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Non Local Hyperbolic Dynamics of Clusters

Rinaldo M. Colombo, Mauro Garavello

Abstract

The formation, movement and gluing of clusters can be described through a system of non local balance laws. Here, the well posedness of this system is obtained, as well as various stability estimates. Remarkably, qualitative properties of the solutions are proved, providing information on stationary solutions and on the propagation speed. In some cases, fragmentation leads to clusters developing independently. Moreover, these equations may serve as an encryption/decryption tool. This poses new analytical problems and asks for improved numerical methods.

Non Local Hyperbolic Dynamics of Clusters

Abstract

The formation, movement and gluing of clusters can be described through a system of non local balance laws. Here, the well posedness of this system is obtained, as well as various stability estimates. Remarkably, qualitative properties of the solutions are proved, providing information on stationary solutions and on the propagation speed. In some cases, fragmentation leads to clusters developing independently. Moreover, these equations may serve as an encryption/decryption tool. This poses new analytical problems and asks for improved numerical methods.

Paper Structure

This paper contains 20 sections, 8 theorems, 81 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Well Posedness and Qualitative Properties
  3. Examples
  4. Fragmentation, or Clusters Formation
  5. On the Role of V
  6. A Stationary Solution
  7. Encryption -- Decryption
  8. The Scalar 1-Dimensional Case
  9. $2$-Dimensonal Examples
  10. Proofs
  11. Claim 1: $\mathcal{T}$ is well defined.
  12. Claim 2: For $T$ small, $\mathcal{T}$ is a contraction.
  13. Claim 3: Extension of $\rho^*$ for all times.
  14. Claim 4: Uniqueness of the Solution to the Cauchy problem.
  15. Claim 5: Definition of the group $\mathcal{G}$.
  16. ...and 5 more sections

Key Result

Theorem 2.2

Let item:6 and item:7 hold. Then eq:1 generates a unique map such that

Figures (7)

  • Figure 2.1: An example of a situation complying with \ref{['eq:32']} of Corollary \ref{['cor:clustering']} in the case $k = 3$. The dashed circumferences may not overlap with the solid ones.
  • Figure 3.1: Solution to \ref{['eq:1']}--\ref{['eq:50']} with $\eta$ as in \ref{['eq:46']}. This evolution is consistent with Corollary \ref{['cor:clustering']}, since the $4$ parts of the solutions in the $4$ corners evolve independently, as well as with Proposition \ref{['prop:stationary']}, since the top left part is stationary.
  • Figure 3.2: Evolution of the solutions to \ref{['eq:1']}--\ref{['eq:55']}--\ref{['eq:46']}. On the left, respectively middle and right column, the vector field is $V^1$, respectively $V^2$ and $V^3$ in \ref{['eq:57']}. Only in the solution in the left column is the total mass conserved. In the middle and right cases, some mass exits the numerical domain.
  • Figure 3.3: Numerical integration of equation \ref{['eq:1']} with parameters and initial datum as in \ref{['eq:46']}--\ref{['eq:45']}. Coherently with Proposition \ref{['prop:stationary']}, the resulting solution is stationary.
  • Figure 3.4: Left, the original data \ref{['eq:42']}; center the encrypted data and, right, the decrypted data. Encryption is obtained through \ref{['eq:1']}--\ref{['eq:36']}, decryption through an integration backward in time. Theorem \ref{['thm:cauchyProblem']} guarantees the feasibility of this procedure.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Corollary 2.6
  • Proposition 2.7
  • Lemma 4.1
  • Lemma 4.2