The equations of Ostwald ripening for dilute systems

We consider a dilute mixture in 3D of a finite number of particles initially close to spherical, but of varying sizes, and representing one of the phases evolving according to the quasistatic dynamics. Under the scaling hypotheses that (1) typical size/typical distance and (2) deviation from sphericity/typical size are small, we associate centers and radii to each particle for the whole evolution and derive rigorously a set of ODEs fo the radii which we relate to the Lifschitz-Slyosov-Wagner theory of coarsening

Motion of a droplet by surface tension along the boundary

We give a description of the ultimate dynamics for the simplest evolution equation compatible with the Van der Waals Free Energy. We establish existence of stable sets of solutions corresponding to the physical motion of a small, almost semicircular interface (droplet) intersecting the boundary of the domain and moving towards a point where the curvature has a local maximum, Our results represent a particular extension of the Equilibrium theory of Modica and Sternberg to the next dynamic level in the Morse decomposition of the flow

Periodic traveling waves and locating oscillating patterns in multidimensional domains

We establish the existence and robustness of layered, time-periodic solutions to a reaction-diffusion equation in a bounded domain in R-n, when the diffusion coefficient is sufficiently small and the reaction term is periodic in time and bistable in the state variable. Our results suggest that these patterned, oscillatory solutions are stable and locally unique. The location of the internal layers is characterized through a periodic traveling wave problem for a related one-dimensional reaction-diffusion equation. This one-dimensional problem is of independent interest and for this we establish the existence and uniqueness of a heteroclinic solution which, in constant-velocity moving coodinates, is periodic in time. Furthermore, we prove that the manifold of translates of this solution is globally exponentially asymptotically stable

Convergence to higher-energy stationary solutions of a bistable equation with nonsmooth reaction term

V tomto článku vyšetřujeme lokální stabilitu kritických bodů o vyšší energii než je globální minimum funkcionálu energie přiřazenému bistabilní rovnici. Předpokládáme, že double-well potenciál není třídy C^2 v bodech sveho globálního minima.V předcházejícím článku jsme dokazali, že funkcionál energie má kontinua kritických bodů o vyšší energii než ground states. Tyto kritické body jsou lokálními minimy a funkcionál je konvexní ve všech směrech, které jsou transversální na směr těchto kontinuí. Pro každé z těchto kontinuí ukazujeme, existenci otevřené podmnožiny bazénu atrakce. Náš výsledek nabízí alternativní vysvětlení pomalé dynamiky podél atraktoru, která byla široce diskutována v odborné literatuře.In this paper, we investigate the local stability of critical points with energy that is higher than the ground-state energy of the functional associated with the bistable equation. We assume that a double-well potential lacks C^2 regularity at the global minimizers. Previous work has shown that, for small diffusion parameter, the energy functional possesses continua of critical points at high energy levels and that the relative interior of these continua are local minimizers. The local geometry of the energy functional at these points is convex in directions perpendicular to the continua, and thus has a trough-like shape locally, with critical points at the base of the trough. For each such continuum, we show that there is an open set containing the interior of the continuum that is a subset of the basin of attraction for the continuum. This stability result allows for a different perspective on the so-called slow dynamics along the attractor