On the spectral problem for three-dimesional bi-Laplacian in the unit sphere
Бiрлiк шардағы үшөлшемдi би-Лапласиан үшiн қойылған спектрлiк есеп туралы
О спектральной задаче для трехмерного би-Лапласиана в единичном шаре
Jenaliyev M.T. Serik A.M.
2024E.A. Buketov Karaganda University Publish house
Bulletin of the Karaganda University. Mathematics Series
2024#114Issue 286 - 104 pp.
In this work, we introduce a new concept of the stream function and derive the equation for the stream function in the three-dimensional case. To construct a basis in the space of solutions of the Navier-Stokes system, we solve an auxiliary spectral problem for the bi-Laplacian with Dirichlet conditions on the boundary. Then, using the formulas employed for introducing the stream function, we find a system of functions forming a basis in the space of solutions of the Navier-Stokes system. It is worth noting that this basis can be utilized for the approximate solution of direct and inverse problems for the Navier-Stokes system, both in its linearized and nonlinear forms. The main idea of this work can be summarized as follows: instead of changing the boundary conditions (which remain unchanged), we change the differential equations for the stream function with a spectral parameter. As a result, we obtain a spectral problem for the bi-Laplacian in the domain represented by a three-dimensional unit sphere, with Dirichlet conditions on the boundary of the domain. By solving this problem, we find a system of eigenfunctions forming a basis in the space of solutions to the Navier-Stokes equations. Importantly, the boundary conditions are preserved, and the continuity equation for the fluid is satisfied. It is also noteworthy that, for the three-dimensional case of the Navier-Stokes system, an analogue of the stream function was previously unknown.
bi-Laplacian , Navier-Stokes system , spectral problem , stream function
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Institute of Mathematics and Mathematical Modeling
Institute of Mathematics and Mathematical Modeling
Al-Farabi KazNU
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