Approach to self-similarity in Smoluchowski's coagulation equations

We consider the approach to self-similarity (or dynamical scaling) in Smoluchowski's equations of coagulation for the solvable kernels $K(x,y)=2$, $x+y$ and $xy$. In addition to the known self-similar solutions with exponential tails, there are one-parameter families of solutions with algebraic decay, whose form is related to heavy-tailed distributions well-known in probability theory. For K=2 the size distribution is Mittag-Leffler, and for $K=x+y$ and $K=xy$ it is a power-law rescaling of a maximally skewed $\alpha$-stable Levy distribution. We characterize completely the domains of attraction of all self-similar solutions under weak convergence of measures. Our results are analogous to the classical characterization of stable distributions in probability theory. The proofs are simple, relying on the Laplace transform and a fundamental rigidity lemma for scaling limits.Comment: Latex2e, 42 pages with 1 figur

Non-self-similar behavior in the LSW theory of Ostwald ripening

The classical Lifshitz-Slyozov-Wagner theory of domain coarsening predicts asymptotically self-similar behavior for the size distribution of a dilute system of particles that evolve by diffusional mass transfer with a common mean field. Here we consider the long-time behavior of measure-valued solutions for systems in which particle size is uniformly bounded, i.e., for initial measures of compact support. We prove that the long-time behavior of the size distribution depends sensitively on the initial distribution of the largest particles in the system. Convergence to the classically predicted smooth similarity solution is impossible if the initial distribution function is comparable to any finite power of distance to the end of the support. We give a necessary criterion for convergence to other self-similar solutions, and conditional stability theorems for some such solutions. For a dense set of initial data, convergence to any self-similar solution is impossible.Comment: 31 pages, LaTeX2e; Revised version, to appear in J. Stat. Phy

Biodegradable polymers based on trimethylene carbonate for tissue engineering applications

In the field of tissue engineering, the search for suitable materials for use in the preparation of scaffolds to host the developing tissue represents a major subject of study. Biodegradable materials show great potential in this area as, by resorbing upon performing their function, they obviate long-term biocompatibility concerns

A host of traveling waves in a model of three-dimensional water-wave dynamics

We describe traveling waves in a basic model for three-dimensional water-wave dynamics in the weakly nonlinear long-wave regime. Small solutions that are periodic in the direction of translation (or orthogonal to it) form an infinite-dimensional family. We characterize these solutions through spatial dynamics, by reducing a linearly ill-posed mixed-type initial-value problem to a center manifold of infinite dimension and codimension. A unique global solution exists for arbitrary small initial data for the two-component bottom velocity, specified along a single line in the direction of translation (or orthogonal to it). A dispersive, nonlocal, nonlinear wave equation governs the spatial evolution of bottom velocity.Comment: 22 pages with 1 figure, LaTeX2e with amsfonts, epsfig package