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

Abstract

International audienceThe incompressible Navier-Stokes equations are considered in the two-dimensional strip R × [0, L], with periodic boundary conditions and no exterior forcing. If the initial velocity is bounded, it is shown that the solution remains uniformly bounded for all time, and that the vorticity distribution converges to zero as t → ∞. This implies, after a transient period, the emergence of a laminar regime in which the solution rapidly converges to a shear flow described by the one-dimensional heat equation in an appropriate Galilean frame. The approach is constructive and provides explicit estimates on the size of the solution and the lifetime of the turbulent period in terms of the initial Reynolds number

Infinite Energy Solutions for Dissipative Euler Equations in $${\mathbb{R}^2}$$ R 2

“…where C depends on u 0 , but is independent of t. Actually, we do not know whether or not the u(t) L 2 b norm can grow as t → ∞. However, it has been recently established in [12] that in the case of damped Navier-Stokes equation in an infinite cylinder (with the periodicity assumption with respect to one variable, say, x 1 ), the corresponding solution remains bounded as t → ∞. We also mention that estimate (4.29) has been recently obtained in [13] based on a slightly different representation of the non-linearity and pressure term in Navier-Stokes equation which allows to avoid the usage of rather delicate interpolation inequality (4.26).…”

Local Energy Weak Solutions for the Navier–Stokes Equations in the Half-Space

“…It is interesting in its own right since one can study nontrivial dynamics generated by the solutions themselves and not driven by a source term. Let us just mention the latest works of Abe and Giga [3,4,1,2] about Stokes and Navier-Stokes equations in L ∞ , of Gallay and Slijepčević [12] about boundedness for 2D Navier-Stokes equations and of Maremonti and Shimizu [27], Kwon and Tsai [22] about global weak solutions with initial data non decaying at space infinity.…”

On the large time $L^{\infty}$-estimates of the Stokes semigroup in two-dimensional exterior domains

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