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

Author

Listed:
  • 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
    Department of Mechanics and Mathematics, Al-Farabi Kaznu National University, Almaty 050040, Kazakhstan)

  • Madi Yergaliyev

    (Department of Differential Equations, Institute of Mathematics and Mathematical Modeling, Almaty 050010, Kazakhstan
    Department of Mechanics and Mathematics, Al-Farabi Kaznu National University, Almaty 050040, Kazakhstan)

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.

Suggested Citation

  • Muvasharkhan Jenaliyev & Akerke Serik & Madi Yergaliyev, 2024. "Navier–Stokes Equation in a Cone with Cross-Sections in the Form of 3D Spheres, Depending on Time, and the Corresponding Basis," Mathematics, MDPI, vol. 12(19), pages 1-17, October.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:19:p:3137-:d:1493520
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