[Paper Review] Eulerian dynamics in multi-dimensions with radial symmetry

Changhui Tan|arXiv (Cornell University)|Jul 30, 2020

Navier-Stokes equation solutions22 references4 citations

TL;DR

This paper introduces a novel pair of scalar quantities, $u_r$ and $u/r$, to replace the 1D velocity gradient $\partial_x u$ in multi-dimensional Eulerian dynamics with radial symmetry. By controlling the spectral gap through these variables, the authors establish sharp threshold conditions for global regularity in the Euler-Poisson and Euler-alignment equations, proving global smooth solutions for large subcritical initial data and identifying precise blowup criteria.

ABSTRACT

We study the global wellposedness of pressure-less Eulerian dynamics in multi-dimensions, with radially symmetric data. Compared with the 1D system, a major difference in multi-dimensional Eulerian dynamics is the presence of the spectral gap, which is difficult to control in general. We propose a new pair of scalar quantities that provides a significant better control of the spectral gap. Two applications are presented. (i) the Euler-Poisson equations: we show a sharp threshold condition on initial data that distinguish global regularity and finite time blowup; (ii) the Euler-alignment equations: we show a large subcritical region of initial data that leads to global smooth solutions.

Motivation & Objective

• To address the challenge of controlling the spectral gap in multi-dimensional pressure-less Eulerian dynamics, which hinders the extension of 1D regularity results.
• To develop a new framework for analyzing global wellposedness in multi-dimensional systems by replacing the 1D velocity gradient with radial-specific scalar quantities.
• To apply the new framework to two classical models: the Euler-Poisson and Euler-alignment equations, with a focus on radial symmetry.
• To derive sharp threshold conditions that distinguish global smooth solutions from finite-time blowup in these systems.
• To extend 1D critical threshold theory to higher dimensions by leveraging radial symmetry and new dynamical variables.

Proposed method

• Introduce the radial velocity gradient $u_r$ and the radial mean velocity $u/r$ as key scalar quantities that capture 1D-like dynamics in radial symmetry.
• Derive evolution equations for $u_r$ and $u/r$ that inherit Riccati-type structures, enabling control over the spectral gap.
• Define auxiliary variables $q = u_r$ and $s = u/r$, and derive a closed system of ODEs for $(q, s)$ that governs the dynamics of the velocity gradient.
• Use the boundedness of $(q, s)$ to control the spectral gap and ensure global regularity of the solution via the relation $\nabla \mathbf{u} = \text{diag}(q, s, \dots, s)$ under radial symmetry.
• Apply the framework to the Euler-Poisson system by introducing a forcing term derived from the Poisson equation, and to the Euler-alignment system via a nonlocal alignment force.
• Establish threshold conditions via comparison with solutions to Riccati-type ODEs, leading to explicit subcritical regions in initial data space.

Experimental results

Research questions

• RQ1 Can a new pair of scalar quantities be constructed to effectively control the spectral gap in multi-dimensional Eulerian dynamics with radial symmetry?
• RQ2 What are the sharp threshold conditions on initial data that determine global regularity versus finite-time blowup in the Euler-Poisson system under radial symmetry?
• RQ3 How can the critical threshold theory from 1D be extended to multi-dimensional systems with nonlocal forces such as in the Euler-alignment model?
• RQ4 What role does the spectral gap play in the formation of singularities, and how can it be controlled using radial symmetry?
• RQ5 Can the new framework be generalized to systems with pressure, swirl, or non-radial perturbations?

Key findings

• The authors introduce $u_r$ and $u/r$ as effective replacements for $\partial_x u$ in multi-dimensional radial flows, enabling Riccati-type analysis.
• For the Euler-Poisson equations, a sharp threshold condition is derived: global regularity holds if initial data satisfy $G_0 \geq \sigma_G^+(C_0)$, where $\sigma_G^+$ is defined by a non-autonomous ODE.
• In the Euler-alignment equations, a large subcritical region is identified where global smooth solutions exist, significantly expanding previous results.
• The threshold curves $\sigma_G^+$ and $\sigma_G^-$ are dimension-dependent, with a growing gap between subcritical and supercritical regions as dimension $n$ increases.
• For $n=1$, the new threshold conditions reduce to the known sharp 1D result, confirming consistency and validity of the framework.
• Global wellposedness is established via boundedness of $\nabla \mathbf{u}$, which follows from uniform boundedness of $q = u_r$ and $s = u/r$, with asymptotic flocking behavior confirmed.