Grant Details

This research project originates from an outstanding unsolved question, related to one of the Clay Institute millennium prize problems: whether the 3D Navier-Stokes equations, whose well posedness is still an open problem, may have better well posedness properties when a suitable noise is added to the equations. The project PI and the unit PI's have substantially contributed to this topic in the past and they represent an ideal team for the prosecution of this investigation. We shall deeply consider this question in the project. However, with the awareness of its extreme difficulty and the relatively short time span of the project compared to such an outstanding problem, the project has been diversified in several side themes of great independent interest, some of them originated precisely from the main question above. The expected contribution of the project will mostly be the advancement of the theories which are behind the main question of well posedness of 3D stochastic Navier-Stokes equations. The project is therefore organized along two main directions summarized in two WorkPackages, WP1 and WP2, with several milestones indicating the expected outcomes of the activity. WP1 is devoted to Regularization by Noise. Some of the milestones are specifically addressed to 3D fluid mechanics and have to do with the impact of a stretching noise on vector fields. Other milestones have to do with more general questions like the regularization properties of rough deterministic inputs, the role of Kolmogorov equation and its regularity theory both in questions of uniqueness and blow-up, application of methodologies like the regularity of Kolmogorov equations to specific models of stochastic ordinary and partial differential equations. WP2 is devoted to a broad range of questions in Stochastic Fluid Mechanics. Some of them are directly linked to the regularization by noise methodologies but find place in WP2 because they are important also for other reasons, like the understanding of turbulence: an example is the concept of eddy viscosity. Other milestones have to do with specific models of fluid mechanics like point vortices, whose dynamics and statistics are very important in turbulence theory and in the approximation of deterministic and stochastic fluid equations. Further ones have to do with properties of the law of the solutions and the invariant measures of equations, touching also problems like the inviscid limit and some questions of boundary layers.