On the wellposedness of the KdV equation on the space of pseudomeasures

Abstract

In this paper we prove a wellposedness result of the KdV equation on the space of periodic pseudo-measures, also referred to as the Fourier Lebesgue space Fℓ ∞ (T, R), where Fℓ ∞ (T, R) is endowed with the weak* topology. Actually, it holds on any weighted Fourier Lebesgue space Fℓ s,∞ (T, R) with −1/2 < s 0 and improves on a wellposedness result of Bourgain for small Borel measures as initial data. A key ingredient of the proof is a characterization for a distribution q in the Sobolev space H −1 (T, R) to be in Fℓ ∞ (T, R) in terms of asymptotic behavior of spectral quantities of the Hill operator −∂ 2 x +q. In addition, wellposedness results for the KdV equation on the Wiener algebra are proved.

Weak Sobolev almost periodic solutions for the 1d NLS

“…If we take for instance I j = j −p , for j ∈ S and I j = 0 for j ∈ S c , then i(ν•) : R → w p is not continuous w.r.t. the strong 16 topology, see also [KM18]. On the other hand, for all x ∈ T, the map t → j∈S j −p e i(jx−νj t) =: v(t, x)…”

“…[Bou93] and [Bou94]). In the context of integrable PDEs there are various results on weak almost-periodic solutions, we mention [KM18] for the KdV, [GH17] for the Szego equation and [GKT] for the Benjamon-Ono. Here we work close to the elliptic fixed point u = 0, we consider (NLS V ) as a small perturbation of the (integrable) linear Schrödinger equation and prove the persistence of almost-periodic solutions.…”

Sharp well-posedness results of the Benjamin–Ono equation in $H^s (\mathbb{T}, \mathbb{R})$ and qualitative properties of its solutions

“…A detailed analysis of the frequencies of these equations allowed to prove in addition to the well-posedness results qualitative properties of solutions of these equations, among them properties corresponding to the ones stated in Theorem 1.7 -see e.g. [19]- [22]. Very recently, sharp global well-posedness results for the cubic NLS, the mKdV, the KdV, and the KdV2 equations on the real line were obtained in [17], [23], and [9], respectively.…”

On the Integrability of theBenjamin‐OnoEquation on the Torus

“…Related work: By similar methods, results of the type stated in Theorem 6 have been obtained for other integrable PDEs such as the KdV equation, the KdV2 equation, and the mKdV equation -see e.g. [7], [8], [5], [6].…”