Evaluation of solutions of one class of finite-dimensional nonlinear equations
DOI:
https://doi.org/10.70474/72n9nt14Keywords:
Differential operator, nonlinear equation, existence of a solution, uniqueness of a solution, a priori estimation of a solutionAbstract
In this article, we obtain a theorem on a priori estimates for solutions of nonlinear equations in a finite-dimensional space. This theorem is proved under certain conditions which are borrowed from the conditions that are satisfied by finite-dimensional approximations of one class of nonlinear initial-boundary-value problems. The main result establishes sufficient conditions for the existence of a solution to A(u)=f, where A is a nonlinear operator. An example is given to illustrate the applicability of the main result to nonlinear analysis and mathematical physics
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References
Fefferman C. Existence and smoothness of the Navier--Stokes equation, Clay Mathematics Institute, Cambridge, MA, 2000, 1–5, http://claymath.org/millennium/Navier-Stokes_Equations/.
Otelbaev M. Existence of a strong solution to the Navier-Stokes equation, Mathematical Journal (Almaty), 2013, 13, 4, 5–104, http://www.math.kz/images/journal/2013-4/Otelbaev_ N-S_ 21_ 12_ 2013.pdf.
Ladyzhenskaya O.A. Solution “in general” of the Navier-Stokes boundary value problem in the case of two spatial variables, Dokl. Akad. Nauk SSSR, 1958, 123, 3, 427–429.
Ladyzhenskaya O.A. The sixth problem of the millennium: Navier-Stokes equations, existence and smoothness, Uspekhi Mat. Nauk, 2003, 58, 2, 45–78. DOI: https://doi.org/10.4213/rm610
Hopf E. Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 1951, 4, 213–231. DOI: https://doi.org/10.1002/mana.3210040121
Otelbaev M. Examples of not strongly solvable in the large equations of the Navier-Stokes type, Math. Notes, 2011, 89, 5, 771–779.
Otelbaev M. Durmagambetov A.A. Seitkulov E.N. Conditions for the existence of a strong solution in the large of one class of nonlinear evolution equations in Hilbert space. II, Siberian Math. J., 2008, 49, 4, 855–864.
Otelbaev M. Conditions for the existence of a strong solution in the large of one class of nonlinear evolution equations in Hilbert space, Proc. Steklov Inst. Math., 2008, 260, 1, 202–212. DOI: https://doi.org/10.1134/S0081543808010148
Otelbaev M. Zhapsarbaeva L.K. Continuous dependence of the solution of a parabolic equation in a Hilbert space on parameters and on initial data, Differ. Equ., 2009, 45, 6, 818–849.
Lions J.-L. Some methods for solving nonlinear boundary value problems, Mir, Moscow, 1972, 586 pp.
Saks R.S. Cauchy problem for the Navier-Stokes equations, Fourier method, Ufa Math. J., 2011, 3, 1, 53–79.
Pokhozhaev S.I. Smooth solutions of the Navier-Stokes equations, Mat. Sb., 2014, 205, 2, 131–144. DOI: https://doi.org/10.4213/sm8226
Koshanov B.D. Otelbaev M.O. Correct Contractions stationary Navier-Stokes equations and boundary conditions for the setting pressure, AIP Conf. Proc., 2016, 1759, http://dx.doi.org/10.1063/1.4959619. DOI: https://doi.org/10.1063/1.4959619
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