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

Abstract

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 $ρ_s>0$, determined from the potentials and diffusion coefficient. For $ρ\leq ρ_s$, there exists an equilibrium solution $c^{\text{eq}}_{(ρ)}$. If $ρ\leρ_s$ the solution converges strongly to $c^{\text{eq}}_{(ρ)}$, while if $ρ> ρ_s$ the solution converges weakly to $c^{\text{eq}}_{(ρ_s)}$. The excess $ρ- ρ_s$ 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 $ρ<ρ_s$ 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.