Geodesic
Analytic first integrals of non-linear parabolic systems of differential equations in the sense of Petrovskii, and applications
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 29 (1974) no. 2, pp. 124-157

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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. 
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     author = {M. I. Vishik and A. V. Fursikov},
     title = {Analytic first integrals of non-linear parabolic systems of differential equations in the sense of {Petrovskii,} and applications},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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M. I. Vishik; A. V. Fursikov. Analytic first integrals of non-linear parabolic systems of differential equations in the sense of Petrovskii, and applications. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 29 (1974) no. 2, pp. 124-157. http://geodesic.mathdoc.fr/item/RM_1974_29_2_a9/