Gradient flows driven by a non-smooth repulsive interaction potential

Bonaschi, Giovanni A.

doi:10.48550/arxiv.1310.3677

Cited by 2 publications

(10 citation statements)

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“…During this work, a purely mathematical problem related to the definition of the Wasserstein subdifferential of W on singular measures came out, which is strictly related to the time reversal issue stated above. Collecting together the results from [1,18,10], one can prove existence and explicit characterization of the sub-differential of the functional W in the case of absolutely continuous measures for both the repulsive and the attractive case, and for concentrated measures in the attractive case. Unfortunately the same arguments cannot be applied when dealing with concentrated measures in the repulsive case.…”

Section: Wasserstein Gradient Flowmentioning

confidence: 99%

“…Wasserstein Gradient Flows. Our starting point is that of the Wasserstein gradient flow in the space of probability measures in the spirit of [1] combined with the recent results from [18,10,19]. In what follows, P 2 (R) is the space of probability measures on R with finite second moment.…”

Section: Three Concepts Of Solutionsmentioning

confidence: 99%

“…convex up to a quadratic perturbation. Therefore, the case W (x) = |x| can be easily covered in view of the convexity of W (see [18]); although W (x) = −|x| is neither convex nor λ-convex, the corresponding functional (see (4) below) is λ-convex in the sense of McCann [36], see [10,19], and therefore the abstract Wasserstein gradient flow theory applies. These well-posedness issues are strongly connected with the behaviour of the equations under time reversal.…”

Section: Introductionmentioning

confidence: 99%

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Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D

et al. 2015

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Abstract. We prove the equivalence between the notion of Wasserstein gradient flow for a onedimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and the notion of entropy solution of a Burgers-type scalar conservation law on the other. The solution of the former is obtained by spatially differentiating the solution of the latter. The proof uses an intermediate step, namely the L 2 gradient flow of the pseudo-inverse distribution function of the gradient flow solution. We use this equivalence to provide a rigorous particle-system approximation to the Wasserstein gradient flow, avoiding the regularization effect due to the singularity in the repulsive kernel. The abstract particle method relies on the so-called wave-front-tracking algorithm for scalar conservation laws. Finally, we provide a characterization of the sub-differential of the functional involved in the Wasserstein gradient flow.

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Section: Wasserstein Gradient Flowmentioning

confidence: 99%

Section: Three Concepts Of Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D

et al. 2015

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“…for s ∈ (0, 1) and t ≥ 0, cf. [11,30,47] for similar computations. In order to give a meaning to the above system in the case of µ or η having atoms, we use the convention sign(0) = 0.…”

Section: General Measures Initial Datamentioning

confidence: 87%

“…In this subsection we consider the case of densities as initial data. Following the approach of [1] combined with the results from [11,12,25,26, 36], we pose system (6) as the gradient flow of the interaction energy functional (4) (that we recall here for the reader's convenience)…”

Section: Absolutely Continuous Initial Datamentioning

confidence: 99%

Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions

Carrillo¹,

Francesco²,

Esposito³

et al. 2018

Preprint

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We prove global-in-time existence and uniqueness of measure solutions of a nonlocal interaction system of two species in one spatial dimension. For initial data including atomic parts we provide a notion of gradient-flow solutions in terms of the pseudo-inverses of the corresponding cumulative distribution functions, for which the system can be stated as a gradient flow on the Hilbert space L 2 (0, 1) 2 according to the classical theory by Brézis. For absolutely continuous initial data we construct solutions using a minimising movement scheme in the set of probability measures. In addition we show that the scheme preserves finiteness of the L m -norms for all m ∈ [1, +∞] and of the second moments. We then provide a characterisation of equilibria and prove that they are achieved (up to time subsequences) in the large time asymptotics. We conclude the paper constructing two examples of non-uniqueness of measure solutions emanating from the same (atomic) initial datum, showing that the notion of gradient flow solution is necessary to single out a unique measure solution.

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Gradient flows driven by a non-smooth repulsive interaction potential

Cited by 2 publications

References 0 publications

Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D

Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D

Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions

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