Dynamical Properties of Spatially Non-Decaying 2D Navier?Stokes Flows with Kolmogorov Forcing in an Infinite Strip

Abstract

The main result of this paper is the global well-posedness of the Cauchy problem to the 2D Navier-Stokes system with the initial data u 0 ∈ BUC(O) and the external force, the fluid flow is supposed to be periodic in one of the spatial directions whereas in the unbounded direction only uniform boundedness is assumed. However, to obtain uniqueness we need to make assumptions which suppresses additional pressure gradients. For this aim Riesz operators on L ∞ (O) are used to define p(t) ∈ BMO(O). For time-independent forces the solutions are shown to grow at most cubically in the time t.Mathematics Subject Classification (2000). 35Q30, 76D03, 76D05.

Spatially Nondecaying Solutions of the 2d Navier-Stokes Equation in a Strip

“…Indeed, assume that c = 0 and g is square integrable g ∈ [L 2 ( )] 2 . Then, instead of (6.25), we will have g L 2 θε,x 0 ( ) ≤ C g L 2 ( ) with the constant C independent of ε. Thus, instead of (8.17), we will have the better estimate…”

“…Indeed, if does not contain boundary, e.g. = ‫ޒ‬ 2 or = ‫ޓ‬ 1 × ‫ޒ‬ where ‫ޓ‬ 1 is a circle (like in the Kolmogorov problem), the maximum principle applied to (1.3) allows us to obtain global a priori estimates for the vorticity ω which, together with the accurate analysis of the explicit formulae for the Helmholtz projectors, allow us to obtain the global in time a priori estimates for the solution u(t) and, thus, to prove the global solvability of the Navier-Stokes equation in the uniformly local phase spaces (see [2] and [12]). Unfortunately, the a priori estimate for vorticity obtained from the maximum principle grows linearly in time, so all of the further estimates will also grow in time (to the best of our knowledge, for the case = ‫ޒ‬ 2 , it gives double exponential (∼e Ce Ct ) growth rate and polynomial (∼t 3 ) growth rate for = ‫ޓ‬ 1 × ‫.…”

Infinite energy solutions of the two-dimensional Navier–Stokes equations

“…Navier-Stokes equations Thierry Gallay (1) ABSTRACT. -These notes are based on a series of lectures delivered by the author at the University of Toulouse in February 2014.…”

“…Dans ce contexte, il n'est pas difficile de montrer que le problème est localement bien posé [18], et des estimations a priori sur le tourbillon impliquent que toutes les solutions sont globales et que leur croissance temporelle est au plus exponentielle [19,39]. En outre, comme l'a récemment montré S. Zelik, on peut utiliser des estimations d'énergie localisées pour obtenir un contrôle beaucoup plus précis sur le champ de vitesse dans l'espace (1) Univ. Grenoble Alpes, CNRS, Institut Fourier, 100 rue des Maths, 38610 Gières, France -Thierry.Gallay@univ-grenoble-alpes.fr…”

Uniform Boundedness and Long-Time Asymptotics for the Two-Dimensional Navier–Stokes Equations in an Infinite Cylinder

“…[1,13,14] For any initial data u 0 ∈ BUC(Ω L ) with div u 0 = 0, the Navier-Stokes equations (1.1) with the canonical choice of the pressure have a unique global mild solution u ∈ C 0 ([0,+∞), BUC(Ω L )) such that u(0) = u 0 . Moreover, there exists a constant C > 0, depending only on the initial Reynolds number R u , such that…”