arXiv subjects

Juan Velazquez

Publications and source records attributed to Juan Velazquez.

2 recordsLinked to original sources

arXiv2025

Solutions of the thin film equation obtained in the limit of vanishing slip

We analyze the evolution of thin liquid droplets in the lubrication approximation with different slip conditions at the liquid-solid interface. Motivated by the classical no-slip paradox which states that the Navier-Stokes equations with a no-slip boundary condition require unphysical infinite dissipation during droplet spreading, we focus on the limit of vanishing slip. We show that in the no-slip limit three fundamentally different classes of limiting solutions are approached, each of them corresponding to a different scaling of the microscopic contact angle as the regularization parameter vanishes. These findings suggest that the thin-film equation with no slip supports a rich family of physically admissible solutions, provided one interprets the no-slip thin film equation as the asymptotic limit of models which regularized slip conditions. Even though the large apparent contact angles in some of these solutions seem incompatible with the lubrication approximation, a refined analysis shows that the underlying physical variables remain consistent with the assumptions for the lubrication approximation.

math.AP↗

arXiv2019

Large Time Behavior of Exchange-driven Growth

Exchange-driven growth (EDG) is a process in which pairs of clusters interact by exchanging single unit with a rate given by a kernel $K(j,k)$. Despite EDG model's common use in the applied sciences, its rigorous mathematical treatment is very recent. In this article we study the large time behaviour of EDG equations. We show two sets of results depending on the properties of the kernel $(i)$ $K(j,k)=b_{j}a_{k}$ and $(ii)$ $K(j,k)=ja_{k}+b_{j} +\varepsilonβ_{j}α_{k}$. For type I kernels, under the detailed balance assumption, we show that the system admits equilibrium solutions up to a critical mass $ρ_{s}$ above which there is no equilibrium. We prove that if the system has an initial mass above $ρ_{s}$ then the solutions converge to critical equilibrium distribution in a weak sense while strong convergence can be shown when initial mass is below $ρ_{s}$. For type II kernels, we make no assumption of detailed balance and equilibrium is obtained via a contraction property. We provide two separate results depending on the monotonicity of the kernel or smallness of the total mass. For the first case we show exponential convergence in the number of clusters norm and for the second we prove exponential convergence in the total mass norm.

math.AP↗