Roughness of generic functions. Part II: prevalence of mixing and enhanced dissipation
I will address the problem of mixing and dissipation properties of generic shear flows. Here genericity is understood in the measure-theoretic sense of “prevalence”. The analysis shows that two different notions of irregularity play a key role in deriving upper bounds for the decay of the relevant norms. Corresponding lower bounds indicate that the results are near-optimal when one focuses on a specific family of norms.
Based on the papers Lucio Galeati and Massimiliano Gubinelli, ‘Prevalence of $\rho$-Irregularity and Related Properties', ArXiv:2004.00872 [Math], 2 April 2020, http://arxiv.org/abs/2004.00872. Lucio Galeati and Massimiliano Gubinelli, ‘Noiseless Regularisation by Noise', ArXiv:2003.14264 [Math], 31 March 2020, http://arxiv.org/abs/2003.14264; Lucio Galeati and Massimiliano Gubinelli, ‘Mixing for Generic Rough Shear Flows', ArXiv:2107.12115 [Math], 26 July 2021, http://arxiv.org/abs/2107.12115.