We say that an ordinary or partial differential equation is regularized by noise if the addition of a suitable noise term restores well-posedness or improves regularity of the solution to the equation. Regularization by noise is by now well understood for ODEs and linear transport-type PDEs, but it is less understood for nonlinear PDEs like Euler and Navier-Stokes equations, with many open questions.

In the first part of this series of lectures, we consider the case of ODEs and associated transport equations with irregular drift: we show that the addition of an additive Brownian noise restores well-posedness of the ODE; we introduce the corresponding transport noise and show that this noise restores well-posedness for the associated transport equation. In the second part, we focus our attention on the effect of a particular transport-type noise, which is divergence-free, Gaussian, white in time and poorly correlated in space (nonsmooth Kraichnan noise). The associated linear transport model, introduced by Kraichnan, has been widely studied due to its peculiar features, like spontaneous stochasticity. In the case of incompressible 2D Euler equations, we show that the addition of nonsmoooth Kraichnan noise improves significantly the well-posedness theory, for example bringing uniqueness among finite enstrophy solutions. If time allows, we show further regularization properties of the Kraichnan noise.

Based on classical works (e.g. the works by Flandoli and coauthors) and on joint works with Marco Bagnara, Michele Coghi, Lucio Galeati, Francesco Grotto.