Higher Sobolev Norm Growth for Half-Wave Maps

Definitions and mathematical statement

The supplied record concerns higher Sobolev norm growth for half-wave maps with nonrational initial data. Its formulations do not uniquely specify a mathematical problem: the reconstructed statement specifies the circle, whereas the retained original passage discusses the real line. The excerpt also does not give the evolution equation denoted by (HWM) or define the rational class being excluded. These qualifications remain unresolved.

The original passage identifies the complex Grassmannian $\operatorname{Gr}_1({\mathbb{C}}^2)$, the space of one-dimensional complex linear subspaces of ${\mathbb{C}}^2$, with $S^2=\{v\in{\mathbb{R}}^3:|v|=1\}$.

For the circle formulation, use the conventional normalization $\mathbb T={\mathbb{R}}/(2\pi{\mathbb{Z}})$. For $s>\tfrac12$, define \[ H^s(\mathbb T;S^2)=\{v\in H^s(\mathbb T;{\mathbb{R}}^3):|v(x)|=1\text{ for almost every }x\}, \qquad \|v\|_{H^s(\mathbb T)}^2=\sum_{k\in{\mathbb{Z}}}(1+k^2)^s|\widehat v(k)|^2, \] where $\widehat v(k)=(2\pi)^{-1}\int_0^{2\pi}v(x)e^{-ikx}\,{\,\mathrm{d}} x$, with the squared modulus summed over the three components.

Conditional on the circle being the intended domain, and on specifying the source's half-wave maps evolution and rational class, the question can be formulated as follows. For each fixed $s>\tfrac12$, determine the possible time behavior of $\|u(t)\|_{H^s(\mathbb T)}$ for solutions with nonrational initial data $u(0)=u_0\in H^s(\mathbb T;S^2)$. In particular, can such initial data produce finite-time loss of boundedness, \[ \limsup_{t\uparrow T}\|u(t)\|_{H^s(\mathbb T)}=+\infty \quad\text{for some }0<T<\infty, \] with the solution having finite $H^s$ norm for $0\leq t<T$? Can a solution having finite $H^s$ norm at every finite positive time instead satisfy \[ \limsup_{t\to+\infty}\|u(t)\|_{H^s(\mathbb T)}=+\infty? \] These formulas describe unbounded norm growth. The source does not specify whether its word "blowup" requires an actual limit to $+\infty$, rather than a divergent limsup, or whether both time directions are intended. The initial finite-norm requirement makes the growth question meaningful but is not explicitly stated in the extracted question.

The real-line context cannot simply be substituted into this formulation. The retained passage attributes to its reference [5] global well-posedness and soliton resolution on ${\mathbb{R}}$ for all initial data in $\dot H^{1/2}({\mathbb{R}};S^2)$. The homogeneous critical seminorm is conventionally given, up to normalization, by \[ [v]_{\dot H^{1/2}({\mathbb{R}})}^2=\int_{{\mathbb{R}}}\int_{{\mathbb{R}}}\frac{|v(x)-v(y)|^2}{|x-y|^2}\,{\,\mathrm{d}} x\,{\,\mathrm{d}} y. \] A sphere-valued map on ${\mathbb{R}}$ has infinite ordinary $L^2({\mathbb{R}})$ norm, so an inhomogeneous $H^s({\mathbb{R}};{\mathbb{R}}^3)$ norm of the map itself is not finite. A real-line version therefore needs an explicit choice of homogeneous higher regularity or a norm of a perturbation relative to a specified reference map, together with any associated conditions at infinity. The supplied record does not make that choice. The real-line theorem context and the circle question are consequently retained as distinct supplied formulations, pending mathematical review.

Short English statement

The question asks whether nonrational half-wave maps can develop unbounded higher Sobolev norms for $s>\tfrac12$, at a finite time or along arbitrarily large times. The supplied circle formulation conflicts with the real-line context of the original excerpt. The equation, rational class, intended real-line norm, and precise meaning of blowup require clarification before a definitive statement can be fixed.

Sources

• Local UnsolvedMath registry: supplied complete record with permanent ID $30006631$, title "Higher Sobolev Norm Growth for Half-Wave Maps," and exact source problem code catalogue entry. The record gives proposed year 2026 and names no proposer.
• Supplied report citation: Nonlinear Waves and Dispersive Equations (2026), Oberwolfach Reports, pp. 611--682, DOI 10.4171/owr/2026/10. https://doi.org/10.4171/owr/2026/10.
• Supplied literature source, labeled "OWR report": https://publications.mfo.de/bitstream/handle/mfo/4444/OWR_2026_10.pdf?sequence=1.
• Supplied literature source, labeled "Gérard--Lenzmann preprint": https://arxiv.org/abs/2602.20985. The retained excerpt attributes global well-posedness on ${\mathbb{R}}$ for all data in $\dot H^{1/2}({\mathbb{R}};S^2)$, together with soliton resolution, to its reference [5]. The supplied record does not provide that reference's full bibliographic entry or explicitly establish its identity with this linked preprint.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem open and states that the cited work establishes global well-posedness at critical $H^{1/2}$, while higher $H^s$ norm behavior for nonrational data remains open. This is supplied dated curation, not an independently verified assessment. Its general critical-space wording should be read alongside the excerpt's explicit real-line homogeneous-space qualification.
• The registry supplies the status open, the statement status corrected_verified, and a statement-verification note saying that the reconstruction was checked against the official report. These are supplied provenance claims; no source visit or independent verification is asserted here.