Cazenave’s open problem for energy-critical complex Ginzburg–Landau equations

Cazenave’s problem asks for global regularity and control of singular limits for energy-critical complex Ginzburg–Landau equations without assuming global bounds for the limiting solution. The new work addresses the focusing and defocusing undamped aligned cases in dimensions $d=3,4$, rather than every formulation of the problem.

Known results

• November 2023: energy-critical CGL converges to the nonlinear heat equation in $d=3,4$, and in one focusing case to nonlinear Schrödinger flow, under threshold and global-bound assumptions.
• February 2024: threshold data are classified relative to the stationary energy $E(W)$; below threshold solutions decay, while above threshold they blow up.
• October 2024: global well-posedness was obtained in an exterior domain, with weak-solution consequences for the defocusing equation.
• 2012: negative-energy solutions were shown to blow up in finite time.

August 25, 2026 preprint

Global regularity and general-coefficient singular limits for energy-critical complex Ginzburg–Landau equations claims persistence of $H^1\cap C_0$ regularity, positive-time smoothness, and singular-limit control in the stated focusing and defocusing cases. The result is presented as resolving the energy-critical cases, but the preprint is unrefereed and its proof has not been independently confirmed.

Current status (as of August 2026): The energy-critical cases in dimensions $d=3,4$ are claimed solved by an unrefereed preprint, while independent verification and broader formulations of Cazenave’s problem remain open.