An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equations. (English) Zbl 0961.35112

Mattingly, J. C.; Sinai, Ya. G.

Commun. Contemp. Math. 1, No. 4, 497-516 (1999).

In this interesting paper, elementary proofs are given for well-known results on existence, uniqueness and smoothness (analyticity) of solutions of the time dependent, two-dimensional nonlinear Navier-Stokes equations under periodic boundary conditions.
The vorticity function \(\omega\) is expanded into a Fourier series, and (a suitable Galerkin approximation of) the corresponding vorticity equation is treated as infinite-dimensional dynamical system for the Fourier coefficients of \(\omega\). Given any regular initial datum, a suitable set in the phase space is constructed, such that it contains this datum and that all points of this set have the desired quantitative properties. At its boundary, the vector field of the dynamical system points inward and hence, the solution will never escape this set.

Reviewer: Hans-Christoph Grunau (Bayreuth)

MSC:

35Q30	Navier-Stokes equations
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids
37N10	Dynamical systems in fluid mechanics, oceanography and meteorology
76D05	Navier-Stokes equations for incompressible viscous fluids

Keywords:

existence; uniqueness; smoothness; Navier-Stokes equations; periodic boundary conditions; vorticity equation

References:

[1]	DOI: 10.1063/1.868526 · Zbl 1023.76513 · doi:10.1063/1.868526
[2]	DOI: 10.1016/0022-1236(89)90015-3 · Zbl 0702.35203 · doi:10.1016/0022-1236(89)90015-3
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[4]	DOI: 10.1007/BF02547354 · JFM 60.0726.05 · doi:10.1007/BF02547354