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Mass concentration in a spatially inhomogeneous coagulation model with fast sedimentation

Iulia Cristian, Juan J. L. Velázquez

TL;DR

The work analyzes a spatially inhomogeneous coagulation equation with a gravity-induced transport term and establishes local-in-time existence of mass-conserving mild solutions under sum-type kernels with $\gamma\in(1,1+\alpha)$. By exploiting a fixed-point iteration and regionwise supersolution constructions, it obtains bounds uniform in the small sedimentation parameter $\varepsilon$, yielding a rigorous existence theory for times of order one. In the fast-sedimentation limit $\varepsilon\to 0$, the solutions concentrate on the curve $y=v^{\alpha}$ and formally converge to a Dirac-measure-supported state, leading to a reduced one-dimensional diagonal-kernel coagulation equation for the marginal $G(v,t)$ with kernel $K_{\text{diag}}(v,w)= C v^{\gamma+1-\alpha} \delta(v-w)$ (homogeneity $\gamma-\alpha<1$). This provides a concrete mechanism by which diagonal-kernel coagulation can emerge from inhomogeneous transport-dominated coagulation models and offers physical intuition for long-time behavior and potential global existence in related systems.

Abstract

We study a spatially inhomogeneous coagulation model that contains a transport term in the spatial variable. The transport term models the vertical motion of particles due to gravity, thereby incorporating their fall into the dynamics. Local existence of mass-conserving solutions for a class of coagulation rates for which in the spatially homogeneous case instantaneous gelation (i.e., instantaneous loss of mass) occurs has been proved in [Cristian-Niethammer-Velázquez, 2024]. In order to obtain some insight into how to prove global existence of solutions, we allow a fast sedimentation speed. For very fast sedimentation speed, we rigorously prove that solutions converge to a Dirac measure in the space variable. We also formally obtain in the limit a one-dimensional coagulation equation with diagonal kernel, i.e., only particles of the same size interact. This provides a physical intuition on how coagulation models with a diagonal kernel emerge.

Mass concentration in a spatially inhomogeneous coagulation model with fast sedimentation

TL;DR

The work analyzes a spatially inhomogeneous coagulation equation with a gravity-induced transport term and establishes local-in-time existence of mass-conserving mild solutions under sum-type kernels with . By exploiting a fixed-point iteration and regionwise supersolution constructions, it obtains bounds uniform in the small sedimentation parameter , yielding a rigorous existence theory for times of order one. In the fast-sedimentation limit , the solutions concentrate on the curve and formally converge to a Dirac-measure-supported state, leading to a reduced one-dimensional diagonal-kernel coagulation equation for the marginal with kernel (homogeneity ). This provides a concrete mechanism by which diagonal-kernel coagulation can emerge from inhomogeneous transport-dominated coagulation models and offers physical intuition for long-time behavior and potential global existence in related systems.

Abstract

We study a spatially inhomogeneous coagulation model that contains a transport term in the spatial variable. The transport term models the vertical motion of particles due to gravity, thereby incorporating their fall into the dynamics. Local existence of mass-conserving solutions for a class of coagulation rates for which in the spatially homogeneous case instantaneous gelation (i.e., instantaneous loss of mass) occurs has been proved in [Cristian-Niethammer-Velázquez, 2024]. In order to obtain some insight into how to prove global existence of solutions, we allow a fast sedimentation speed. For very fast sedimentation speed, we rigorously prove that solutions converge to a Dirac measure in the space variable. We also formally obtain in the limit a one-dimensional coagulation equation with diagonal kernel, i.e., only particles of the same size interact. This provides a physical intuition on how coagulation models with a diagonal kernel emerge.

Paper Structure

This paper contains 26 sections, 23 theorems, 447 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Connection with coagulation models with diagonal kernel
  4. Self-consistent argument for behavior of solutions as $t\rightarrow\infty$
  5. Connection with previous results for spatially inhomogeneous coagulation models
  6. Connection with coagulation models with fast fusion
  7. Local in time existence
  8. Setting and main results
  9. Justification of the main results
  10. Approximation of the model by a transport term in the region $y\leq (\frac{v}{3})^{\alpha}$
  11. Mathematical toolbox
  12. Plan of the proof of Theorem \ref{['proposition existence via fixed point']}
  13. Region $y\geq (\frac{v}{3})^{\alpha}$
  14. Continuity argument
  15. Iteration argument
  16. ...and 11 more sections

Key Result

Theorem 1.3

Let $K$ be as in sum kernel or rain kernel intro with $\alpha\in(0,1), \gamma\in[0,1+\alpha)$. Let $T>0$ be sufficiently small. Let $f_{\textup{in}}\in \textup{C}^{1}(\mathbb{R}\times\mathbb{R}_{> 0})$ such that for some $A>0$, $x\in\mathbb{R}$, $v\in(0,\infty)$, and where $m=\alpha p$, $m$ even. Then there exists a mass-conserving solution $f$ for the equation (original equation).

Theorems & Definitions (68)

  • Remark 1.2
  • Theorem 1.3: Local existence of mass-conserving solutions for small times
  • Remark 1.4
  • Remark 2.3
  • Definition 2.4: Mild solutions
  • Definition 2.4: Mild solutions
  • Definition 2.5
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8
  • ...and 58 more