Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Marina A. Ferreira, Sakari Pirnes

TL;DR

This work establishes global existence of measure-valued solutions for the multicomponent Smoluchowski coagulation equation under mild moment assumptions, using an abstract Arzelà–Ascoli framework in uniform spaces to handle a broad class of kernels with power-law upper bounds and region-dependent singularities. It proves mass-conservation for kernels with $\gamma_2\le 1$ and demonstrates finite-time gelation when a lower bound with $\gamma_{gel}\in(1,2)$ holds, extending results to multicomponent and measure-valued settings beyond previous one-component theory. The authors construct regularized problems on a compact annulus, obtain uniform-in-$\epsilon$ bounds, and pass to the limit to obtain a weak solution to the original coagulation equation, with corollaries ensuring strong discrete solutions for the discrete model. The paper also explores localization in mass-conserving solutions, showing under suitable homogeneity and moment bounds that mass concentrates along specific directions in size space, thereby enriching the understanding of long-time behavior in multicomponent coagulation systems. Overall, the results provide a robust well-posedness framework for a broad family of kernels, including both mass-conserving and gelling dynamics, with applications to discrete and continuous multicomponent coagulation models.

Abstract

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.

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

TL;DR

This work establishes global existence of measure-valued solutions for the multicomponent Smoluchowski coagulation equation under mild moment assumptions, using an abstract Arzelà–Ascoli framework in uniform spaces to handle a broad class of kernels with power-law upper bounds and region-dependent singularities. It proves mass-conservation for kernels with and demonstrates finite-time gelation when a lower bound with holds, extending results to multicomponent and measure-valued settings beyond previous one-component theory. The authors construct regularized problems on a compact annulus, obtain uniform-in- bounds, and pass to the limit to obtain a weak solution to the original coagulation equation, with corollaries ensuring strong discrete solutions for the discrete model. The paper also explores localization in mass-conserving solutions, showing under suitable homogeneity and moment bounds that mass concentrates along specific directions in size space, thereby enriching the understanding of long-time behavior in multicomponent coagulation systems. Overall, the results provide a robust well-posedness framework for a broad family of kernels, including both mass-conserving and gelling dynamics, with applications to discrete and continuous multicomponent coagulation models.

Abstract

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.

Paper Structure

This paper contains 17 sections, 27 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Aim of the paper
  3. Notations
  4. Main results
  5. Plan of the paper and key ideas for proofs
  6. Definitions and auxiliary results
  7. The regularized problem
  8. Step 1: regularization
  9. Step 2: solving the regularized problem
  10. Proof of the existence result
  11. Step 3: sequential compactness
  12. Step 4: the limiting measure-valued function satisfies the coagulation equation
  13. Proof of the existence result
  14. Existence of strong discrete solutions
  15. Proof of mass-conservation
  16. ...and 2 more sections

Key Result

Theorem 1.1

(Existence) Suppose that $K$ is as in eq:condK_cont and satisfies the upper bound eq:condK_sym2 with $-\beta\in{\mathbb R}$, and $\gamma,\lambda\in {\mathbb R}^2$ satisfying $-\lambda_j\leq\gamma_j+\lambda_j$ and $\gamma_j+\lambda_j<1$ for $j=1,2$. Suppose that a given initial data $f_0 \in \mathcal

Theorems & Definitions (35)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 3.2
  • ...and 25 more