Energy Thresholds for Landau–Lifshitz Dynamics

Definitions and mathematical statement

Write $u_0=u(0,\cdot)$. The supplied Landau--Lifshitz equation is \[ \partial_tu=a_1\left(\Delta u+|\nabla u|^2u\right)+a_2u\times\Delta u. \] Here $\nabla$ and $\Delta$ denote spatial differentiation and the spatial Laplacian, and $\times$ denotes the usual three-dimensional cross product. The spatial domain and its dimension, the target constraint on $u$, boundary conditions or conditions at infinity, and the permitted values of $a_1,a_2$ are not specified in the supplied record. In particular, the record does not explicitly prescribe a sphere-valued target or an energy formula and normalization.

The requested dichotomy can therefore only be recorded conditionally. Let $\mathcal A$ stand for the intended, unspecified set of coefficient pairs, and let $\mathcal D_{a_1,a_2}$ stand for the intended admissible initial data. Let $E(u_0)$ denote the unspecified energy. If a solution theory and continuation criterion have been fixed, write $T_{\max}(u_0;a_1,a_2)$ for the maximal forward existence time in that theory. These are placeholders for missing specifications, not additional hypotheses supplied by the source.

The intended question is whether, for every $(a_1,a_2)\in\mathcal A$, both assertions hold: \[ \begin{aligned} &\forall u_0\in\mathcal D_{a_1,a_2},\qquad E(u_0)\leq4\pi \ \Longrightarrow\ T_{\max}(u_0;a_1,a_2)=\infty;\\ &\forall e\in{\mathbb{R}},\ e>4\pi,\quad \exists u_0\in\mathcal D_{a_1,a_2}:\qquad E(u_0)=e,\quad T_{\max}(u_0;a_1,a_2)<\infty, \end{aligned} \] where the finite maximal time in the second assertion is intended to represent finite-time blow-up in the chosen solution class. The source does not specify the norm or regularity whose breakdown constitutes blow-up, nor does it supply a continuation theorem identifying blow-up with finite maximal existence time.

The equality case $E(u_0)=4\pi$ is explicitly included in the proposed global-existence assertion. The phrase "at every energy above $4\pi$" is represented by existence at each exact energy $e>4\pi$. The source explicitly includes $a_1=0$, calling this the Schrödinger-map case, but does not specify the accompanying restriction on $a_2$, or whether $(a_1,a_2)=(0,0)$ is excluded. It describes the question as an extension of a heat-flow and wave-map dichotomy without providing the hypotheses of those comparison results. Consequently, this is an unresolved formulation of the supplied question, not a fully specified conjecture or an assertion of the comparison results.

Short English statement

The supplied question asks whether the full Landau--Lifshitz family, including the stated Schrödinger-map case $a_1=0$, has global evolution for every admissible datum of energy at most $4\pi$, and some finite-time blow-up datum at each exact energy above $4\pi$. The domain, target, coefficient range, energy, data class, and meaning of blow-up must be specified before this becomes a precise mathematical conjecture.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001592$, exact source problem code catalogue entry, title "Energy Thresholds for Landau--Lifshitz Dynamics," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is 2010.
• Supplied report citation: Nonlinear Waves and Dispersive Equations (2011), Oberwolfach Reports, pp. 2393--2463, DOI 10.4171/owr/2010/41. Supplied link: https://doi.org/10.4171/owr/2010/41. The record attributes the problem to workshop open-problem discussions documented by Oberwolfach Reports; it supplies no problem-specific page or speaker attribution.
• Additional supplied literature-source link: https://ems.press/content/serial-article-files/32986?nt=1.
• Literature assessment dated exactly 2026-08-21: the supplied status is "partially solved." The supplied assessment states that later work establishes global, stability, and blow-up results in several equivariant and target-specific regimes, but not the stated dichotomy for the full Landau--Lifshitz family. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The supplied statement status is corrected_verified, with a recorded manual semantic, typographic, and LaTeX review asserting a meaning-preserving, self-contained cleanup. This is source metadata and does not resolve the missing mathematical specifications identified here.