A non-local problem for the Fokker-Planck equation related to the Becker-D\"{o}ring model

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 $c^{\text{eq}}_{(\rho)}$, while if $\rho > \rho_s$ the solution converges weakly to $c^{\text{eq}}_{(\rho_s)}$. The excess $\rho - \rho_s$ gets lost due to the formation of larger and larger clusters. In this regard, the model behaves similarly to the classical Becker-D\"oring equation. The system possesses a free energy, strictly decreasing along the evolution, which establishes the long time behavior. In the subcritical case $\rho<\rho_s$ 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-D\"oring 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.Comment: Minor revised version accepted for publication in Discrete & Continuous Dynamical Systems -

Strong Convergence to the Homogenized Limit of Parabolic Equations with Random Coefficients

This paper is concerned with the study of solutions to discrete parabolic equations in divergence form with random coefficients, and their convergence to solutions of a homogenized equation. It has previously been shown that if the random environment is translational invariant and ergodic, then solutions of the random equation converge under diffusive scaling to solutions of a homogenized parabolic PDE. In this paper point-wise estimates are obtained on the difference between the averaged solution to the random equation and the solution to the homogenized equation for certain random environments which are strongly mixing.Comment: 46 page

Young stars at large distances from the galactic plane: mechanisms of formation

We have collected from the literature a list of early-type stars, situated at large distances from the galactic plane, for which evidence of youth seems convincing. We discuss two possible formation mechanisms for these stars: ejection from the plane by dynamical interactions within small clusters, and formation away from the plane, via induced shocks created by spiral density waves. We identify the stars that could be explained by each mechanism. We conclude that the ejection mechanism can account for about two thirds of the stars, while a combination of star formation at z = 500-800 pc from the plane and ejection, can account for 90 percent of these stars. Neither mechanism, nor both together, can explain the most extreme examples.Comment: 6 pages, No figures. Sixth Pacific Rim Conference on Stellar Astrophysics - A tribute to Helmut Abt, (Kluwer