  The paper improves the classical uniqueness result for the Euler system in
the $n$ dimensional case assuming that $\nabla u^E \in L_1(0,T;BMO(\Omega))$,
only. Moreover the rate of the convergence for the inviscid limit of solutions
to the Navier-Stokes equations is obtained, provided the same regularity of the
limit Eulerian flow. A key element of the proof is a logarithmic inequality
between the Hardy and $L_1$ spaces which is a consequence of the basic
properties of the Zygmund space $\LlnL$.
