• noise on flows
• regularization by noise
• transition point

We studied a one-dimensional model inspired by the Euler equations. In this deterministic model, the blowup is fully studied, being present only in Holder regularity and particular cases. By adding noise on the flow equation (noise on the characteristics), we aim to study if there's regularization, or blowup prevention, for this equation. The regularization result is presented in a rather informal way, supported by quantitative results that point towards a positive outcome, that we weren't able to fully demonstrate.
The main technique used is based on the "transition point": in fact, the blowup is used when holder singularities are around certain explosion points. While analyzing the motion of these points, we distinguished between two regimes, one where the stochastic fluctuations dominate this motion, the other where the deterministic drift is the one leading the movement. We show that with positive probability we exit from the fluctuation regime, and with positive probability we move further from the explosion points than a certain fixed value. We then iterate this procedure, to conclude that no holder point is definitely near an explosion point, thus suggesting a positive outcome.