This book is devoted to stochastic Navier-Stokes equations and more generally to stochasticity in fluid mechanics. The two opening chapters describe basic material about the existence and uniqueness of solutions: first in the case of additive noise treated pathwise and then in the case of state-dependent noise. The main mathematical techniques of these two chapters are known and given in detail for using the book as a reference for advanced courses. By contrast, the third and fourth chapters describe new material that has been developed in very recent years or in works now in preparation. The new material deals with transport-type noise, its origin, and its consequences on dissipation and well-posedness properties. Finally, the last chapter is devoted to the physical intuition behind the stochastic modeling presented in the book, giving great attention to the question of the origin of noise in connection with small-scale turbulence, its mathematical form, and its consequenceson large-scale properties of a fluid.

Autorentext

Franco Flandoli

Franco Flandoli is a full professor in probability at Scuola Normale Superiore of Pisa since 2017; before, he has been a researcher and professor at the University of Torino and the University of Pisa. He has been a visiting professor at several international institutions and gave various series of advanced lectures, like those in Waseda, Tokyo, related to this book, a CIME Summer School in 2007 and the Saint Flour lectures in 2010, related to other Springer volumes. He got a Medal at the World Congress in Probability and Statistics, Istanbul 2012, and a Medal of the Accademia Nazionale delle Scienze, 2020. His main research activity is concerned with stochastic analysis and stochastic partial differential equations applied to fluid mechanics and turbulence, a subject where he holds the ERC advanced grant "NoisyFluid" and drives a group of young researchers.

Eliseo Luongo

Eliseo Luongo is a PhD candidate at Scuola Normale Superioreof Pisa. After graduating in Mathematical Engineering at Politecnico of Milan in 2020, he moved to Scuola Normale Superiore of Pisa, where he is currently a member of the research group of Franco Flandoli. His research activity is concerned with stochastic partial differential equations applied to fluid mechanics, in particular models where turbulence and the boundary of the domain play a role.