On the orthogonality of a system of solenoidal functions in a three-dimensional cube
DOI:
https://doi.org/10.70474/aggv2f03Keywords:
spectral problem, fourth-order differential operator, system of solenoidal functions, orthogonality propertyAbstract
Previously, we constructed a system of orthonormal functions (SOF) as a solution to a spectral problem for a fourth-order operator in a three-dimensional cube. Using a three-dimensional curl operator applied to SOF, we derived a system of solenoidal functions (SSF), which are crucial in the study of incompressible fluid dynamics and the theory of Navier-Stokes equations. However, the SSF obtained in this way did not possess the orthogonality property, which is often desirable in theoretical analysis and numerical applications. The main result of this work is the construction of a new system of solenoidal functions, based on the original SOF, which is shown to be almost orthogonal. This property makes the system suitable for use in spectral methods and other analytical approaches where near-orthogonality ensures better convergence and stability. The methodology developed in this study can be generalized to other types of boundary value problems involving higher-order differential operators and may contribute to the development of more efficient computational schemes in fluid mechanics.
Downloads
References
Ladyzhenskaya, O. A. On the construction of bases in spaces of solenoidal vector fields Journal of Mathematical Sciences 130 4 4827 2005 10.1007/s10958-005-0379-5 DOI: https://doi.org/10.1007/s10958-005-0379-5
Saks, R. S. Spectral problems for rotor and Stokes operators Journal of Mathematical Sciences 136 2 3794 2006 10.1007/s10958-006-0201-z
Saks, R. S. Cauchy Problem for Navier-Stokes Equations. Fourier Method Ufa Mathematical Journal 3 1 51 77 2011
Ladyzhenskaya, O. A. Mathematical theory of viscous incompressible flow Gordon and Breach Science Publisher 1987
Lions, J.-L. Quelques methodes de resolution des problemes aux limited non lineaires Dunod 1969
Temam, R. Navier-Stokes equations. Theory and numerical analysis North-Holland Publishing Company 1979
Jenaliyev, M.; Ramazanov, M.; Yergaliyev, M. On the numerical solution of one inverse problem for a linearized two-dimensional system of Navier-Stokes equations Opuscula Mathematica 42 5 709 725 2022 10.7494/OpMath.2022.42.5.709 DOI: https://doi.org/10.7494/OpMath.2022.42.5.709
Weinstein, A. Étude des spectres des equations aux dérivées partielles de la théorie des plaques élastiques Mémoires des Sciences Mathématiques 88 Gauthier-Villars 1937
Gould, S. H. Variational Methods for Eigenvalue Problems. An Introduction to the Weinstein Method of Intermediate Problems. 2nd edition Oxford University Press 1966
Henrot, A. Extremum problems for eigenvalues of elliptic operators Birkhauser Verlag 2006
Jenaliyev, M. T.; Serik, A. M. On the spectral problem for three-dimensional bi-Laplacian in the unit sphere Bulletin of the Karaganda University. Mathematics series 114 2 86 104 2024 10.31489/2024m2/86-104 DOI: https://doi.org/10.31489/2024m2/86-104
Jenaliyev, M.; Serik, A.; Yergaliyev, M. Navier–Stokes Equation in a Cone with Cross-Sections in the Form of 3D Spheres, Depending on Time, and the Corresponding Basis Mathematics 12 19 3137 2024 10.3390/math12193137 DOI: https://doi.org/10.3390/math12193137
Kabanikhin, S. I. Inverse and Ill-Posed Problems. Theory and Applications de Gruyter 2012
Prilepko, A. I.; Orlovsky, D. G.; Vasin, I. A. Methods for Solving Inverse Problems in Mathematical Physics Marcel Dekker 2000

Additional Files
Published
Issue
Section
License
Copyright (c) 2025 Kazakh Mathematical Journal

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
One can find the license terms "CC Attribution-NonCommercial-NoDerivatives 4.0" here.