Energy Bounds for the Two-Dimensional Navier-Stokes Equations in an Infinite Cylinder

Abstract

International audienceWe consider the incompressible Navier-Stokes equations in the cylinder R × T, with no exterior forcing, and we investigate the long-time behavior of solutions arising from merely bounded initial data. Although we do not prove that such solutions stay uniformly bounded for all times, we show that they converge in an appropriate sense to the family of spatially homogeneous equilibria as t → ∞. Convergence is uniform on compact subdomains, and holds for all times except on a sparse subset of the positive real axis. We also improve the known upper bound on the L ∞ norm of the solutions, although our results in this direction are not optimal. Our approach is based on a detailed study of the local energy dissipation in the system, in the spirit of a recent work devoted to a class of dissipative partial differential equations with a formal gradient structure [5]. Keywords: Navier-Stokes equations, localized energy estimates, extended dissipative system. MSC(2010): 35Q30, 35B40, 76D0

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

“…If u ∈ BUC(Ω L ) is divergence-free and if the associated vorticity ω is bounded, one can show that the elliptic equation −Δp = ρ div(u · ∇)u has a bounded solution p : Ω L → R such that p(x 1 , 0) = p(x 1 , L) for all x 1 ∈ R. Moreover, there exists C > 0 such that [13,Lemma 2.3] and Sect. 2.4 below.…”

“…[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…”

“…The specific situation where the flow is periodic in one space direction was considered by Afendikov and Mielke [1], with the motivation of understanding the transition to instability in Kolmogorov flows. In the particular case where no exterior forcing is applied, the results of [1] give an upper bound on u(·, t) L ∞ which grows linearly in time, and can be improved with little extra effort to provide estimate (1.4), see [13]. A further progress was made in [13], where the authors proved that u(·, t) L ∞ cannot grow faster than t 1/6 as t → ∞.…”

“…In the particular case where no exterior forcing is applied, the results of [1] give an upper bound on u(·, t) L ∞ which grows linearly in time, and can be improved with little extra effort to provide estimate (1.4), see [13]. A further progress was made in [13], where the authors proved that u(·, t) L ∞ cannot grow faster than t 1/6 as t → ∞. Moreover, several results were obtained in [13,Theorem 1.3] which strongly suggest that solutions of (1.1) in Ω L should stay uniformly bounded for all time.…”

“…A further progress was made in [13], where the authors proved that u(·, t) L ∞ cannot grow faster than t 1/6 as t → ∞. Moreover, several results were obtained in [13,Theorem 1.3] which strongly suggest that solutions of (1.1) in Ω L should stay uniformly bounded for all time. For instance, for all T > 0, we have the following estimate…”