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

Capiński, Marek; Cutland, Nigel J.

Ann. Appl. Probab. 3, No. 1, 212-227 (1993).

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.

Reviewer: W.Kotarski (Katowice)

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)

Keywords:

invariant measures; Euler equations; nonstandard construction of a global flow; existence of statistical solutions