peyré, G., Cuturi, M., et al.: Computational optimal transport. Found. Trends Mach. Learn. 11(5–6), 355–607 (2019)

    Article  Google Scholar 

  35. Piccoli, B., Rossi, F.: Generalized Wasserstein distance and its application to transport equations with source. Arch. Ration. Mech. Anal. 211(1), 335–358 (2014)

    Article  MathSciNet  Google Scholar 

  36. Piccoli, B., Rossi, F.: On properties of the generalized Wasserstein distance. Arch. Ration. Mech. Anal. 222(3), 1339–1365 (2016)

    Article  MathSciNet  Google Scholar 

  37. Profeta, A., Sturm, K.-T.: Heat flow with Dirichlet boundary conditions via optimal transport and gluing of metric measure spaces. arXiv preprint arXiv:1809.00936, (2018)

  38. Rockafellar, R.: Duality and stability in extremum problems involving convex functions. Pac. J. Math. 21(1), 167–187 (1967)

    Article  MathSciNet  Google Scholar 

  39. Rockafellar, R.: Integrals which are convex functionals II. Pac. J. Math. 39(2), 439–469 (1971)

    Article  MathSciNet  Google Scholar 

  40. Santambrogio, F.: Optimal Transport for Applied Mathematicians, vol. 55(58-63), p. 94. Birkäuser, New York (2015)

    Book  Google Scholar 

  41. Villani, C.: Topics in Optimal Transportation, vol. 58. American Mathematical Soc, Providence (2003)

    MATH  Google Scholar 

  42. Villani, C.: Optimal Transport: Old and New, vol. 338. Springer, Berlin (2008)

    MATH  Google Scholar 

  43. Zurek, A.: Problèmes à interface mobile pour la dégradation de matériaux et la croissance de biofilms: analyse numérique et modélisation. Ph.D. thesis (2019)

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Acknowledgements

The author warmly thanks C. Cancès and B. Merlet for discussions pertaining to this work, and D. Vorotnikov for his help with the proof of Theorem 4. This work was funded by the Portuguese Science Foundation through FCT Project PTDC/MAT-STA/28812/2007 SchröMoka.

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Authors and Affiliations

  1. Institut Élie Cartan de Lorraine, Université de Lorraine, Site de Nancy, B.P. 70239, 54506, Vandoeuvre-lès-Nancy Cedex, France

    Léonard Monsaingeon

  2. Grupo de Física Matemática, GFMUL, Faculdade de Ciências, Universidade de Lisboa, 1749-016, Lisbon, Portugal

    Léonard Monsaingeon

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  1. Léonard Monsaingeon
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Correspondence to Léonard Monsaingeon.

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Communicated by A. Malchiodi.

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Monsaingeon, L. A new transportation distance with bulk/interface interactions and flux penalization. Calc. Var. 60, 101 (2021). https://doi.org/10.1007/s00526-021-01946-2

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