On the Helicity conservation for the incompressible Euler equations

Abstract

In this work we investigate the helicity regularity for weak solutions of the incompressible Euler equations. To prove regularity and conservation of the helicity we will treat the velocity u u and its curl ⁡ u \operatorname {curl} u as two independent functions and we mainly show that the helicity is a constant of motion assuming u ∈ L t 2 r ( C x θ ) u \in L^{2r}_t(C^\theta _x) and curl ⁡ u ∈ L t κ ( W x α , 1 ) \operatorname {curl} u \in L^{\kappa }_t(W^{\alpha ,1}_x) , where r , κ r,\kappa are conjugate Hölder exponents and 2 θ + α ≥ 1 2\theta +\alpha \geq 1 . Using the same techniques we also show that the helicity has a suitable Hölder regularity even in the range where it is not necessarily constant.