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29 March 2024
 
  » arxiv » 1812.0678

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On the Helicity conservation for the incompressible Euler equations
Luigi De Rosa ;
Date 3 Dec 2018
AbstractIn 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 use two different approaches: in the first we threat the velocity $u$ and its $curl, u$ as two indipendent functions and we mainly show that the helicity is a constant of motion assuming $u in L^{2r}_t(C^ heta_x)$ and $curl ,u in L^{kappa}_t(W^{alpha,1}_x)$ where $r,kappa $ are conjugate H"older exponents and $2 heta+alpha geq 1$; while in the second approach we show the conservation assuming $u in L^3_t(W^{ heta,3}_x)$ for $ heta geq frac{2}{3}$. 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. We also add a simple remark about the critical Onsager’s exponent $ heta =frac{1}{3}$ in Sobolev spaces.
Source arXiv, 1812.0678
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