Uniqueness of a solution to the Cauchy problem for the Hopf equation in the two-dimensional case

We consider the Cauchy problem for the Hopf equation corresponding to the two-dimensional Navier–Stokes equations. The global uniqueness of a solution is proved under the condition that the mean energy of the initial measure is finite, without any additional conditions, i.e., under the same assumptions as in the existence theorem. Thereby we receive a sharp improvement of Foias’s result obtained in the 1970’s about the uniqueness of a spatial statistical solution. Bibliography: 12 titles.