Critical Well-Posedness for Derivative Nonlinear Schrödinger Equations

Definitions and mathematical statement

Endpoint question. Is the one-dimensional derivative nonlinear Schrödinger Cauchy problem\[i\partial_tu+\partial_x^2u=i\partial_x(|u|^2u),\qquad (t,x)\in(-T,T)\times{\mathbb{R}},\qquad u(0,x)=u_0(x),\] locally well posed for complex-valued data $u_0\in\widehat H^{1/2}_1({\mathbb{R}})$? The endpoint parameters are $s=1/2$, $r=1$, and the spatial domain is ${\mathbb{R}}$, not a periodic domain. The original prose only names an endpoint for the nonlinear Schrödinger problem; the curated field supplies this precise derivative equation and reports source checking.

With a common Fourier--Lebesgue convention,\[\|f\|_{\widehat H^s_r}=\|\langle\xi\rangle^s\widehat f(\xi)\|_{L^{r'}({\mathbb{R}})},\qquad \frac1r+\frac1{r'}=1,\qquad \langle\xi\rangle=(1+|\xi|^2)^{1/2},\] for tempered distributions whose Fourier transform has the stated weighted integrability. Then $r=1$ means $r'=\infty$ and $\|f\|_{\widehat H^{1/2}_1}=\operatorname*{ess\,sup}_{\xi}\langle\xi\rangle^{1/2}|\widehat f(\xi)|$; it does not mean a weighted Fourier $L^1$ norm. The record does not explicitly define this convention or say whether the endpoint space is the full weighted Fourier $L^\infty$ space or the closure of Schwartz functions in that norm. Those spaces differ, and strong time continuity of the free Schrödinger group is an additional endpoint issue that must be checked in the chosen data space.

A standard local-well-posedness target, once the endpoint topology and solution class are fixed, is existence on some time interval about zero for every admissible datum, uniqueness in a stated spacetime class, persistence in the data space, and continuous dependence on initial data on a common local interval near each datum. Strong $C_t\widehat H^{1/2}_1$ continuity, weaker time continuity, auxiliary spacetime conditions making $|u|^2u$ meaningful, and uniqueness only in a contraction space are different possible requirements. None is supplied explicitly in the short record, so the endpoint space or topology is not silently repaired.

The supplied triage checked 2026-08-21 labels the endpoint open and reports that later results still use $r>1$ or Sobolev spaces. Their precise hypotheses and relevance to the chosen endpoint convention are not reproduced. This dated assessment requires independent verification and is not an asserted solution or ill-posedness result.

Short English statement

Determine local well-posedness of the derivative Schrödinger equation at the stated Fourier--Lebesgue endpoint. Under the usual dual-exponent convention the Fourier norm is weighted $L^\infty$, so the full space versus Schwartz closure and the required time-continuity topology are essential missing details.

Sources

• Local UnsolvedMath registry, permanent ID 30000148, code catalogue entry; Critical Well-Posedness for Derivative Nonlinear Schrödinger Equations, Oberwolfach Reports Open Problems.
• Nonlinear Waves and Dispersive Equations (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/50, pp. 2653--2728. https://doi.org/10.4171/owr/2004/50. Proposed year and year tag: 2004; no proposer named.
• Supplied literature label original OWR report: https://doi.org/10.4171/owr/2004/50.
• Supplied literature label Grunrock--Herr: https://arxiv.org/abs/0704.2996.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. The corrected_verified field reports official-report inspection, not independently checked here.