Navier–Stokes Equation in a Cone with Cross-Sections in the Form of 3D Spheres, Depending on Time, and the Corresponding Basis

Muvasharkhan Jenaliyev, Akerke Serik and Madi Yergaliyev ()
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Muvasharkhan Jenaliyev: Department of Differential Equations, Institute of Mathematics and Mathematical Modeling, Almaty 050010, Kazakhstan
Akerke Serik: Department of Differential Equations, Institute of Mathematics and Mathematical Modeling, Almaty 050010, Kazakhstan
Madi Yergaliyev: Department of Differential Equations, Institute of Mathematics and Mathematical Modeling, Almaty 050010, Kazakhstan

Mathematics, 2024, vol. 12, issue 19, 1-17

Abstract: The work establishes the unique solvability of a boundary value problem for a 3D linearized system of Navier–Stokes equations in a degenerate domain represented by a cone. The domain degenerates at the vertex of the cone at the initial moment of time, and, as a consequence of this fact, there are no initial conditions in the problem under consideration. First, the unique solvability of the initial-boundary value problem for the 3D linearized Navier–Stokes equations system in a truncated cone is established. Then, the original problem for the cone is approximated by a countable family of initial-boundary value problems in domains represented by truncated cones, which are constructed in a specially chosen manner. In the limit, the truncated cones will tend toward the original cone. The Faedo–Galerkin method is used to prove the unique solvability of initial-boundary value problems in each of the truncated cones. By carrying out the passage to the limit, we obtain the main result regarding the solvability of the boundary value problem in a cone.

Keywords: Navier–Stokes; a priori estimates; Galerkin method; degenerate domain (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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