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Existence for the Discrete Nonlinear Fragmentation Equation with Degenerate Diffusion

Saumyajit Das, Ram Gopal Jaiswal

Abstract

A mathematical model for the discrete nonlinear fragmentation (collision-induced breakage) equation with diffusion is studied. The existence of global weak solutions is established in arbitrary spatial dimensions without assuming a strictly positive lower bound on the diffusion coefficients, extending previous results that were restricted to one-dimensional domains and relied on uniformly positive diffusion. The analysis is carried out under boundedness assumptions on the collision and breakage kernels. The proof is based on the construction of a suitable regularized system, combined with weak $L^2$ a priori estimates and compactness arguments in $L^1$, which allow the passage to the limit in the nonlinear fragmentation operator.

Existence for the Discrete Nonlinear Fragmentation Equation with Degenerate Diffusion

Abstract

A mathematical model for the discrete nonlinear fragmentation (collision-induced breakage) equation with diffusion is studied. The existence of global weak solutions is established in arbitrary spatial dimensions without assuming a strictly positive lower bound on the diffusion coefficients, extending previous results that were restricted to one-dimensional domains and relied on uniformly positive diffusion. The analysis is carried out under boundedness assumptions on the collision and breakage kernels. The proof is based on the construction of a suitable regularized system, combined with weak a priori estimates and compactness arguments in , which allow the passage to the limit in the nonlinear fragmentation operator.
Paper Structure (15 sections, 15 theorems, 234 equations)

This paper contains 15 sections, 15 theorems, 234 equations.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. Notation
  4. Truncated regularized approximate system
  5. Regularized system
  6. Uniform estimates and compactness of the regularized solutions
  7. Control of the Regularized Term via a Truncation Procedure
  8. Definition of the truncation
  9. Justification of weak time and spatial derivatives
  10. Differential inequality satisfied by the truncation
  11. Weak formulations of the truncated inequality
  12. Passing to the limit delta to zero
  13. Passing to the limit epsilon to zero
  14. Weak supersolution to the nonlinear fragmentation equation \ref{['NFE']}
  15. Proof of Theorem \ref{['existence result']}

Key Result

Theorem 1.4

Let the coagulation kernel $a_{i,j}$, the breakage kernel $b_{i,j}^k$, and the diffusion coefficients $d_i$ satisfy either Assumption weaker assumption or Assumption strong assumption for all $i,j,k \in \mathbb{N}$. Suppose that the initial data satisfy Then the system NFE admits a global-in-time nonnegative weak solution $\{f_i\}_{i\in\mathbb{N}}$ such that, for every $i \in \mathbb{N}$,

Theorems & Definitions (27)

  • Definition 1.3: Weak solution
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6: Positivity of solutions, Pierre2010
  • Theorem 1.7: desvillettes2007global
  • Proposition 2.1: Existence and regularity for the truncated system
  • proof
  • Proposition 3.1: Existence and mass conservation for the infinite regularized system
  • proof
  • Proposition 4.1
  • ...and 17 more