"Life after death" in ordinary differential equations with a non-Lipschitz singularity

TL;DR

This paper studies differential equations with non-Lipschitz singularities, introducing a regularization method to understand solution continuation past blowup, revealing attractor-based dynamics and conditions for unique solutions.

Contribution

It develops a regularization approach for non-Lipschitz ODEs, linking pre- and post-blowup dynamics via renormalization and attractors, and identifies conditions for unique continuation.

Findings

Regularization via smoothing yields a well-defined limit for solution continuation.

Post-blowup dynamics are characterized by different attractors in the renormalized system.

Under certain conditions, the regularization selects a unique global solution.

Abstract

We consider a class of ordinary differential equations in $d$-dimensions featuring a non-Lipschitz singularity at the origin. Solutions of such systems exist globally and are unique up until the first time they hit the origin, $t = t_b$, which we term `blowup'. However, infinitely many solutions may exist for longer times. To study continuation past blowup, we introduce physically motivated regularizations: they consist of smoothing the vector field in a $\nu$--ball around the origin and then removing the regularization in the limit $\nu\to 0$. We show that this limit can be understood using a certain autonomous dynamical system obtained by a solution-dependent renormalization procedure. This procedure maps the pre-blowup dynamics, $t < t_b$, to the solution ending at infinitely large renormalized time. In particular, the asymptotic behavior as $t \nearrow t_b$ is described by an attractor. The post-blowup dynamics, $t > t_b$, is mapped to a different renormalized solution starting infinitely far in the past. Consequently, it is associated with another attractor. The $\nu$-regularization establishes a relation between these two different "lives" of the renormalized system. We prove that, in some generic situations, this procedure selects a unique global solution (or a family of solutions), which does not depend on the details of the regularization. We provide concrete examples and argue that these situations are qualitatively similar to post-blowup scenarios observed in infinite-dimensional models of turbulence.