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Stability results, almost global generalized Beltrami fields and applications to vortex structures in the Euler equations. (English) Zbl 1397.35192

This very comprehensive work deals with so-called Beltrami fields , that is, vector fields in three dimensions whose curl is the product of the field itself by a constant factor. In particular they are the kind of stationary solutions of the Euler equations where one has been able to show the existence of vortex structures of arbitrarily complicated topology.
The main objective of the present paper is to study the existence, regularity and stability results of generalized Beltrami fields . These vector fields play a fundamental role in the understanding of turbulence.
A further aim of the paper consists in showing that, although generalized Beltrami fields are indeed rare , one is still able to prove some kind of partial stability result. This article contains a lot of preliminary material with many given proofs. The bibliography contains 45 items.

MSC:

35Q31 Euler equations
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76M15 Boundary element methods applied to problems in fluid mechanics
35J46 First-order elliptic systems
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B35 Stability in context of PDEs
76F25 Turbulent transport, mixing
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