The Euler equation: A uniform nonstandard construction of a global flow, invariant measures and statistical solutions.(English)Zbl 0779.35084

The following Euler equations for $$v: [0,\infty)\times \mathbb{R}^ 2\to \mathbb{R}^ 2$$ $\partial v/\partial t+\langle v,\nabla\rangle v+\nabla p=0, \qquad \text{div } v=0$ are considered. For those equations a nonstandard construction of a global flow and some classes of measures invariant with respect to that flow, including examples of non-Gaussian ones, are presented.
Existence of statistical solutions of the Euler equations for a wide class of initial measures is also obtained.

MSC:

 35Q05 Euler-Poisson-Darboux equations 28E05 Nonstandard measure theory 35R60 PDEs with randomness, stochastic partial differential equations 58J70 Invariance and symmetry properties for PDEs on manifolds 26E35 Nonstandard analysis 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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