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

This article is cited in 11 scientific papers (total in 11 papers)

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
Received: 02.10.1973

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  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath
    3. M. I. Vishik, A. V. Fursikov, “The Cauchy problem for nonlinear Schrödinger-type equations”, Math. USSR-Sb., 25:3 (1975), 429–440  mathnet  crossref  mathscinet  zmath
    4. A. V. Fursikov, “Formuly dlya nekotorykh funktsionalov ot gladkikh reshenii odnogo klassa sistem kvazilineinykh uravnenii”, UMN, 31:1(187) (1976), 265–266  mathnet  mathscinet  zmath
    5. A. V. Babin, “Fractional powers of a nonlinear analytic differential operator”, Math. USSR-Sb., 37:1 (1980), 9–38  mathnet  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  mathscinet  zmath
    7. M. I. Vishik, A. I. Komech, A. V. Fursikov, “Some mathematical problems of statistical hydromechanics”, Russian Math. Surveys, 34:5 (1979), 149–234  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath
    9. S. B. Kuksin, “Diffeomorphisms of function spaces corresponding to quasilinear parabolic equations”, Math. USSR-Sb., 45:3 (1983), 359–378  mathnet  crossref  mathscinet  zmath
    10. D. A. Khrychev, “On approximate solution of systems of moment equations”, Math. USSR-Sb., 54:1 (1986), 81–98  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath
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