Large-data equicontinuity for the derivative NLS

Abstract

We consider the derivative NLS equation in one spatial dimension, which is known to be completely integrable. We prove that the orbits of L 2 bounded and equicontinuous sets of initial data remain bounded and equicontinuous, not only under this flow, but under the entire hierarchy. This allows us to remove the small-data restriction from prior conservation laws and global well-posedness results.

Global well-posedness for the generalized derivative nonlinear Schrödinger equation

“…Moreover, although H 1 2 regularity is necessary for uniform continuity of the solution map, it is believed [42] that complete integrability will help lower the global well-posedness regularity threshold, possibly all the way to the critical Sobolev space L 2 . Indeed, the well-posedness threshold has already been reduced to H 1 6 in [26], substantially improving all previous results. On the other hand, blowup for (DNLS) on non-standard domains (for example, the half-line with the Dirichlet boundary condition) is known to be possible [66,73].…”

“…Global well-posedness, however, is considerably harder, as the problem is L 2 critical. For this reason, Bahouri and Perelman (as well as Harrop-Griffiths, Killip, Ntekoume and Vişan [26,42] in their subsequent work) crucially rely on the complete integrability of (DNLS). In the case σ < 1, the main difficulties are reversed.…”

$H^s$ Bounds for the Derivative Nonlinear Schrödinger Equation

“…Klaus-Schippa [17] gave H s a priori estimates for 0 < s < 1 2 in the case of small mass, Killip-Ntekoume-Vis ¸an [14] improved the small mass assumption to 4π and furthermore proved a global wellposedness result in H s (R), 1 6 ≤ s < 1 2 , for initial data with mass less than 4π. Very recently, Harrop-Griffiths, Killip, and Vis ¸an [9] have removed the small mass assumption both from their H s a priori bounds, 0 < s < 1 2 , as well as from their global wellposedness result in H s (R) with 1 6 ≤ s < 1 2 . In this paper, we are concerned with the global-in-time boundedness of solutions to the DNLS equation in H s spaces.…”

“…The main idea is to build off of the H s bounds with 0 < s < 1 2 from [9] and to take advantage of the complete integrability of the equation. As in [2], [9], the present work relies heavily on the conservation of the transmission coefficient for the spectral problem associated to the DNLS equation.…”

“…The main idea is to build off of the H s bounds with 0 < s < 1 2 from [9] and to take advantage of the complete integrability of the equation. As in [2], [9], the present work relies heavily on the conservation of the transmission coefficient for the spectral problem associated to the DNLS equation. This property has already been used in many other works; of particular relevance to us are the papers of Gérard [5], Killip-Vis ¸an-Zhang [16], Killip-Vis ¸an [15], and Koch-Tataru [18], on the cubic NLS and KdV equations.…”