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

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.

27 pages, 8 figures

References in corpus (11)

• From the butterfly effect to intrinsic randomness: the spontaneous growth of singular shear flows
• Wave-breaking and generic singularities of nonlinear hyperbolic equations
• A Lagrangian fluctuation-dissipation relation for scalar turbulence, I. Flows with no bounding walls
• Spontaneously stochastic solutions in one-dimensional inviscid systems
• Chaotic blowup in the 3D incompressible Euler equations on a logarithmic lattice
• Rayleigh-Taylor turbulence with singular nonuniform initial conditions
• Spontaneous stochasticity of velocity in turbulence models
• Toward analytic theory of the Rayleigh-Taylor instability: lessons from a toy model
• Bifurcations of blowup in inviscid shell models of convective turbulence
• Chaotic and regular instantons in helical shell models of turbulence
• Regularisation for Planar Vector Fields