$L^\infty $ estimates for the JKO scheme in parabolic-elliptic Keller-Segel systems

Carrillo, José A.; Santambrogio, Filippo

doi:10.1090/qam/1493

Cited by 15 publications

(12 citation statements)

References 36 publications

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“…There, it was also shown that the free energy functional E is ξ-convex for m ≥ 1 − 1 d in the set of bounded densities L ∞ (R d ) ∩ P 2 (R d ) with a given fixed bound allowing the use of the recent theory of ξ-convex gradient flows in [39]. Summarizing, the recent results for the Newtonian attractive kernel [39,30,33] allow for a rigorous gradient flow structure of the Newtonian attractive kernel case for m ≥ 1 − 1 d with initial data in L ∞ (R d ) ∩ P 2 (R d ).…”

Section: 4mentioning

confidence: 93%

“…[39] has shown that the gradient flow is well-posed if the energy E is ξ-convex, where ξ is a modulus of convexity. [33] has recently shown that for (1.1) with attractive Newtonian potential, for any ρ 0 in L ∞ (R d ) ∩ P 2 (R d ), there is a local-in-time gradient flow solution. The authors show that there are local in time L ∞ bounds at the discrete variational level allowing for local in time well defined gradient flow solutions.…”

Section: 4mentioning

confidence: 99%

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Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics

et al. 2019

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We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs. This balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrization techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations as t → ∞.

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Section: 4mentioning

confidence: 93%

Section: 4mentioning

confidence: 99%

Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics

et al. 2019

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“…From the perspective of approximating gradient flows, which are solutions of diffusive partial differential equations (3), such regularity and positivity can be guaranteed as long as the initial data are smooth and positive and either the diffusion is sufficiently strong or the drift and interaction terms do not cause loss of regularity. On the other hand, developing conditions on the energy and initial data that ensure such regularity and positivity holds at the level of the JKO scheme, for minimizers of Problem 1, remains largely open: results on the propagation of L p (R d ) or BV bounds along the scheme have only recently emerged [17,50,63].…”

Section: Remarkmentioning

confidence: 99%

Primal Dual Methods for Wasserstein Gradient Flows

et al. 2021

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Combining the classical theory of optimal transport with modern operator splitting techniques, we develop a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media, materials science, and biological swarming. Our method proceeds as follows: first, we discretize in time, either via the classical JKO scheme or via a novel Crank–Nicolson-type method we introduce. Next, we use the Benamou–Brenier dynamical characterization of the Wasserstein distance to reduce computing the solution of the discrete time equations to solving fully discrete minimization problems, with strictly convex objective functions and linear constraints. Third, we compute the minimizers by applying a recently introduced, provably convergent primal dual splitting scheme for three operators (Yan in J Sci Comput 1–20, 2018). By leveraging the PDEs’ underlying variational structure, our method overcomes stability issues present in previous numerical work built on explicit time discretizations, which suffer due to the equations’ strong nonlinearities and degeneracies. Our method is also naturally positivity and mass preserving and, in the case of the JKO scheme, energy decreasing. We prove that minimizers of the fully discrete problem converge to minimizers of the spatially continuous, discrete time problem as the spatial discretization is refined. We conclude with simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our approach, including higher-order convergence our novel Crank–Nicolson-type method, when compared to the classical JKO method.

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“…For general Φ, we use some ideas from the proof of [16,Theorem 1]. Let us approximate S with a sequence (S ε ) ε>0 of smooth convex functions such that S ε c ε > 0 for any ε > 0 with S ε (0+) = −∞.…”

Section: ] For Instance)mentioning

confidence: 99%

Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients

Kwon

Mészáros

2021

J. London Math. Soc.

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This paper is devoted to the study of some nonlinear parabolic equations with discontinuous diffusion intensities. Such problems appear naturally in physical and biological models. Our analysis is based on variational techniques and in particular on gradient flows in the space of probability measures equipped with the distance arising in the Monge-Kantorovich optimal transport problem. The associated internal energy functionals in general fail to be differentiable, therefore classical results do not apply directly in our setting. We study the combination of both linear and porous medium type diffusions and we show the existence and uniqueness of the solutions in the sense of distributions in suitable Sobolev spaces. Our notion of solution allows us to give a fine characterization of the emerging critical regions, observed previously in numerical experiments. A link to a three phase free boundary problem is also pointed out. Contents

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$L^\infty $ estimates for the JKO scheme in parabolic-elliptic Keller-Segel systems

Cited by 15 publications

References 36 publications

Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics

Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics

Primal Dual Methods for Wasserstein Gradient Flows

Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients

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