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

TL;DR

This paper analyzes long-time behavior of coagulation equations with injection, identifying conditions under which self-similar solutions exist or do not, based on the homogeneity parameters of the kernel.

Contribution

It establishes the existence or non-existence of self-similar solutions for coagulation equations with injection, depending on the kernel parameters, and characterizes the mass transport mechanisms.

Findings

Self-similar solutions exist for certain parameter ranges.

Mass transport is driven by different mechanisms depending on parameters.

No self-similar solutions exist outside specific parameter regimes.

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.