Invariant and Quasi-invariant Measures for Equations in Hydrodynamics

  • Ana Bela CruzeiroEmail author
  • Alexandra Symeonides
Conference paper
Part of the CIM Series in Mathematical Sciences book series (CIMSMS)


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].


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.GFMUL and Dep. Mat. Instituto Superior Técnico Univ. LisboaLisboaPortugal
  2. 2.GFMUL and Dep. Mat. Faculdade de Ciências Univ. LisboaLisboaPortugal

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