Seminários de métodos numéricos

Resumo: Shell models are multiscale, infinite-dimensional systems of ordinary differential equations that are widely used as toy models for nonlinear energy cascades in hydrodynamic turbulence.

For certain parameter regimes, some of these models are known to develop finite-time singularities through self-similar blowup. Previous numerical studies conjectured the universality of the blowup dynamics. In this talk, we present a new numerical study revealing the existence of non-unique self-similar blowup profiles in shell models. We derive the equations governing self-similar profiles exhibiting finite-time power-law spectral behavior and treat the spectral slope as a free parameter. Combining techniques from dynamical systems with basic machine learning optimization, we discover the existence of infinitely many self-similar profiles associated with different spectral slopes. Direct numerical simulations of the shell model indicate that all of these blowup profiles are unstable except for the one previously conjectured to be universal. This is a joint work with Eric Simonnet and Simon Thalabard.