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

Abstract

These notes are based on a series of lectures delivered by the author at the University of Toulouse in February 2014. They are entirely devoted to the initial value problem and the long-time behavior of solutions for the two-dimensional incompressible Navier-Stokes equations, in the particular case where the domain occupied by the fluid is the whole plane R 2 and the velocity field is only assumed to be bounded. In this context, local well-posedness is not difficult to establish [17], and a priori estimates on the vorticity distribution imply that all solutions are global and grow at most exponentially in time [18,38]. Moreover, as was recently shown by S. Zelik, localized energy estimates can be used to obtain a much better control on the uniformly local energy norm of the velocity field [44]. The aim of these notes is to present, in an explanatory and self-contained way, a simplified and optimized version of Zelik's argument which, in combination with a new formulation of the Biot-Savart law for bounded vorticities, allows one to show that the L ∞ norm of the velocity field grows at most linearly in time. The results do not rely on the viscous dissipation, and remain therefore valid for the so-called "Serfati solutions" of the two-dimensional Euler equations [2]. Finally, a recent work by S. Slijepčević and the author shows that all solutions remain uniformly bounded in the viscous case if the velocity field and the pressure are periodic in one space direction [14,15].

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

“…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).…”

“…, see also the recent work [13] where the analogous linear growth estimate has been established for the infinite energy solutions of the Navier-Stokes equations.…”

Well-posedness of the 2D Euler equations when velocity grows at infinity

“…Hence, our bound on a(T ) is doubly exponential (it would be worse for unbounded velocities). It is shown in [9] (extending [19]) for bounded velocity, however, that u 1 L ∞ (0,T ;S 1 ) can be bounded linearly in time, which means that C 0 actually only increases singly exponentially in time.…”

Incompressible Euler Equations and the Effect of Changes at a Distance

“…Such a bound is never possible for a compactly supported function: this inability is the fundamental reason why we cannot use the "cleaner" localization that a compactly supported function would provide. This same issue shows up, for instance, in [8] (see Section 3.3), though it is addressed in a different manner.…”

“…If p = ∞ and d = 2, then Theorem 1 above addresses Serfati (bounded vorticity, bounded velocity) solutions to the Euler equations in the plane (see [15]), and supplies an alternate proof of uniqueness to those in [15,1,17]. Moreover, it is shown in [8] that for Serfati solutions,…”