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Blow-up rate of solution to generalised Blasius equation

Guillaume Blanc, Alice Contat

TL;DR

This work determines the blow-up rate for the maximal solution of the generalized Blasius-type equation $y^{(d+1)}=y\cdot y^{(d)}$ with zero initial lower derivatives, establishing finite-time blow-up and precise asymptotics for derivatives when $d\le10$ and Theta-bounds for all $d$. The authors reduce the problem to a Lotka–Volterra system through a sequence of time changes built from $y$ and its derivatives, and analyze the LV dynamics using Lyapunov/average Lyapunov methods to obtain convergence to a bulk fixed point in low dimensions and time-average convergence in general. A phase transition in dimension arises: the positive-definiteness-based Lyapunov approach holds for $d\le10$, while for $d\ge11$ numerical evidence suggests oscillations around the fixed point, leaving the strong blow-up rate question open in higher dimensions. As a corollary, the identified blow-up rate informs a probabilistic Poissonian burning model in $\mathbb{R}^d$, yielding almost sure complete burning of space and motivating a future discrete scaling-limit analysis.

Abstract

We identify the blow-up rate of a solution to a generalised Blasius equation, that we came across while studying a probabilistic model of "Poissonian burning" in Euclidean space. Our proof involves the study of the long-time behaviour of solutions to a Lotka--Volterra system.

Blow-up rate of solution to generalised Blasius equation

TL;DR

This work determines the blow-up rate for the maximal solution of the generalized Blasius-type equation with zero initial lower derivatives, establishing finite-time blow-up and precise asymptotics for derivatives when and Theta-bounds for all . The authors reduce the problem to a Lotka–Volterra system through a sequence of time changes built from and its derivatives, and analyze the LV dynamics using Lyapunov/average Lyapunov methods to obtain convergence to a bulk fixed point in low dimensions and time-average convergence in general. A phase transition in dimension arises: the positive-definiteness-based Lyapunov approach holds for , while for numerical evidence suggests oscillations around the fixed point, leaving the strong blow-up rate question open in higher dimensions. As a corollary, the identified blow-up rate informs a probabilistic Poissonian burning model in , yielding almost sure complete burning of space and motivating a future discrete scaling-limit analysis.

Abstract

We identify the blow-up rate of a solution to a generalised Blasius equation, that we came across while studying a probabilistic model of "Poissonian burning" in Euclidean space. Our proof involves the study of the long-time behaviour of solutions to a Lotka--Volterra system.

Paper Structure

This paper contains 14 sections, 8 theorems, 72 equations, 3 figures.

Table of Contents

  1. Blow-up rate of $y$.
  2. Probabilistic motivation.
  3. Strategy of the proof of Theorem \ref{['thm:main']} and organisation of the paper.
  4. Phase transition in $d$.
  5. Acknowledgements.
  6. Poissonian burning
  7. Preliminary considerations
  8. Absolute monotonicity
  9. Finite time blow-up
  10. Rough bounds
  11. Reduction to a Lotka--Volterra system
  12. Time changes.
  13. Back to $y$.
  14. Study of the Lotka--Volterra system

Key Result

Theorem 1

Fix $d\in\mathbb{N}^*$, and let $y:{[0,T[}\rightarrow\mathbb{R}$ be the maximal solution of eq:cauchy. This solution blows up in finite time: we have $T<\infty$, and $y(t)\rightarrow\infty$ as $t\to T^-$. Moreover, we identify its blow-up rate in the following sense:

Figures (3)

  • Figure 1: Simulation of the model of Poissonian burning for $d=1$, in the window $[-5,5]$. Space is depicted horizontally, whereas time evolves vertically. At each time, so for each horizontal layer, we color the points that are burned with the same color as the first point of the process that set them on fire.
  • Figure 2: Euclidean distance to the bulk stationary point for four generic solutions of the Lotka--Volterra system \ref{['eq:lotkavolterra']} for $d=11$.
  • Figure 3: Oscillation around the bulk stationary point for some generic solution of the Lotka--Volterra system \ref{['eq:lotkavolterra']} for $d=11$, coordinate per coordinate.

Theorems & Definitions (20)

  • Theorem 1
  • Corollary 1
  • proof : Proof of Corollary \ref{['cor:probabilistic']}
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Claim 1
  • ...and 10 more