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 Uspekhi Mat. Nauk, 1974, Volume 29, Issue 2(176), Pages 123–153 (Mi umn4359)

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

M. I. Vishik, A. V. Fursikov

Abstract: 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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English version:
Russian Mathematical Surveys, 1974, 29:2, 124–157

Bibliographic databases:

UDC: 517.9
MSC: 35K55, 35K40, 35Q30, 76D06, 60H30, 35Bxx

Citation: M. I. Vishik, A. V. Fursikov, “Analytic first integrals of non-linear parabolic systems of differential equations in the sense of Petrovskii, and applications”, Uspekhi Mat. Nauk, 29:2(176) (1974), 123–153; Russian Math. Surveys, 29:2 (1974), 124–157

Citation in format AMSBIB
\Bibitem{VisFur74} \by M.~I.~Vishik, A.~V.~Fursikov \paper Analytic first integrals of non-linear parabolic systems of differential equations in the sense of Petrovskii, and applications \jour Uspekhi Mat. Nauk \yr 1974 \vol 29 \issue 2(176) \pages 123--153 \mathnet{http://mi.mathnet.ru/umn4359} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=415075} \zmath{https://zbmath.org/?q=an:0294.35043} \transl \jour Russian Math. Surveys \yr 1974 \vol 29 \issue 2 \pages 124--157 \crossref{https://doi.org/10.1070/RM1974v029n02ABEH003844} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. M. I. Vishik, A. V. Fursikov, “Some questions in the theory of nonlinear elliptic and parabolic equations”, Math. USSR-Sb., 23:2 (1974), 287–318
2. M. I. Vishik, A. V. Fursikov, “Asymptotic expansion of moment functions of solutions of nonlinear parabolic equations”, Math. USSR-Sb., 24:4 (1974), 575–591
3. M. I. Vishik, A. V. Fursikov, “The Cauchy problem for nonlinear Schrödinger-type equations”, Math. USSR-Sb., 25:3 (1975), 429–440
4. A. V. Fursikov, “Formuly dlya nekotorykh funktsionalov ot gladkikh reshenii odnogo klassa sistem kvazilineinykh uravnenii”, UMN, 31:1(187) (1976), 265–266
5. A. V. Babin, “Fractional powers of a nonlinear analytic differential operator”, Math. USSR-Sb., 37:1 (1980), 9–38
6. L. A. Belousov, “The asymptotic behaviour for large $t$ of the Fourier coefficients of solutions of a system of Navier–Stokes equations”, Russian Math. Surveys, 34:4 (1979), 155–156
7. M. I. Vishik, A. I. Komech, A. V. Fursikov, “Some mathematical problems of statistical hydromechanics”, Russian Math. Surveys, 34:5 (1979), 149–234
8. A. V. Fursikov, “Control problems and theorems concerning the unique solvability of a mixed boundary value problem for the three-dimensional Navier–Stokes and Euler equations”, Math. USSR-Sb., 43:2 (1982), 251–273
9. S. B. Kuksin, “Diffeomorphisms of function spaces corresponding to quasilinear parabolic equations”, Math. USSR-Sb., 45:3 (1983), 359–378
10. D. A. Khrychev, “On approximate solution of systems of moment equations”, Math. USSR-Sb., 54:1 (1986), 81–98
11. A. V. Fursikov, “On uniqueness of the solution of the chain of moment equations corresponding to the three-dimensional Navier–Stokes system”, Math. USSR-Sb., 62:2 (1989), 465–490
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