散焦三次非线性薛定谔方程在调制空间 $M^{\infty,1}(\mathbb{R})$ 中的全局适定性
Global well-posedness of defocusing cubic NLS in $M^{\infty,1}(\mathbb{R})$
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中文总结 AI 辅助
本文证明了散焦三次NLS方程在调制空间中的全局适定性，通过构造满足局部守恒律的非负密度并协调局部化尺度与NLS缩放来控制范数，适用于任意数据包括准周期轮廓。
中文摘要 AI 辅助
我们证明了在调制空间 $M^{\infty,1}(\mathbb{R})$ 中一维散焦三次非线性薛定谔方程的全局适定性。该空间不施加空间衰减条件，包含 $C_b^2(\mathbb{R})$ 以及所有绝对收敛的平面波和。该结果适用于该空间中的任意数据，包括大的光滑准周期轮廓及其局部扰动。证明通过前向 Weyl 比构造了一个满足局部守恒律的非负密度，前向 Weyl 比由相关谱问题的半直线平方可积解定义。该密度的局部化积分的适当非线性组合控制调制范数。最后，协调选择空间局部化尺度和 NLS 缩放使得累积边界通量足够小，从而可以将每个温和解全局延拓。
英文摘要
We prove global well-posedness of the one-dimensional defocusing cubic nonlinear Schrödinger equation in the modulation space $M^{\infty,1}(\mathbb{R})$. This space imposes no spatial decay and contains $C_b^2(\mathbb{R})$ as well as all absolutely convergent sums of plane waves. The result applies to arbitrary data in this space, including large smooth quasiperiodic profiles and their localized perturbations. The proof constructs a nonnegative density satisfying a local conservation law from forward Weyl ratios, which are defined through half-line square-integrable solutions of the associated spectral problem. A suitable nonlinear combination of localized integrals of this density controls the modulation norm. Finally, choosing the spatial localization scale and NLS scaling in a coordinated way makes the accumulated boundary flux small enough to continue every mild solution globally.
