"Life after death" in ordinary differential equations with a nonLipschitz singularity
Abstract
We consider a class of ordinary differential equations in $d$dimensions featuring a nonLipschitz 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 solutiondependent renormalization procedure. This procedure maps the preblowup 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 postblowup 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 postblowup scenarios observed in infinitedimensional models of turbulence.
 Publication:

arXiv eprints
 Pub Date:
 June 2018
 arXiv:
 arXiv:1806.09001
 Bibcode:
 2018arXiv180609001D
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematical Physics;
 Mathematics  Dynamical Systems
 EPrint:
 27 pages, 8 figures