Boundary and inverse problems for Navier-Stokes equations of homogeneous, inhomogeneous fluids, thermal convection, and Kelvin-Voigt fluids

Relevance: The Navier–Stokes equations describing the flow of a viscous incompressible fluid have for many decades attracted the attention of scientists working on the solvability of partial differential equations and of specialists in numerical analysis because of their numerous applications. Despite this interest, the question of the existence and uniqueness of solutions of the non-stationary Navier–Stokes equations in the case of three spatial variables remains open. Although several thousand, or tens of thousands, of works have been devoted to the theoretical study of boundary value problems for the stationary and non-stationary Navier–Stokes equations, starting with the famous works [1–3] of the French mathematician J. Leray published in the 1930s, the theory still has many "blank spots". Researchers have still not been able to identify a function class in which global (i.e. on any time interval [0, T]) solvability and uniqueness of the solution of an initial-boundary value problem — for example, the Dirichlet problem on the boundary of the flow domain for the three-dimensional non-stationary Navier–Stokes equations — could be proven. This problem has proved so important, complex and intricate that it was included in the famous list of the seven most important problems of the 21st century (the Millennium Prize Problems). The project intends to prove global theorems on the existence and uniqueness of the solution of the initial-boundary value problem by first appropriately modifying the generalized Günther–Lichtenstein method of successive approximations and considering classical solutions of the problem. This requires a priori estimates sufficient to construct the Green operator using potential estimates and the corresponding S.L. Sobolev embedding theorems. The mathematical model of the problem contains a Euler-type equation for an incompressible fluid, for which, as is known, the global existence and uniqueness theorem has not yet been proven in the three-dimensional case. For the system considered here, a significant simplification and improvement of results compared with the Euler system is expected thanks to a modification of the Günther–Lichtenstein successive approximation methods that we have tested in the theory of other problems and in computational practice. This is due to the better smoothing properties of the control operator, which here is expressed through the Green operator of the Stokes system. Research on the topic is mainly theoretical and fundamental, and its scientific significance stems precisely from the deep fundamental level of the results. In addition, the scientific significance of the proposed research is due to the possible application of advanced modern results of the theory of differential operators to modeling technological processes and to the creation of new original methods of research and analysis. In recent years, Kazakhstan has seen a rapid growth in social demand for fundamental research, and this project responds to that demand. The national and international significance of the project lies in the fact that the objects of study — the Navier–Stokes equations for homogeneous and inhomogeneous fluids — are, on the one hand, important for mathematics itself and for mechanics, physics, geophysics, chemistry and other natural sciences, and, on the other hand, of substantial interest from a purely mathematical point of view. The results are therefore relevant and will be understood by researchers all over the world, who can use them for further research. This makes it possible to speak of the international significance of the project. The project is a direct continuation of the authors' earlier research under program AP05132041 (Theory and methods for solving direct and inverse problems for Navier–Stokes equations and partial differential equations, 2018–2020), funded by a grant of the Ministry of Education and Science of the Republic of Kazakhstan.

1. Antontsev S.N. Khompysh Kh. An inverse problem for generalized Kelvin–Voigt equation with p-Laplacian and damping term // Inverse Problems, 2021. 37(8), 085012 (Scopus percentile 86%, Web of Science Q1, SJR 1.005) https://doi.org/10.1088/1361-6420/ac1362
2. Aitzhanov S.E., Kusherbayeva U.R., Bekenayeva K.S. Solvability of pseudoparabolic equation with Caputo fractional derivative // Chaos, Solitons & Fractals, Volume 160, July 2022, 112-193. https://doi.org/10.1016/j.chaos.2022.112193 (Scopus percentile 99%, Web of Science Q1, SJR 1.647)
3. Aitzhanov S.E., Tileuberdi Zh., Sanat G. Solvability of an initial-boundary value problem for a nonlinear pseudoparabolic equation with degeneration // Bulletin of the Karaganda University Mathematics series. -2022. -№ 1(105). -P. 4-12. DOI 10.31489/2022M1/4-12
4. B.E. Kanguzhin, B.D. Koshanov. Uniqueness Criteria for Solving a Time Nonlocal Problem for a High-Order Differential Operator Equation l(.)-A with a Wave Operator with Displacement // Symmetry 2022, 14, 1239. DOI https://doi.org/10.3390/sym14061239 (Scopus percentile 93%, Web of Science Q1, SJR 0.544)