The paper considers precisely solvable problems for the Euler and Navier–Stokes equations for the case of an incompressible fluid. The analysis of the causes of ill-posed problems in a set of harmonic functions is carried out. Exact solutions based on classical eigenvalue problems (linear and nonlinear) are given. The layered structure of the proposed hydrodynamic flows is distinguished in the case of stationary solutions of the Euler system and nonstationary solutions of the Navier–Stokes system in an infinite cylindrical domain.
Keywords: incompressible fluid equations, solutions of Euler equations, Navier–Stokes system
Received December 16, 2025
Revised January 22, 2026
Accepted January 26, 2026
Funding Agency: This work was supported by a state assignment from the Federal State Institution Research Center "Kurchatov Institute NIISI – Fundamental scientific research project GP 47 on topic № 0580-2021-0007 “Development of methods for mathematical modeling of distributed systems and corresponding computational methods”.
Valerii Alekseevich Galkin, Dr. Phys.-Math. Sci., Prof., Surgut branch of the Federal State Institution Research Center “Kurchatov Institute” – NIISI, HMAO, Surgut, 628406, Russia, e-mail: val-gal@yandex.ru
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Cite this article as: V.A. Galkin. On some solutions of the Euler and Navier–Stokes Equations for an incompressible fluid Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 1, pp. 77–87.
