Analytic first integrals of non-linear parabolic systems of differential equations in the sense of Petrovskii, and applications

First integrals are constructed for non-linear parabolic systems (in the sense of Petrovskii) of differential equations with periodic boundary conditions; these are functionals $G(t,u)$ taking a constant value with respect to $t$ on any solution $u(t,x)$ of the original system: $G(t,u(t,\,\cdot\,))=\mathrm{const}$. First integrals are looked for as solutions of a certain first order partial differential equation in infinitely many variables. It is proved that the Cauchy problem for this equation in the case of analytic initial values has a unique solution that is analytic in $u$ and defined in a neighbourhood of zero of the corresponding function space. The result is used for the construction of moment functions and the characteristic functional of a statistical solution of the original parabolic system. All the results of this article are valid also for the Navier–Stokes system.