Anomalous Regularization and Blow-up Prevention in Stochastic Fluid Dynamics Models

This PhD thesis is concerned with two distinct phenomena induced by stochastic perturbations, both capable of improving well-posedness theory in stochastic fluid dynamics models.The first phenomenon, which is anomalous regularisation, is the main topic of the first part of this thesis. It arises when considering a rough Kraichnan noise in transport-like form. Such noise, which is white in time and poorly correlated in space, is a phenomenological model for turbulence and displays interesting features like splitting and anomalous dissipation. Heuristically speaking, splitting is at the core of a fractional diffusion mechanism which gives rise to anomalous regularisation. First, we will show how the generalised SQG model perturbed by the Kraichnan noise in transport form, gains pathwise uniqueness for Lp-valued solutions. Afterwards, we consider the Kazantsev-Kraichnan model, i.e. a passive divergence-free vector field which is both transported and stretched by the Kraichnan velocity. In this case, the spatial roughness of the driving velocity precludes the use of the Alfvén representation formula and it induces an instantaneous blow-up of the L2-norm. Nevertheless, in a certain regime of parameters, anomalous regularisation takes place and makes it possible to prove well-posedness and a fractional regularity gain.The second phenomenon treated in this thesis is the prevention of blow-up via strong superlinear noise. The goal is to show that there exist stochastic perturbations capable of preventing blow-up in a large class of SPDEs which include the 3D Euler equations. The idea is to design a suitable diffusion coefficient able to compensate for the nonlinearity in the PDE that may cause blow-up in the deterministic setting. In the case of Ito noise, the goal can be reached by a single source of randomness (a one-dimensional Wiener process is sufficient) provided it multiplies a sufficiently strong diffusion coefficient. The strategy is ultimately based on employing a logarithmic Lyapunov function, exploiting its concavity to ensure that the second-order Ito correction term is negative, and a Gronwall-type argument. Then, the case of Stratonovich noise is considered. In this setting the same diffusion considered before will not work because the Ito-Stratonovich correction promptly compensates for the second order Ito term. However, a different diffusion coefficient, which requires the driving Wiener process to be truly multidimensional, can still be found.