Abstract:
This paper describes in a rigorous manner how the dynamics of the characteristic curves for the Lifshitz-Slyozov-Wagner (LSW) model of coarsening transforms a class of noncompactly supported initial data in functions that behave in a self-similar manner for long times.

Abstract:
In this paper the existence of a class of self-similar solutions of the Einstein-Vlasov system is proved. The initial data for these solutions are not smooth, with their particle density being supported in a submanifold of codimension one. They can be thought of as intermediate between smooth solutions of the Einstein-Vlasov system and dust. The motivation for studying them is to obtain insights into possible violation of weak cosmic censorship by solutions of the Einstein-Vlasov system. By assuming a suitable form of the unknowns it is shown that the existence question can be reduced to that of the existence of a certain type of solution of a four-dimensional system of ordinary differential equations depending on two parameters. This solution starts at a particular point $P_0$ and converges to a stationary solution $P_1$ as the independent variable tends to infinity. The existence proof is based on a shooting argument and involves relating the dynamics of solutions of the four-dimensional system to that of solutions of certain two- and three-dimensional systems obtained from it by limiting processes.

Abstract:
We consider mass-conserving self-similar solutions of Smoluchowski's coagulation equation with multiplicative kernel of homogeneity $2l\lambda \in (0,1)$. We establish rigorously that such solutions exhibit a singular behavior of the form $x^{-(1+2\lambda)}$ as $x \to 0$. This property had been conjectured, but only weaker results had been available up to now.

Abstract:
In this paper we prove the existence of a large class of periodic solutions of the Vlasov-Poisson in one space dimension that decay exponentially as t goes to infinity. The exponential decay is well known for the linearized version of the Landau damping problem and it has been proved in [4] for a class of solutions of the Vlasov-Poisson system that behaves asymptotically as free streaming solutions and are sufficiently flat in the space of velocities. The results in this paper enlarge the class of possible asymptotic limits, replacing the fatness condition in [4] by a stability condition for the linearized problem.

Abstract:
In this paper we prove global existence for solutions of the Vlasov-Poisson system in convex bounded domains with specular boundary conditions and with a prescribed outward electrical field at the boundary.

Abstract:
The existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation has so far only been established for the solvable and the diagonal kernel. In this paper we prove the existence of such self-similar solutions for continuous kernels $K$ that are homogeneous of degree $\gamma \in [0,1)$ and satisfy $K(x,y) \leq C (x^{\gamma} + y^{\gamma})$. More precisely, for any $\rho \in (\gamma,1)$ we establish the existence of a continuous weak self-similar profile with decay $x^{-(1{+}\rho)}$ as $x \to \infty$.

Abstract:
We consider self-similar solutions with finite mass to Smoluchowski's coagulation equation for rate kernels that have homogeneity zero but are possibly singular such as Smoluchowski's original kernel. We prove pointwise exponential decay of these solutions under rather mild assumptions on the kernel. If the support of the kernel is sufficiently large around the diagonal we also proof that $\lim_{x \to \infty} \frac{1}{x} \log \Big(\frac{1}{f(x)}\Big)$ exists. In addition we prove properties of the prefactor if the kernel is uniformly bounded below.

Abstract:
Modelling the Calvin cycle of photosynthesis leads to various systems of ordinary differential equations and reaction-diffusion equations. They differ by the choice of chemical substances included in the model, the choices of stoichiometric coefficients and chemical kinetics and whether or not diffusion is taken into account. This paper studies the long-time behaviour of solutions of several of these systems, concentrating on the ODE case. In some examples it is shown that there exist two positive stationary solutions. In several cases it is shown that there exist solutions where the concentrations of all substrates tend to zero at late times and others (runaway solutions) where the concentrations of all substrates increase without limit. In another case, where the concentration of ATP is explicitly included, runaway solutions are ruled out.

Abstract:
We explore the self-organization dynamics of a set of entities by considering the interactions that affect the different subgroups conforming the whole. To this end, we employ the widespread example of coagulation kinetics, and characterize which interaction types lead to consensus formation and which do not, as well as the corresponding different macroscopic patterns. The crucial technical point is extending the usual one species coagulation dynamics to the two species one. This is achieved by means of introducing explicitly solvable kernels which have a clear physical meaning. The corresponding solutions are calculated in the long time limit, in which consensus may or may not be reached. The lack of consensus is characterized by means of scaling limits of the solutions. The possible applications of our results to some topics in which consensus reaching is fundamental, like collective animal motion and opinion spreading dynamics, are also outlined.

Abstract:
In this work we study the stochastic process of two-species coagulation. This process consists in the aggregation dynamics taking place in a ring. Particles and clusters of particles are set in this ring and they can move either clockwise or counterclockwise. They have a probability to aggregate forming larger clusters when they collide with another particle or cluster. We study the stochastic process both analytically and numerically. Analytically, we derive a kinetic theory which approximately describes the process dynamics. One of our strongest assumptions in this respect is the so called well-stirred limit, that allows neglecting the appearance of spatial coordinates in the theory, so this becomes effectively reduced to a zeroth dimensional model. We determine the long time behavior of such a model, making emphasis in one special case in which it displays self-similar solutions. In particular these calculations answer the question of how the system gets ordered, with all particles and clusters moving in the same direction, in the long time. We compare our analytical results with direct numerical simulations of the stochastic process and both corroborate its predictions and check its limitations. In particular, we numerically confirm the ordering dynamics predicted by the kinetic theory and explore properties of the realizations of the stochastic process which are not accessible to our theoretical approach.