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...ction remained unanswered up to now, because the important conceptual and technical difficulties associated with the nonlinear equation have severely hampered the results obtained in this setting: in =-=[3, 8]-=- only the existence of some damped solutions was proven. We fill this gap in a recent work [16], whose main result we shall now describe. 2. Main result If f is a function defined on Td × Rd , we note...

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...rimental efforts, no rigorous justification of the Landau damping has been given in a nonlinear dynamical sense. In the past decade, there has been renewed interest [26] [37] [28] [18] [19] [50] [13] =-=[25]-=- [38] [47] as well as controversy about the Landau damping. In [13] [25], it was shown that there exist certain analytical perturbations for which electric fields decay exponentially in the nonlinear ...

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... was only bridged recently by the ground-breaking work of Mouhot and Villani, who showed that the phase-mixing indeed persists in the nonlinear Vlasov equations for small perturbations [68] (see also =-=[18, 44]-=-). The algebraic decay of the velocity field for solutions to (1.3) predicted by Orr can be most readily understood as a consequence of vorticity mixing driven by the shear flow, and hence can be cons...

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...ma physics, which is the origin of the term inviscid damping [9, 48, 11, 3]. The nonlinear theory of Landau damping was greatly advanced only recently by the work of Mouhot and Villani [41] (see also =-=[12, 26]-=-), who showed that the Vlasov equations undergo Landau damping similar to that predicted on the linear level by Landau in 1946 [31]. The mathematical relationship between our work and [41] is discusse...

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...for a large class of Vlasov equations that contains the Vlasov-Poisson system by Mouhot and Villani [13] (see also the recent simplified proof [5]). Some earlier partial results were obtained in [6], =-=[9]-=-. The related problem of the stability of the Couette flow in the two-dimensional Euler equation has been also studied recently [4]. The question left open in these papers is the possibility of nonlin...

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...t . Re1/3. See [9] and the references therein for an in-depth discussion of the phenomenon. The term ‘inviscid damping’ appeared after it was noticed to be the hydrodynamic analogue of Landau damping =-=[39, 49, 18, 32, 45, 10]-=- in the kinetic description of plasmas which are sufficiently collisionless (the analogue of inviscid) to be well-described by the Vlasov equations. See e.g. [14, 50, 16, 3, 45, 4, 9] and the referenc...

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...r) equation poses important conceptual and technical problems. As a consequence, up to now the only existing results were proving existence of some damped solutions with prescribed behavior as t → ±∞ =-=[2, 5]-=-. We fill this gap in a recent work [9], whose main result we shall now describe. 2. Main result If f is a function defined on Td × Rd, we note, for any k ∈ Zd and η ∈ Rd, f̂(k, v) = ∫ Td f(x, v) e−2i...