Papers

• On an inhomogeneous coagulation model with a differential sedimentation kernel,
  together with Barbara Niethammer and Juan J. L. Velázquez,
  submitted [ pdf ]
  Show Abstract
  
  We study an inhomogeneous coagulation equation that contains a transport term in the spatial variable modeling the sedimentation of clusters. We prove local existence of mass conserving solutions for a class of coagulation kernels for which in the space homogeneous case instantaneous gelation (i.e., instantaneous loss of mass) occurs. Our result holds true in particular for sum-type kernels of homogeneity greater than one, for which solutions do not exist at all in the spatially homogeneous case. Moreover, our result covers kernels that in addition vanish on the diagonal, which have been used to describe the onset of rain and the behavior of air bubbles in water.
  
  
• Fast fusion in a two-dimensional coagulation model,
  together with Juan J. L. Velázquez,
  Journal des Mathématiques Pures et Appliquées, 184: 27p. (2024)
  [ arXiv ] [ DOI ]
  Show Abstract
  
  In this work, we study a particular system of coagulation equations characterized by two values, namely volume $v$ and surface area $a$. Compared to the standard one-dimensional models, this model incorporates additional information about the geometry of the particles. We describe the coagulation process as a combination between collision and fusion of particles. We prove that we are able to recover the standard one-dimensional coagulation model when fusion happens quickly and that we are able to recover an equation in which particles interact and form a ramified-like system in time when fusion happens slowly.
  
  
• Long-time asymptotics for coagulation equations with injection that do not have stationary solutions,
  together with Marina A. Ferreira, Eugenia Franco, and Juan J. L. Velázquez,
  Archive for Rational Mechanics and Analysis, 247(6): 61p. (2023)
  [ arXiv ] [ DOI ]
  Show Abstract
  
  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 $\gamma<1$, such that $K(x,y)$ is approximately $x^{\gamma+\lambda}y^{-\lambda}$, when $x$ is larger than $y$. We restrict the analysis to the case $\gamma+2\lambda\geq 1$. 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 $\gamma+2\lambda<1$. In that case, the transport of mass toward infinity is due to the collision between particles of comparable sizes. In the case $\gamma+2\lambda\geq 1$, 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 $\gamma$ and $\lambda$ for this class of coagulation equations with transport. We prove that for the complementary case such self-similar solutions do not exist.
  
  
• Coagulation equations for non-spherical clusters,
  together with Juan J. L. Velázquez,
  submitted [ pdf ]
  Show Abstract
  
  In this work, we study the long time asymptotics of a coagulation model which describes the evolution of a system of particles characterized by their volume and surface area. The aggregation mechanism takes place in two stages: collision and fusion of particles. During the collision stage, the two particles merge at a contact point. The newly formed particle has volume and area equal to the sum of the respective quantities of the two colliding particles. After collision, the fusion phase begins and during it the geometry of the interacting particles is modified in such a way that the volume of the total system is preserved and the surface area is reduced. During their evolution, the particles must satisfy the isoperimetric inequality. Therefore, the distribution of particles in the volume and area space is supported in the region where $a\geq (36\pi)^{\frac{1}{3}}v^{\frac{2}{3}}$. We assume the coagulation kernel has a weak dependence on the area variable. We prove existence of self-similar profiles for some choices of the functions describing the fusion rate for which the particles have a shape that is close to spherical. On the other hand, for other fusion mechanisms, we show that the particle distribution describes a system of ramified-like particles.