Uniform estimates for solutions of a class of nonlinear equations in a finite-dimensional space
DOI:
https://doi.org/10.26577/JMMCS2023v120i4a2Keywords:
finite-dimensional Hilbert space, nonlinear equations, initial-boundary value problem, weak solution, strong solution, a priori estimates of the solutionAbstract
The need to study boundary value problems for elliptic parabolic equations is dictated by numerous practical applications in the theoretical study of the processes of hydrodynamics, electrostatics, mechanics, heat conduction, elasticity theory, quantum physics.
Let H (dimH ≥ 1) – a finite-dimensional real Hilbert space with inner product ⟨·,·⟩ and norm ∥ · ∥. We will study the equation of the following form
u + L (u) = g ∈ H,
where L(·) is a non-linear continuous transformation, g is an element of the space H, u is the required solution of the problem from H.
In this paper, we obtain two theorems on a priori estimates for solutions of nonlinear equations in a finite-dimensional Hilbert space. The work consists of four items.
The conditions of the theorems are such that they can be used in the study of a certain class of initial-boundary value problems to obtain strong a priori estimates. This is the meaning of these theorems.
References
Fefferman Ch., Existence and smoothness of the Navier-Stokes equation, Clay Mathematics Institute. (2000), 1-5.
Otelbaev M.O., Existence of a strong solution of the Navier-Stokes equation, Mathematical journal, 13(50), (2013), 5-104.
Ladyzhenskaya O.A., Solution ”in the whole” of the Navier-Stokes boundary value problem in the case of two space variables, Reports of the Academy of Sciences of the USSR, 123 (3), (1958), 427-429.
Ladyzhenskaya O.A., The sixth problem of the millennium: the Navier-Stokes equations, existence and smoothness, Advances in Mathematical Sciences, 58(2), (2003), 45-78.
Hopf E., Uber die Anfangswertaufgabe fur die Hydrodinamischen Grundgleichungen., Math. Nachr.(4), (1951), 213-231.
Otelbaev M.O., Examples of equations of the Navier-Stokes type that are not strongly solvable in general, Math notes, 89(5), (2011), 771-779.
Otelbaev M.O., Durmagambetov A.A., Seytkulov E.N., Conditions for the existence of a strong solution in the whole of one class of non-linear evolution equations in a Hilbert space, Siberian Mathematical Journal, 49(4), (2008), 855-864.
Otelbaev M.O., Zhapsarbayeva L.K., Continuous dependence of the solution of a parabolic equation in Hilbert space on parameters and initial data, Differential Equations, 45(6), (2009), 818-849.
Lyons J.-L., Some methods for solving nonlinear boundary value problems, Moscow: Mir (1972).
Saks R.S., Cauchy problem for the Navier-Stokes equations, Fourier method, Ufa Mathematical Journal, 3(1), (2011), 53-79.
Pokhozhayev S.I., Smooth solutions of the Navier-Stokes equations, Mathematical collection, 205(2), (2014), 131-144.
Koshanov B.D, Otelbaev M.O., Correct Contractions stationary Navier-Stokes equations and boundary conditions for the setting pressure, AIP Conference Proceedings, 1759. http://dx.doi.org/10.1063/1.4959619.
Kozhanov A.I., Koshanov B.D., Sultangazieva Zh.B., New boundary value problems for fourth-order quasi-hyperbolic equations,Siberian Electronic Mathematical Reports, 16 (2019), 1383-1409. http://dx.doi.org/10.33048/semi.2019.16.098.
Kozhanov A.I., Koshanov B.D., Sultangazieva Zh.B., Emir Kady oglu A.N., Smatova G.D., The spectral problem for nonclassical differential equations of the sixth order. Bulletin of the Karaganda University, series Mathematics, 97(1), (2020), 79-86. https://doi.org/10.31489/2020M1/79-86
Koshanov B.D., Green’s functions and correct restrictions of the polyharmonic operator. Journal of Mathematics, Mechanics and Computer Science, 109(1) (2021), 34-54. https://doi.org/10.26577/JMMCS.2021.v109.i1.03
Koshanov B.D., Koshanova M.D. On the representation of the Green function of the Dirichlet Problem and their properties for the polyharmonic equations, AIP Conference Proceedings, 1676, (2015), http://dx.doi.org/10.1063/1.4930446
Koshanov B.D., Soldatov A.P., Boundary value problem with normal derivatives for a higher order elliptic equation on the plane, Differential Equations, 52:12 (2016), 1594-1609. https://doi.org/10.1134/S0012266116120077.
Koshanov B.D., Kuntuarova A.D., Equivalence of the Fredholm solvability condition for the Neumann problem to the complementarity condition, Journal of Mathematics, Mechanics and Computer Science, 111:3 (2021), 39-51. https://doi.org/10.26577/JMMCS.2021.v111.i3.04
Koshanov B.D., Soldatov A.P., Generalized Neumann problem for an elliptic equation, Complex Variables and Elliptic Equations, (2022), https://doi.org/10.1080/17476933.2021.1958797
Kanguzhin B.E., Koshanov B.D., Criteria for the uniqueness of the solution of a time-nonlocal problem for a high-order differential-operator equation l(·) − A with a shifted wave operator A, Sibirian Mathematical Journal, 63:6 (2022), 1083- 1090. https://doi.org/10.1134/S0037446622060088
Kanguzhin B.E., Koshanov B.D., Uniqueness Criteria for Solving a Time Nonlocal Problem for a High-Order Differential Operator Equation l(·) − A with a Wave Operator with Displacement, Simmetry, 14:6, 1239 (2022), https://www.mdpi.com/2073-8994/14/6/1239
Kanguzhin B.E., Koshanov B.D., Solution Uniqueness Criteria in a Time Nonlocal Problem for the Operator Differential Equation l(·) − A with the Tricomi Operator A, Differential Equations, 59:1 (2023), 1-12. https://doi.org/10.1134/S0012266123010019
Koshanov B.D., Kakharman N., Segizbayeva R.U., Sultangaziyeva Zh.B., Two theorems on estimates for solutions of one class of nonlinear equations in a finite-dimensional space, Bulletin of the Karaganda University, series Mathematics, (2022), 70-84. https://mathematics-vestnik.ksu.kz/apart/2022-107-3/7.pdf
