The Euler Equations and Non-Local Conservative Riccati Equations
Peter Constantin
TL;DR
The paper constructs an infinite-dimensional family of exact, incompressible 3D Euler solutions with infinite energy that blow up in finite time using an Eulerian-Lagrangian ansatz $\mathbf{u}=(u, z\gamma)$. It reduces the problem to a non-local conservative Riccati equation for $\gamma$ and solves it along characteristics, yielding an explicit formula for $\gamma$ in terms of the initial data and a time-reparametrization $\tau(t)$. The authors establish finite-time blow-up for mean-zero initial data with negative minimum, derive the blow-up time and robust asymptotics, and show that no one-sided blow-up can occur; the results hold under variations of the curl and even with zero initial vorticity. This work provides explicit, analyzable solutions that illuminate nonlocal Riccati dynamics within the Euler equations and the nature of finite-time singularities.
Abstract
We present an infinite dimensional family of of exact solutions of the incompressible three-dimensional Euler equations. These solutions, proposed by Gibbon and Ohkitani, have infinite kinetic energy and blow up in finite time.