L.G. Loitsianskii derived an integral invariant for the decay of the turbulence by taking the fourth moment of the Kármán–Howarth equation in 1939, i.e.,
If  decays faster than  as  and also in this limit, if we assume that  vanishes, we have the quantity,
which is invariant. Lev Landau and Evgeny Lifshitz showed that this invariant is equivalent to conservation of angular momentum. However, Ian Proudman and W.H. Reid showed that this invariant does not hold always since  is not in general zero, at least, in the initial period of the decay. In 1967, Philip Saffman showed that this integral depends on the initial conditions and the integral can diverge under certain conditions.