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On the Helicity conservation for the incompressible Euler equations | Luigi De Rosa
; | Date: |
3 Dec 2018 | 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 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 | Services: | Forum | Review | PDF | Favorites |
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