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 $u$ and its $\operatorname {curl} u$ as two independent functions and we mainly show that the helicity is a constant of motion assuming $u \in L^{2r}_t(C^\theta _x)$ and $\operatorname {curl} u \in L^{\kappa }_t(W^{\alpha ,1}_x)$, where $r,\kappa$ are conjugate Hölder exponents and $2\theta +\alpha \geq 1$. 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.