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 threat the velocity $u$ and its $curl\, u$ 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 $curl \,u \in L^{ \kappa}_t(W^{ \alpha,1}_x)$ where $r,\kappa $ are conjugate H\"older exponents and $2\theta+\alpha \geq 1$. Using the same techniques we also show that the helicity has a suitable H\"older regularity even in the range where it is not necessarily constant.
Categories: math.AP