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Steady States of Transport-Coagulation-Nucleation Models

Julia Delacour, Marie Doumic, Carmela Moschella, Christian Schmeiser

Abstract

To model the dynamics of polymers formed through nucleation, elongated by polymerisation, shortened by depolymerisation and subject to aggregation reactions, we study a nonlinear integro-differential equation. Growth and shrinkage are described by transport terms, nucleation by a positive boundary condition, and aggregation by a Smoluchowski coagulation kernel. Our main result is the existence of steady states for the multiplicative coagulation kernel despite this kernel producing gelation in finite time for the pure coagulation equation. This is made possible by a sufficiently strong decay rate for large polymers. Beyond the existence result, the qualitative properties of the steady states are illustrated through explicit examples and numerical experiments. The analytical results connect the growth behaviour of the transport velocity and of the coagulation kernel to the decay properties of steady states.

Steady States of Transport-Coagulation-Nucleation Models

Abstract

To model the dynamics of polymers formed through nucleation, elongated by polymerisation, shortened by depolymerisation and subject to aggregation reactions, we study a nonlinear integro-differential equation. Growth and shrinkage are described by transport terms, nucleation by a positive boundary condition, and aggregation by a Smoluchowski coagulation kernel. Our main result is the existence of steady states for the multiplicative coagulation kernel despite this kernel producing gelation in finite time for the pure coagulation equation. This is made possible by a sufficiently strong decay rate for large polymers. Beyond the existence result, the qualitative properties of the steady states are illustrated through explicit examples and numerical experiments. The analytical results connect the growth behaviour of the transport velocity and of the coagulation kernel to the decay properties of steady states.
Paper Structure (14 sections, 3 theorems, 51 equations, 2 figures)

This paper contains 14 sections, 3 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The choice of the transport velocity
  3. Constant transport velocity and constant coagulation rate:
  4. Constant transport velocity and multiplicative coagulation rate:
  5. Choosing a transport velocity with changing sign:
  6. The inverse problem -- explicit examples of steady state solutions
  7. Exponential steady states:
  8. Steady states with algebraic decay:
  9. Existence and properties of steady states
  10. Qualitative properties of steady states
  11. behaviour of solutions of \ref{['eq_1s']}, \ref{['BC-infty']} close to $x=x_0$:
  12. Numerical results
  13. Comparison with an explicitly known steady state:
  14. The behaviour close to TEXT:

Key Result

Lemma 1

Let assumption eq_3s and $0<\varepsilon < 1/x_0$ hold. Then problem eq_1-reg has a solution $j_\varepsilon\in C^1([0,x_0])\times C^1([x_0,1/\varepsilon])$ with $j_\varepsilon/v_\varepsilon\ge 0$, $M_1[j_\varepsilon/v_\varepsilon]=\mathcal{M}_1$, and

Figures (2)

  • Figure 1: Snapshots of the numerical solution for the case of an explicitly known steady state. On the left are depicted the initial data (green), an intermediate time (black), and the numerical steady state (blue). The intermediate state shows that the initial peak is moved to the left by transport, and additional peaks are created by coagulation. The kink at $x=x_0$ of the numerical steady state is a numerical artefact due to the singularity of the steady state equation. On the right, a comparison between the numerical steady state (blue) and the exact steady state (red dashed) is displayed.
  • Figure 2: behaviour of numerical steady states near $\boldsymbol{x_0}$. Numerical steady states for different values of $c$, computed on grids with $\Delta x = 0.034$ (blue solid line), $\Delta x = 0.017$ (green dashed line), and $\Delta x = 0.008$ (red dotted line). (a)$c < 1$, (b)$c = 1$, (c)$c > 1$. (d) Peak values of the numerical steady states for different values of $\Delta x$, for $c < 1$ (blue), $c = 1$ (red), and $c > 1$ (black).

Theorems & Definitions (6)

  • Definition 1
  • Lemma 1
  • Remark 1
  • Theorem 4.1
  • Lemma 2
  • Remark 2