Some remarks on the radius of spatial analyticity for the Euler equations

Abstract

We consider the Euler equations on T d with analytic data and prove lower bounds for the radius of spatial analyticity ε(t) of the solution using a new method based on inductive estimates in standard Sobolev spaces. Our results are consistent with similar previous results proved by Kukavica and Vicol, but give a more precise dependence of ε(t) on the radius of analyticity of the initial datum.

On the dissipative solutions for the inviscid Boussinesq equations

“…Remark 2.1. In the case θ = 0, Theorem 2.1 recovers the result of Kukavica and Vicol [16] and Cappiello and Nicola [5] for the incompressible Euler equation.…”

“…where the restriction |β|+|δ| < |α|+|γ| is due to the fact that u•∇∂ α+γ u, ∂ α+γ u = 0 because u is divergence free. By [5], we have…”

“…(4.12) For R 33 , in a similar way we have (4.15) We can take C 0 ≥ C in (4.15), so that Grownwall inequality [Lemma 2.1 in [5]] gives…”

“…Combining the estimates I 1 and I 2 , we have from (4.3) (4.15) We can take C 0 ≥ C in (4.15), so that Grownwall inequality [Lemma 2.1 in [5]] gives E N [(u, θ)(t)] ≤ exp C 0 (N − 1)…”

“…The approach used in [16,17,18,9] relies on the energy method in infinite order Gevrey-Sobolev spaces. Recently, Cappiello and Nicola [5] developed a new inductive method to simplify the proof of [16,17]. In this paper, we shall apply this inductive method to study the analyticity of smooth solution for the inviscid Boussinesq equations.…”