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Existence of knotted vortex tubes in steady Euler flows. (English) Zbl 1317.35184
The purpose of this paper is the proof of a theorem which states that knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation in \(\mathbb{R}^3\) exist. The proof consists of three steps, which are gradually improved in the paper.
The construction of a local Beltrami field, which satisfies the Beltrami equation \(\mathrm{curl}~v=\lambda v\), and has a set of certain invariant tori.
It is proved that these invariant tori are “robust” in a certain sense.
It is proved that the local Beltrami field can be approximated by a global field, which satisfies the Beltrami equation in \(\mathbb{R}^3\), and drops off at infinity in an optimal way.
The proofs use Lyapunov stability, Cauchy-Schwartz, Poincaré, Jensen and Sobolev inequalities, Hodge decomposition, Riesz representation theorem, Fredholm alternative, Poincaré map, Hahn-Banach theorem, Riesz-Markov theorem, Poissons equation and spherical Bessel functions. Finally, applications to Navier-Stokes equation are briefly discussed.

MSC:
35Q31 Euler equations
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
57M25 Knots and links in the \(3\)-sphere (MSC2010)
35J25 Boundary value problems for second-order elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
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