Invariant and Quasi-invariant Measures for Equations in Hydrodynamics

Abstract

Statistical solutions for differential equations, as opposed to pointwise classical ones, are solutions where the initial data is taken in a measure one set in some probability space. The statistical approach to differential equations was partly motivated by A.N. Kolmogorov’s ideas on turbulence ( [28]), a subject where the interest relies on computation of mean quantities, and is suitable for situations where initial conditions are not precisely known. The use of probability theory to describe the concept of ensemble average, as far as Hydrodynamical equations are concerned, can be traced back to the work of E. Hopf [23]. Later on statistical solutions were introduced by C. Foias [20, 21] and studied also by M.I. Vishik and A.V. Fursikov [33].