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Abstract: |
Preliminary to the main topic, there has been an intriguing result of L. Arnold, H. Crauel and V. Wisthutz (1984) about stabilization by noise. M. Capinsky, during a talk in Arnold group in Bremen in 1987, conjectured that such result could have a counterpart for SPDEs of fluid mechanics and could lead to a result of improved mixing rate due to transport noise.
Recently (2019), L. Galeati, D. Luo and myself have proved a result in this direction, which will be described. The noise used in this model may be motivated by arguments about the interaction between small and large scales, formalized by suitable limit theorems (F. and Pappalettera 2020). This motivation hints to a link with turbulence and to the speculation that turbulence and its role on mixing. Some experts also claim that fully developed turbulence prevents the creation of singularities, an open problem in the theory of 3D Navier-Stokes equations. With D. Luo 2020, we have identified a particular version of stochastic 3D Navier-Stokes equations where this fact can be proven. Several questions remain open but some links between turbulence, mixing and SPDEs have been identified. |
