Ground States in the Diffusion-Dominated Regime

Carrillo, José A.; Hoffmann, Franca; Mainini, Edoardo; Volzone, Bruno

doi:10.48550/arxiv.1705.03519

Cited by 4 publications

(5 citation statements)

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“…Remark 5.2. The previous is result is a particular case of [41, Theorem II.1]; see also [27] for more comments. As a main difference, our proof exploits the extra rigidity stemming from the radially decreasing rearrangement, bypassing the need of exploiting the subadditivity of the energy of minimizers at different mass.…”

Section: 14)mentioning

confidence: 88%

Existence of ground states for aggregation-diffusion equations

2019

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We analyze free energy functionals for macroscopic models of multi-agent systems interacting via pairwise attractive forces and localized repulsion. The repulsion at the level of the continuous description is modeled by pressure-related terms in the functional making it energetically favorable to spread, while the attraction is modeled through nonlocal forces. We give conditions on general entropies and interaction potentials for which neither ground states nor local minimizers exist. We show that these results are sharp for homogeneous functionals with entropies leading to degenerate diffusions while they are not sharp for fast diffusions. The particular relevant case of linear diffusion is totally clarified giving a sharp condition on the interaction potential under which the corresponding free energy functional has ground states or not.

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Section: 14)mentioning

confidence: 88%

Existence of ground states for aggregation-diffusion equations

2019

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“…Stability/instability of these stationary solutions together with qualititative properties for the Euler-Poisson system have been analyzed in [45,46,31,43,44,29,49]. A generalization of this case for homogeneous interaction kernels K(x) = |x| k /k, k ∈ (−d, 0), in the whole d-dimensional space without confinement has been recently obtained in [17,8,9,20]. The balance of forces happen in general for the range m > 1 − k d regardeless of the mass of the initial data, known as the diffusion-dominated case.…”

Section: Introductionmentioning

confidence: 99%

On long-time asymptotics for viscous hydrodynamic models of collective behavior with damping and nonlocal interactions

Carrillo

Wróblewska-Kamińska

Zatorska

2019

Math. Models Methods Appl. Sci.

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Hydrodynamic systems arising in swarming modelling include nonlocal forces in the form of attractive-repulsive potentials as well as pressure terms modelling strong local repulsion. We focus on the case where there is a balance between nonlocal attraction and local pressure in presence of confinement in the whole space. Under suitable assumptions on the potentials and the pressure functions, we show the global existence of weak solutions for the hydrodynamic model with viscosity and linear damping. By introducing linear damping in the system, we ensure the existence and uniqueness of stationary solutions with compactly supported density, fixed mass and center of mass. The associated velocity field is zero in the support of the density. Moreover, we show that global weak solutions converge for large times to the set of these stationary solutions in a suitable sense. In particular cases, we can identify the limiting density uniquely as the global minimizer of the free energy with the right mass and center of mass.

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“…They are the natural Liapunov functionals associated to aggregation-diffusion equations with homogeneous entropies and interaction potentials. For the range k ∈ (−d, 0), these equations have been studied in detail in [10,9,14]. The case k > 0 was briefly analyzed in [10], where the fair-competition (m = 1 − k d ) and diffusion dominated ranges (m < 1 − k d ) were considered.…”

Section: Introductionmentioning

confidence: 99%

Free Energies and the Reversed HLS Inequality

Carrillo,

Delgadino

2018

Preprint

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We prove reversed Hardy-Littlewood-Sobolev inequalities by carefully studying the natural associated free energies with direct methods of calculus of variations. Tightness is obtained by a dyadic argument, which quantifies the relative strength of the entropy functional versus the interaction energy. The existence of optimizers is shown in the class of P(R d ). With respect to their regularity, we study conditions for optimizers to be bounded functions. In a related model, we show the condensation phenomena, which suggests that optimizers are not in general regular.

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Ground States in the Diffusion-Dominated Regime

Cited by 4 publications

References 0 publications

Existence of ground states for aggregation-diffusion equations

Existence of ground states for aggregation-diffusion equations

On long-time asymptotics for viscous hydrodynamic models of collective behavior with damping and nonlocal interactions

Free Energies and the Reversed HLS Inequality

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