Global regularity and general-coefficient singular limits for energy-critical complex Ginzburg--Landau equations

Abstract

We study energy-critical complex Ginzburg--Landau equations with a linear damping term $Ru$, $R\geq 0$. For the undamped aligned equation in dimensions $d=3,4$, we treat the focusing and defocusing cases in a unified way and prove persistence of $H^1\cap C_0$ regularity and smoothness for positive times. In particular, this resolves the energy-critical cases of Cazenave's open problem. In dimensions $3\le d\le6$, we develop a coefficient-uniform critical stability framework for the zero-dispersion and inviscid limits. It applies to independent normalized complex coefficient paths in both the focusing and defocusing cases. From limiting data in the natural energy space $H^1$, we obtain convergence on every compact subinterval of the maximal lifespan of the limiting solution. Higher regularity is required only for explicit linear coefficient-error estimates, and the limits do not use global well-posedness, scattering, or global spacetime bounds for the limiting solution. At the inviscid limit, we establish coefficient-uniform homogeneous and retarded Strichartz estimates; a key technical ingredient is the retarded double-endpoint estimate required by the critical forcing space.