Gradient flow formulation and longtime behaviour of a constrained Fokker–Planck equation

Eberle, Simon; Niethammer, Barbara; Schlichting, André

doi:10.1016/j.na.2017.04.009

Cited by 9 publications

(5 citation statements)

References 24 publications

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“…Proof. The proof of the lemma is inspired by [16,Lemma 3.7,(3.10)]. Let P n be the optimal plan for the Wasserstein distance W 2 (ρ n ,ρ n−1 ) between ρ n−1 and ρ n .…”

Section: Proof Of Theorem 42mentioning

confidence: 99%

Wasserstein gradient flow formulation of the time-fractional Fokker–Planck equation

Duong

Jin

2020

Communications in Mathematical Sciences

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Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document. When citing, please reference the published version. Take down policy While the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been uploaded in error or has been deemed to be commercially or otherwise sensitive.

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“…Proof. The proof of the lemma is inspired by [16,Lemma 3.7,(3.10)]. Let P n be the optimal plan for the Wasserstein distance W 2 (ρ n ,ρ n−1 ) between ρ n−1 and ρ n .…”

Section: Proof Of Theorem 42mentioning

confidence: 99%

Wasserstein gradient flow formulation of the time-fractional Fokker–Planck equation

Duong

Jin

2020

Communications in Mathematical Sciences

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“…Our natural boundary condition is the third-order one appearing in (1.1). General gradient flows with time-dependent constraints have been studied in [6]. The general methodology of these papers is a form of Lyapunov-Schmitt reduction and originates from Ye [20].…”

Section: Introductionmentioning

confidence: 99%

On the area-preserving Willmore flow of small bubbles sliding on a domain's boundary

Metsch¹

2022

Preprint

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We consider the area-preserving Willmore evolution of surfaces φ that are close to a half-sphere with a small radius, sliding on the boundary S of a domain Ω while meeting it orthogonally. We prove that the flow exists for all times and keeps a 'half-spherical' shape. Additionally, we investigate the asymptotic behaviour of the flow and prove that for large times the barycenter of the surfaces approximately follows an explicit ordinary differential equation. Imposing additional conditions on the mean curvature of S, we then establish convergence of the flow.

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“…This approach was first implemented for the classical Becker-Döring equation in [35] and recently generalized in [13]. Moreover, a similar strategy was recently applied to obtain rates of convergence to equilibrium for Fokker-Planck equation with constraints [23].…”

mentioning

confidence: 99%

A non-local problem for the Fokker-Planck equation related to the Becker-Döring model

Conlon¹,

Schlichting²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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This paper concerns a Fokker-Planck equation on the positive real line modeling nucleation and growth of clusters. The main feature of the equation is the dependence of the driving vector field and boundary condition on a non-local order parameter related to the excess mass of the system.The first main result concerns the well-posedness and regularity of the Cauchy problem. The well-posedness is based on a fixed point argument, and the regularity on Schauder estimates. The first a priori estimates yield Hölder regularity of the non-local order parameter, which is improved by an iteration argument.The asymptotic behavior of solutions depends on some order parameter ρ depending on the initial data. The system shows different behavior depending on a value ρs > 0, determined from the potentials and diffusion coefficient. For ρ ≤ ρs, there exists an equilibrium solution c eq (ρ) . If ρ ≤ ρs the solution converges strongly to c eq (ρ) , while if ρ > ρs the solution converges weakly to c eq (ρs) . The excess ρ − ρs gets lost due to the formation of larger and larger clusters. In this regard, the model behaves similarly to the classical Becker-Döring equation.The system possesses a free energy, strictly decreasing along the evolution, which establishes the long time behavior. In the subcritical case ρ < ρs the entropy method, based on suitable weighted logarithmic Sobolev inequalities and interpolation estimates, is used to obtain explicit convergence rates to the equilibrium solution.The close connection of the presented model and the Becker-Döring model is outlined by a family of discrete Fokker-Planck type equations interpolating between both of them. This family of models possesses a gradient flow structure, emphasizing their commonality.2010 Mathematics Subject Classification. Primary: 35Q84; secondary: 35F05, 35K55, 37D35, 82C26, 82C70.

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Gradient flow formulation and longtime behaviour of a constrained Fokker–Planck equation

Cited by 9 publications

References 24 publications

Wasserstein gradient flow formulation of the time-fractional Fokker–Planck equation

Wasserstein gradient flow formulation of the time-fractional Fokker–Planck equation

On the area-preserving Willmore flow of small bubbles sliding on a domain's boundary

A non-local problem for the Fokker-Planck equation related to the Becker-Döring model

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