an:06428138
Zbl 1317.35184
Enciso, Alberto; Peralta-Salas, Daniel
Existence of knotted vortex tubes in steady Euler flows
EN
Acta Math. 214, No. 1, 61-134 (2015).
00342569
2015
j
35Q31 37N10 57M25 35J25 35J40
Euler equation; invariant tori; KAM theory; knots; Beltrami fields; Runge-type approximation
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. {\parindent=4mm \begin{itemize} \item[--] 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. \item [--] It is proved that these invariant tori are ``robust'' in a certain sense. \item [--] 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.
\end{itemize}} The proofs use Lyapunov stability, Cauchy-Schwartz, Poincar??, Jensen and Sobolev inequalities, Hodge decomposition, Riesz representation theorem, Fredholm alternative, Poincar?? map, Hahn-Banach theorem, Riesz-Markov theorem, Poissons equation and spherical Bessel functions. Finally, applications to Navier-Stokes equation are briefly discussed.
Thomas Ernst (Uppsala)