Global well-posedness for the derivative nonlinear Schrödinger equation in $L^2(\mathbb{R})$

Abstract

We prove that the derivative nonlinear Schrödinger equation in one space dimension is globally well-posed on the line in L 2 (R), which is the scaling-critical space for this equation.

Norm inflation for the derivative nonlinear Schrödinger equation

“…More precisely, by using a ternary-quinary tree expansion of the Duhamel formula we prove norm inflation in Sobolev spaces below the (scaling) critical regularity for the gauged DNLS. This ill-posedness result is sharp since DNLS is known to be globally well-posed in L 2 (R) [13]. The main novelty of our approach is to control the derivative loss from the cubic nonlinearity by the quintic nonlinearity with carefully chosen initial data.…”

“…Regarding well-posedness, Conjecture 1.1 has seen some recent progress culminating in the breakthrough work [13] which proves global well-posedness of (1.1) in L 2 (R). We also mention the following works on the well-posedness theory for (1.1) [29,30,17,28,20,14,13].…”

“…Remark 1.2. Since (1.1) is globally well-posed in L 2 (R) [13], norm inflation cannot occur in H s (R) for s ≥ 0 in view of the continuity of the Gauge transform (1.2) on smooth functions (i.e. from C [0, T ]; L 2 (R) to itself, for any T > 0 see [16] for a proof in the periodic setting).…”

A remark on the well-posedness of the modified KdV equation in $L^2$

“…Subsequently, this method was extended in the line setting in [5,12] to incorporate dispersion through localsmoothing estimates, which led to impressive results regarding the well-posedness of the fifth-order KdV, and the cubic NLS and complex-valued mKdV equations on R (as discussed above), respectively. We also mention the very recent breakthroughs [23,13,14], using these methods and more, to the low-regularity and large data well-posedness of the derivative NLS.…”

Refined probabilistic local well-posedness for a cubic Schrödinger half-wave equation

“…Another model is the derivative Schrödinger equation on the line, which is L 2 -critical with respect to the scaling, but the flow map fails to be uniformly continuous in H s (R) when s < 1 2 [4,37]. Nevertheless, [30] recently proved from integrable techniques that the equation is globally well-posed in L 2 (R). Previously, integrable techniques were combined with concentration-compactness argument in [1] to prove global well-posedness in H 1 2 (R).…”