Orthogonality relations for a stationary flow of an ideal fluid

For a real solution $(u,p)$ to the Euler stationary equations for an ideal fluid, we derive an infinite series of the orthogonality relations that equate some linear combinations of $m$th degree integral momenta of the functions $u_iu_j$ and $p$ to zero ($m=0,1,\dots$). In particular, the zeroth degree orthogonality relations state that the components ui of the velocity field are $L^2$-orthogonal to each other and have coincident $L^2$-norms. Orthogonality relations of degree $m$ are valid for a solution belonging to a weighted Sobolev space with the weight depending on $m$.