Dynamic optimal transport on networks

Burger, Martin; Humpert, Ina; Pietschmann, Jan‐Frederik

doi:10.48550/arxiv.2101.03415

Cited by 2 publications

(3 citation statements)

References 16 publications

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“…In turn, [1, Lemma 8.2.1] provides the existence of a continuous representative for distributional solutions of continuity equations with velocity fields in L 1 ([0, T ]; L 1 (ρ t )). In particular, for test functions time-independent, we get formulation (3), where we also used that ∇W ε is odd. Note that this formulation overtakes the loss of regularity at 0 for ∇W ε , as already noticed in [8].…”

Section: Definition 22 (Weak Measure Solution To (Nlie)) a Narrowly C...mentioning

confidence: 99%

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Porous medium equation as limit of nonlocal interaction

Burger¹,

Esposito²

2022

Preprint

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This paper studies the convergence of solutions of a nonlocal interaction equation to the solution of the quadratic porous medium equation in the limit of a localising interaction kernel. The analysis is carried out at the level of the (nonlocal) partial differential equations and we use the gradient flow structure of the equations to derive bounds on energy, second order moments, and logarithmic entropy. The dissipation of the latter yields sufficient regularity to obtain compactness results and pass to the limit in the localised convolutions. The strategy we propose relies on a discretisation scheme which could be slightly modified in order to extend our result to PDEs with no gradient flow structure. Our analysis allows to treat the case of limiting weak solutions of the non-viscous porous medium equation at relevant low regularity, assuming the initial value to have finite energy and entropy. However, the latter excludes particle solutions of the nonlocal interaction equation.

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Section: Definition 22 (Weak Measure Solution To (Nlie)) a Narrowly C...mentioning

confidence: 99%

“…The equation above is also significant in the context of networks, where Wasserstein-type metrics have been derived recently (cf. [3,18]).…”

Section: Further Perspectivesmentioning

confidence: 99%

Porous medium equation as limit of nonlocal interaction

Burger¹,

Esposito²

2022

Preprint

Self Cite

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“…Gradient structures on metric graphs, which can be seen inbetween those of Example A and B are recently studied in [EFMM21,BHP21].…”

Section: Introductionmentioning

confidence: 99%

Cosh gradient systems and tilting

Peletier¹,

Schlichting²

2022

Preprint

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We review a class of gradient systems with dissipation potentials of hyperboliccosine type. We show how such dissipation potentials emerge in large deviations of jump processes, multi-scale limits of diffusion processes, and more. We show how the exponential nature of the cosh derives from the exponential scaling of large deviations and arises implicitly in cell problems in multi-scale limits.We discuss in-depth the role of tilting of gradient systems. Certain classes of gradient systems are tilt-independent, which means that changing the driving functional does not lead to changes of the dissipation potential. Such tilt-independence separates the driving functional from the dissipation potential, guarantees a clear modelling interpretation, and gives rise to strong notions of gradient-system convergence.We show that although in general many gradient systems are tilt-independent, certain cosh-type systems are not. We also show that this is inevitable, by studying in detail the classical example of the Kramers high-activation-energy limit, in which a diffusion converges to a jump process and the Wasserstein gradient system converges to a cosh-type system. We show and explain how the tilt-independence of the pre-limit system is lost in the limit system. This same lack of independence can be recognized in classical theories of chemical reaction rates in the chemical-engineering literature.We illustrate a similar lack of tilt-independence in a discrete setting. For a class of 'two-terminal' fast subnetworks, we give a complete characterization of the dependence on the tilting, which strongly resembles the classical theory of equivalent electrical networks.

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Dynamic optimal transport on networks

Cited by 2 publications

References 16 publications

Porous medium equation as limit of nonlocal interaction

Porous medium equation as limit of nonlocal interaction

Cosh gradient systems and tilting

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