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A variational formulation for the Navier-Stokes equation. (English) Zbl 1080.37083

Summary: We prove a new variational principle for the Navier-Stokes equation which asserts that its solutions are critical points of a stochastic control problem in the group of area-preserving diffeomorphisms. This principle is a natural extension of the results by V. I. Arnol’d [Ann. Inst. Fourier 16, 319–361 (1966; Zbl 0148.45301)], and D. G. Ebin and J. Marsden [Ann. Math. (2) 92, 102–163 (1970; Zbl 0211.57401)] concerning the Euler equation.

MSC:

37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
76M30 Variational methods applied to problems in fluid mechanics
93C20 Control/observation systems governed by partial differential equations
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