arXiv subjects

Marina A. Ferreira

Publications and source records attributed to Marina A. Ferreira.

13 recordsLinked to original sources

arXiv2025

Global existence of measure-valued solutions to the multicomponent Smoluchowski coagulation equation

Global solutions to the multicomponent Smoluchowski coagulation equation are constructed for measure-valued initial data with minimal assumptions on the moments. The framework is based on an abstract formulation of the Arzelà-Ascoli theorem for uniform spaces. The result holds for a large class of coagulation rate kernels, satisfying a power-law upper bound with possibly different singularities at small-small, small-large and large-large coalescence pairs. This includes in particular both mass-conserving and gelling kernels, as well as interpolation kernels used in applications. We also provide short proofs of mass-conservation and gelation results for any weak solution, which extends previous results for one-component systems.

math.AP↗

arXiv2024

Smoluchowski Coagulation Equation with a Flux of Dust Particles

We construct a time-dependent solution to the Smoluchowski coagulation equation with a constant flux of dust particles entering through the boundary at zero. The dust is instantaneously converted into particles and flux solutions have linearly increasing mass. The construction is made for a general class of non-gelling coagulation kernels for which stationary solutions, so-called constant flux solutions, exist. The proof relies on several limiting procedures on a family of solutions of equations with sources supported on ever smaller sizes. In particular, uniform estimates on the fluxes of these solutions are derived in order to control the singularity produced by the flux at zero. We further show that, up to the multiplication by a scalar, flux solutions averaged in size and integrated in time, are bounded from above by the explicit solution, $x^{-\frac{γ+3}{2}}$, of the constant flux equation, where $γ$ is the homogeneity of the bounds of the coagulation rate kernel. Flux solutions are expected to converge to a constant flux solution in the large time limit. We show that this is indeed true in the particular case of the constant kernel with zero initial data.

math.AP↗

arXiv2023

Coagulation equations with source leading to anomalous self-similarity

We study the long-time behaviour of the solutions to Smoluchowski coagulation equations with a source term of small clusters. The source drives the system out-of-equilibrium, leading to a rich range of different possible long-time behaviours, including anomalous self-similarity. The coagulation kernel is non-gelling, homogeneous, with homogeneity $γ\leq -1 $, and behaves like $x^{γ+λ} y^{-λ} $ when $y \ll x$ with $γ+2λ> 1 $. Our analysis shows that the long-time behaviour of the solutions depends on the parameters $γ$ and $λ$. More precisely, we argue that the long-time behaviour is self-similar, although the scaling of the self-similar solutions depends on the sign of $γ+λ$ and on whether $γ=-1$ or $γ< -1$. In all these cases, the scaling differs from the usual one that has been previously obtained when $γ+2λ<1$ or $γ+2λ\geq 1, γ>-1$. In the last part of the paper, we present some conjectures supporting the self-similar ansatz also for the critical case $γ+2λ=1, γ\leq -1 $.

math-ph↗

arXiv2022

Long-time asymptotics for coagulation equations with injection that do not have stationary solutions

In this paper we study a class of coagulation equations including a source term that injects in the system clusters of size of order one. The coagulation kernel is homogeneous, of homogeneity $γ< 1$, such that $K(x,y)$ is approximately $x^{γ+ λ} y^{-λ}$, when $x$ is larger than $y$. We restrict the analysis to the case $γ+ 2 λ\geq 1 $. In this range of exponents, the transport of mass toward infinity is driven by collisions between particles of different sizes. This is in contrast with the case when $γ+ 2 λ<1$. In that case, the transport of mass toward infinity is due to the collision between particles of comparable sizes. In the case $γ+2λ\geq 1$, the interaction between particles of different sizes leads to an additional transport term in the coagulation equation that approximates the solution of the original coagulation equation with injection for large times. We prove the existence of a class of self-similar solutions for suitable choices of $γ$ and $λ$ for this class of coagulation equations with transport. We prove that for the complementary case such self-similar solutions do not exist.

math.AP↗

arXiv2022

Non-power law constant flux solutions for the Smoluchowski coagulation equation

It is well known that for a large class of coagulation kernels, Smoluchowski coagulation equations have particular power law solutions which yield a constant flux of mass along all scales of the system. In this paper, we prove that for some choices of the coagulation kernels there are solutions with a constant flux of mass along all scales which are not power laws. The result is proved by means of a bifurcation argument.

math.AP↗

arXiv2022

On the self-similar behaviour of coagulation systems with injection

In this paper we prove the existence of a family of self-similar solutions for a class of coagulation equations with a constant flux of particles from the origin. These solutions are expected to describe the longtime asymptotics of Smoluchowski's coagulation equations with a time independent source of clusters concentrated in small sizes. The self-similar profiles are shown to be smooth, provided the coagulation kernel is also smooth. Moreover, the self-similar profiles are estimated from above and from below by $x^{-(γ+3)/2}$ as $x \to 0$, where $γ<1 $ is the homogeneity of the kernel, and are proven to decay at least exponentially as $x \to \infty$.

math.AP↗

arXiv2022

Asymptotic localization in multicomponent mass conserving coagulation equations

In this paper we prove that the time dependent solutions of a large class of Smoluchowski coagulation equations for multicomponent systems concentrate along a particular direction of the space of cluster compositions for long times. The direction of concentration is determined by the initial distribution of clusters. These results allow to prove the uniqueness and global stability of the self-similar profile with finite mass in the case of coagulation kernels which are not identically constant, but are constant along any direction of the space of cluster compositions.

math.AP↗

arXiv2021

Coagulation equations for aerosol dynamics

Binary coagulation is an important process in aerosol dynamics by which two particles merge to form a larger one. The distribution of particle sizes over time may be described by the so-called Smoluchowski's coagulation equation. This integrodifferential equation exhibits complex non-local behaviour that strongly depends on the coagulation rate considered. We first discuss well-posedness results for the Smoluchowski's equation for a large class of coagulation kernels as well as the existence and nonexistence of stationary solutions in the presence of a source of small particles. The existence result uses Schauder fixed point theorem, and the nonexistence result relies on a flux formulation of the problem and on power law estimates for the decay of stationary solutions with a constant flux. We then consider a more general setting. We consider that particles may be constituted by different chemicals, which leads to multi-component equations describing the distribution of particle compositions. We obtain explicit solutions in the simplest case where the coagulation kernel is constant by using a generating function. Using an approximation of the solution we observe that the mass localizes along a straight line in the size space for large times and large sizes.

math-ph↗

arXiv2021

Localization in stationary non-equilibrium solutions for multicomponent coagulation systems

We consider the multicomponent Smoluchowski coagulation equation under non-equilibrium conditions induced either by a source term or via a constant flux constraint. We prove that the corresponding stationary non-equilibrium solutions have a universal localization property. More precisely, we show that these solutions asymptotically localize into a direction determined by the source or by a flux constraint: the ratio between monomers of a given type to the total number of monomers in the cluster becomes ever closer to a predetermined ratio as the cluster size is increased. The assumptions on the coagulation kernel are quite general, with isotropic power law bounds. The proof relies on a particular measure concentration estimate and on the control of asymptotic scaling of the solutions which is allowed by previously derived estimates on the mass current observable of the system.

math-ph↗

arXiv2021

Multicomponent coagulation systems: existence and non-existence of stationary non-equilibrium solutions

We study multicomponent coagulation via the Smoluchowski coagulation equation under non-equilibrium stationary conditions induced by a source of small clusters. The coagulation kernel can be very general, merely satisfying certain power law asymptotic bounds in terms of the total number of monomers in a cluster. The bounds are characterized by two parameters and we extend previous results for one-component systems to classify the parameter values for which the above stationary solutions do or do not exist. Moreover, we also obtain criteria for the existence or non-existence of solutions which yield a constant flux of mass towards large clusters.

math.AP↗

arXiv2020

Stationary non-equilibrium solutions for coagulation systems

We study coagulation equations under non-equilibrium conditions which are induced by the addition of a source term for small cluster sizes. We consider both discrete and continuous coagulation equations, and allow for a large class of coagulation rate kernels, with the main restriction being boundedness from above and below by certain weight functions. The weight functions depend on two power law parameters, and the assumptions cover, in particular, the commonly used free molecular and diffusion limited aggregation coagulation kernels. Our main result shows that the two weight function parameters already determine whether there exists a stationary solution under the presence of a source term. In particular, we find that the diffusive kernel allows for the existence of stationary solutions while there cannot be any such solutions for the free molecular kernel. The argument to prove the non-existence of solutions relies on a novel power law lower bound, valid in the appropriate parameter regime, for the decay of stationary solutions with a constant flux. We obtain optimal lower and upper estimates of the solutions for large cluster sizes, and prove that the solutions of the discrete model behave asymptotically as solutions of the continuous model.

math-ph↗

arXiv2017

A new continuum theory for incompressible swelling materials

Swelling media (e.g. gels, tumors) are usually described by mechanical constitutive laws (e.g. Hooke or Darcy laws). However, constitutive relations of real swelling media are not well known. Here, we take an opposite route and consider a simple packing heuristics, i.e. the particles can't overlap. We deduce a formula for the equilibrium density under a confining potential. We then consider its evolution when the average particle volume and confining potential depend on time under two additional heuristics: (i) any two particles can't swap their position; (ii) motion should obey some energy minimization principle. These heuristics determine the medium velocity consistently with the continuity equation. In the direction normal to the potential level sets the velocity is related with that of the level sets while in the parallel direction, it is determined by a Laplace-Beltrami operator on these sets. This complex geometrical feature cannot be recovered using a simple Darcy law.

math.AP↗

arXiv2016

Damped Arrow-Hurwicz algorithm for sphere packing

We consider algorithms that, from an arbitrarily sampling of $N$ spheres (possibly overlapping), find a close packed configuration without overlapping. These problems can be formulated as minimization problems with non-convex constraints. For such packing problems, we observe that the classical iterative Arrow-Hurwicz algorithm does not converge. We derive a novel algorithm from a multi-step variant of the Arrow-Hurwicz scheme with damping. We compare this algorithm with classical algorithms belonging to the class of linearly constrained Lagrangian methods and show that it performs better. We provide an analysis of the convergence of these algorithms in the simple case of two spheres in one spatial dimension. Finally, we investigate the behaviour of our algorithm when the number of spheres is large.

math.NA↗