No abstract

We provide a result on the rate of convergence to equilibrium for solutions of the Becker-Döring equations. Our strategy is to use the energy/energy-dissipation relation. The main difficulty is the structure of the equilibria of the Becker-Döring equations, which do not correspond to a gaussian measure, such that a logarithmic Sobolevinequality is not available. We prove a weaker inequality which still implies for fast decaying data that the solution converges to equilibrium as e −ct 1/3 .

The existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation has so far only been established for the solvable kernels and the diagonal one. In this paper we prove the existence of such self-similar solutions for continuous kernels K that are homogeneous of degree γ ∈ [0, 1) and satisfy K(x, y) ≤ C(x γ + y γ ). More precisely, for any ρ ∈ (γ, 1) we establish the existence of a continuous weak nonnegative self-similar profile with decay x −(1+ρ) as x → ∞.For the proof we consider the time-dependent problem in self-similar variables with the aim to use a variant of Tykonov's fixed point theorem to establish the existence of a stationary profile. This requires to identify a weakly compact subset that is invariant under the evolution. In our case we define a set of nonnegative measures which encodes the desired decay behaviour in an integrated form. The main difficulty is to establish the invariance of the lower bound under the evolution. Our key idea is to choose as a test function in the time dependent problem the solution of the associated backward dual problem.

We are interested in the coarsening of a spatial distribution of two phases, driven by the reduction of interfacial energy and limited by diffusion, as described by the Mullins-Sekerka model. We address the regime where one phase covers only a small fraction of the total volume and consists of many disconnected components ("particles"). In this situation, the energetically more advantageous large particles grow at the expense of the small ones, a phenomenon called Ostwald ripening. Lifshitz, Slyozov and Wagner formally derived an evolution for the distribution of particle radii. We extend their derivation by taking into account that only particles within a certain distance, the screening length, communicate. Our arguments are rigorous and are based on a homogenization within a gradient flow structure.

The LSW theory of Ostwald ripening concerns the time evolution of the size distribution of a dilute system of particles that evolve by diffusional mass transfer with a common mean field. We prove global existence, uniqueness and continuous dependence on initial data for measure-valued solutions with compact support in particle size. These results are established with respect to a natural topology on the space of size distributions, one given by the Wasserstein metric which measures the smallest maximum volume change required to rearrange one distribution into another.