We consider initial-value problems for a system of motion equations of a viscous incompressible fluid in Lagrangian variables with n = 2 or n = 3. We show that the fluid motion is independent of pressure. In the absence of external forces, the pressure is constant and the fluid is in free motion. This motion is purely turbulent and is described by quasi-linear equations of parabolic type. We prove existence and uniqueness of the classical solution of the initial-value problem in a bounded region and in the entire space. Necessary conditions of solvability are given. Steady-state equations of fluid motion are derived. Applied problems involving fluid flow in a pipe, onset of turbulence, and existence of Taylor vortices in a solid torus are solved.
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L. D. Landau and E. M. Lifshitz, Fluid Mechanics. Course of Theoretical Physics, vol. 6, Pergamon Press, Oxford (1959).
L. Caffarelli, R. Kohn, and L. Nirenberg, “Partial regularity of suitable weak solutions of the Navier–Stokes equations,” Comm. Pure Appl. Math., 35, 771-837 (1982).
H. Amann, “On the strong solvability of the Navier–Stokes equations,” J. Math. Fluid Mech., 2, 16–98 (2000).
H. Koch and D. Tataru, “Well-posedness for the Navier–Stokes equations,” Adv. Math., 157(1), 22–35 (2001).
I. Gallagher, “Critical function spaces for the well-posedness of the Navier–Stokes initial value problem,” in: Y. Giga and A. Novotny (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham (2016); http://doi.org/https://doi.org/10.1007/978-3-319-10151-4 12-1.
A. Argenziano, M. Cannone, and M. Sammartino, Navier–Stokes Equations in the Half Space with Non Compatible Data, arXiv preprint, arXiv:2202.09415 (2022).
M. Cannone, “Harmonic analysis tools for solving the incompressible Navier–Stoker equations,” in: S. J. Friedlander and S. Serre (eds.), Handbook of Mathematical Fluid Dynamics, vol. III, 161–235, Elsevier (2004).
O. A. Ladyzhenskaya, Theory of Viscous Incompressible Flow, Gordon and Breah, New York (1963).
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI (1967).
S. A. Nazarov and K. Pileckas, “On steady Stokes and Navier–Stokes problems with zero velocity at infinity in a three-dimensional exterior domain,” J. Math. Kyoto Univ., 40, 475–492 (2000).
V. Sverak, “On Landau’s solutions of the Navier–Stoker equations,” J. Math. Sciences, 179(1), 208–228 (2011).
C. Bjorland, L. Brandolese, D. Iftimie, and M. Schondek, “Lp-solutions of the steady-state Navier–Stokes equations with rough external forces,” Comm. Partial Diff. Equations, 36, 216–246 (2011).
J. Guillod, Steady Solutions of the Navier–Stokes Equations in the Plane, arXiv:1511.03938v1 [math.AP] (12 Nov. 2015).
S. Smale, “Mathematical problems for the next century,” Math. Intelligencer, 20(2), 7–15 (1998).
C. L. Fefferman, Existence and Uniqueness of the Navier–Stokes Equation (2002); http://www.claymth.org/millenium/Navier-StokesEquations.
A. Friedman, Differential Equations of Parabolic Type, Dover, Mineola, NY (1964).
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1972).
S. Yu. Dobrokhotov and A. I. Shafarevich, “On the behavior at infinity of the velocity field of an incompressible fluid,” Izv. RAN MZhG, 4, 38–42 (1996).
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Translated from Problemy Dinamicheskogo Upravleniya, Issue 71, 2022, pp. 4–23.
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Baev, A.V. Solving the Navier-Stokes Equation for a Viscous Incompressible Fluid in an n-Dimensional Bounded Region and in the Entire Space ℝn. Comput Math Model 33, 255–272 (2022). https://doi.org/10.1007/s10598-023-09570-9
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DOI: https://doi.org/10.1007/s10598-023-09570-9