Modern problems of mathematical physics
November 27, 2024, 14:00. A joint seminar on Modern Problems of Mathematical Physics: The Institute of mathematics named after V.I. Romanovsky AS RUz and the Institute of Applied Mathematics and Automation KBSC RAS.
Speaker: Muvasharhan Dzhenaliev, Doctor of Physics and Mathematics, Professor, Chief Researcher of the Institute of Mathematics and Mathematical Modeling, Kazakhstan.
Topic of the report: On the construction of fundamental systems in the space of solenoidal vector fields.
Abstract: In a number of works academician O.A. Ladyzhenskaya pointed out the importance of constructing a fundamental system in the solenoidal vectors field for such simplest domain as a cube,
sphere, etc.
Theoretically, such system existence does not need a prove, this is well known. The experts actively use this fact when proving existence theorems for 2-D and 3-D Navier-Stokes equations (in both linear and nonlinear cases) and for further analysis of the qualitative properties of the solution proven to exist. However, for the numerical solution to boundary value problems for the system of equations both Stox and Nautier-Stox there is a need to construct the above fundamental system.
Thus, the main goal of the presented report is to construct these systems in the space of solenoidal
vector fields for square and cubic domains in respect with spatial variables!
In the report, firstly, using a specially modified rotor operator, we introduce the concept of an analogue of the “current function” for the 3-D case. Previously, this concept was known only for the 2-D case.
Here we obtain a generalized spectral problem for a fourth-order differential operator, which is not
solvable in quadratures of tabulated functions.
Secondly, instead of the obtained operator, we introduce a new fourth-order differential operator (perturbed 2-D and 3-D biharmonic operators), and constructed a fundamental system of generalized eigenfunctions for it in the space of scalar current functions and the corresponding eigenvalues.
Thirdly, 2-D and 3-D cases should be considered separately. Applying the formulas by which we introduced the current function for 2-D case, to the built fundamental system in terms of current function, we get some 2-D vector-functions. We show that this system will be the major in the space of solenoidal vectors.
The 3D is slightly different from 2-D case. Here, using the construction of generalized eigenfunctions, we compose three-dimensional vector functions. The sum of them is fundamental in the direct product of three spaces for scalar “current functions”. Applying the formulas by which we introduced the current function for 3-D case, to the built fundamental system in terms of current function, we get some 3-D vector-functions.
Finally, we prove that the resulting system of 3-D vector-functions is fundamental in the space of solenoid vector-field.
Thus, we managed to construct a fundamental system in the solenoidal vectors field for square and cubic domains, the importance of which was pointed out by O.A. Ladyzhenskaya.
Join the Zoom conference:
https://us06web.zoom.us/j/88273567781?pwd=6TMH3UOjALVQmLsLlm6vTMPo5YpCYr.1
Conference ID: 882 7356 7781
Access code: 151325
Seminar leaders: Sh. A. Alimov, academician, A. V. Pskhu, professor, R. R. Ashurov, professor.
Seminar secretary: R. T. Zunnunov.