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Linearization method and sharp thresholds for spherically symmetric multidimensional pressureless Euler-Poisson equations

Olga S. Rozanova, Marko K. Turzynsky

TL;DR

This work develops a linearization framework to study singularity formation in radially symmetric multidimensional pressureless Euler–Poisson equations. By applying the Radon lemma, the authors transform the nonlinear derivative system into a linear ODE problem driven by an auxiliary function q(t); the existence of zeros of q(t) precisely signals loss of smoothness. In key cases (notably d = 1 and d = 4) the criteria can be expressed analytically via special functions, while in general they reduce to numerically tractable checks of q(t) or q(M) using the derived linear systems. The paper also provides phase-plane classifications, sufficient blow-up conditions via separatrices, and numerical demonstrations of the smoothness domain's structure, offering a practical route to assess threshold behavior in these nonlocal models.

Abstract

We show that the question about the criterion of a singularity formation for radially symmetric solutions to the Cauchy problem for a fairly wide class of equations related to the pressureless Euler-Poisson equations can be reduced to the study of solutions to a linear homogeneous ordinary differential equation. In some cases, such a criterion can be obtained in terms of the initial data. In the remaining cases, it is possible to construct a simple numerical procedure, on the basis of which the question about preserving smoothness for any set of initial data can be solved.

Linearization method and sharp thresholds for spherically symmetric multidimensional pressureless Euler-Poisson equations

TL;DR

This work develops a linearization framework to study singularity formation in radially symmetric multidimensional pressureless Euler–Poisson equations. By applying the Radon lemma, the authors transform the nonlinear derivative system into a linear ODE problem driven by an auxiliary function q(t); the existence of zeros of q(t) precisely signals loss of smoothness. In key cases (notably d = 1 and d = 4) the criteria can be expressed analytically via special functions, while in general they reduce to numerically tractable checks of q(t) or q(M) using the derived linear systems. The paper also provides phase-plane classifications, sufficient blow-up conditions via separatrices, and numerical demonstrations of the smoothness domain's structure, offering a practical route to assess threshold behavior in these nonlocal models.

Abstract

We show that the question about the criterion of a singularity formation for radially symmetric solutions to the Cauchy problem for a fairly wide class of equations related to the pressureless Euler-Poisson equations can be reduced to the study of solutions to a linear homogeneous ordinary differential equation. In some cases, such a criterion can be obtained in terms of the initial data. In the remaining cases, it is possible to construct a simple numerical procedure, on the basis of which the question about preserving smoothness for any set of initial data can be solved.

Paper Structure

This paper contains 15 sections, 10 theorems, 64 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The behavior along characteristics
  3. Linearization of system \ref{['sys3']}
  4. Criterion of the singularity formation in terms of auxiliary function
  5. Phase curves of the system \ref{['sysch']} and classification of equilibria
  6. Oscillatory case (the equilibrium is a center), $k>0$, $c>0$.
  7. Behavior of derivatives, formal scheme
  8. Non-oscillatory case I (the equilibrium is a saddle-node), $c=0$.
  9. $d\geqslant 3$, ${\bf V}_0=0$
  10. $d=4$
  11. Non-oscillatory case II (the stable equilibrium is a node), $k<0$, $c>0$.
  12. Sufficient condition for the blow-up
  13. $d=4$
  14. $d\geqslant 3$, numerical result
  15. Discussion

Key Result

Theorem 3.1

A matrix Riccati equation ($W=W(t)$ is a matrix $(n\times m)$, $M_{21}$ is a matrix $(n\times m)$, $M_{22}$ is a matrix $(m\times m)$, $M_{11}$ is a matrix $(n\times n)$, $M_{12}$ is a matrix $(m\times n)$ ) is equivalent to the homogeneous linear matrix equation (${\mathcal{Y}}={\mathcal{Y}}(t)$ is a matrix $(n\times (n+m))$, $M$ is a matrix $((n+m)\times (n+m))$ ) in the following sense. Let o

Figures (3)

  • Figure 1: The behavior of solution of \ref{['sysch']} for $d=3$. Left: oscillatory case, $k=1$, $c=1$, $m=0$. Center: oscillatory case I, $k=1$, $c=0$, $m=0$. Right: non-oscillatory case II, $k=-1$, $c=1$, $m=0$.
  • Figure 2: Separatrices of saddle equilibrium and the direction field for system \ref{['sysch']} at $d=3$, $c=0$, $k=-1$, $G<\frac{c}{d}$. The blow-up region is below the separatrix emerging from the unstable node.
  • Figure 3: The structure of the set, corresponding to a singularity formation (dash), for $d=3$, $k=-1$, $c=1$. Left: $F_0=u_0=0$ (${\bf V}_0=0$). Center: $G_0=v_0=0$ (${\bf E}_0=0$). Right: $F_0=0$, $G_0=0.2$.

Theorems & Definitions (10)

  • Theorem 3.1: The Radon lemma (1927)
  • Theorem 4.1
  • Proposition 1
  • Proposition 2
  • Theorem 6.1
  • Theorem 8.1
  • Corollary 1
  • Corollary 2
  • Proposition 3
  • Theorem 9.1