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A non-local problem for the Fokker-Planck equation related to the Becker-D\"{o}ring model. (arXiv:1711.00782v2 [math.AP] UPDATED)
来源于:arXiv
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 H\"older
regularity of the non-local order parameter, which is improved by an iteration
argument.
The asymptotic behavior of solutions depends on some order parameter $\rho$
depending on the initial data. The system shows different behavior depending on
a value $\rho_s>0$, determined from the potentials and diffusion coefficient.
For $\rho \leq \rho_s$, there exists an equilibrium solution
$c^{\text{eq}}_{(\rho)}$. If $\rho\le\rho_s$ the solution converges strongly to
$c^{\text{eq}} 查看全文>>