发表机构
Academy of Mathematics and Systems Science, Chinese Academy of Sciences; Morningside Center of Mathematics; School of Mathematics and Statistics, Beijing Institute of Technology（中国科学院数学与系统科学研究院; 晨兴数学中心; 北京理工大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文证明了三维聚焦能量临界NLS在基态能量和梯度阈值以下的全局适定性与散射，证实了阈值猜想，并通过改进的Morawetz恒等式和频率局域化估计实现了这一结果。
AI 中文摘要
我们证明了在基态能量和梯度阈值以下的三维聚焦能量临界非线性薛定谔方程（NLS）的全局适定性与散射，从而在三维情形下证实了阈值猜想。受我们关于三维散焦三次NLS在H^s(R^3)（s>1/2）中工作的启发，我们从不假设有限质量的相互作用Morawetz恒等式推导出频率局域化的L^4_{t,x}估计。我们将相互作用权重适应于投影密度，以在聚焦情形下获得正的四次时空项，并利用非线性抵消将频率截断误差吸收到该项中。
英文摘要
We prove global well-posedness and scattering below the ground-state energy and gradient thresholds for the three-dimensional focusing energy-critical nonlinear Schrödinger equation. The key step is to combine a frequency-localized interaction Morawetz estimate with a forcing estimate for a hypothetical critical element. Together they give high-frequency $L^4_{t,x}$ control and place the nonlinearity in $L^2_tL^1_x$, without assuming finite mass. Directional coercivity from the strict threshold gaps makes the principal term coercive for an explicit convex weight. The radius of the weight is determined by the high-frequency density, and the continuity equation controls the negative part of the resulting moving-radius term in the interaction identity. Nonlinear cancellations and long-time Strichartz estimates give a truncation-error bound with a coefficient that vanishes as the cutoff frequency tends to zero. A finite-interval absorption argument closes these estimates, after which the no-waste Duhamel formula rules out the critical element.
Comments30 pages. Comments are welcome!
