严格凸域内薛定谔方程的色散：一般情形
Dispersion for the Schrödinger equation inside strictly convex domains: the general case
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中文总结 AI 辅助
本文针对严格凸域上的半经典薛定谔方程，构造尖锐局部参数化并得到色散估计，证明衰减指数损失1/4为最优，进而获得三维流形上聚焦三次非线性薛定谔方程的全局适定性及Sobolev范数增长界。
中文摘要 AI 辅助
我们考虑维度$d\geq2$的一般光滑有界严格凸域$\Omega\subset\mathbb{R}^d$，并描述具有Dirichlet边界条件的半经典薛定谔方程的色散。我们的结果更一般地适用于具有光滑严格凸边界的紧致光滑黎曼流形。具体而言，我们构造了一个在（半经典）时间上尖锐的局部参数化（parametrix），进而获得色散估计：我们的Green函数固定时间衰减率相对于无边界情形，在$h/t$的指数上损失了$1/4$。该损失对于色散而言是尖锐的，正如第一作者先前在模型凸域情形中所证明的那样。在具有光滑严格凸边界的三维紧致流形上，所得的谱局部化Strichartz估计给出了能量空间中聚焦三次非线性薛定谔方程的全局适定性，从而匹配了Burq-Gérard-Tzvetkov关于无边界流形的相应结果。此外，我们将Planchon-Visciglia-Tzvetkov关于更高Sobolev范数时间增长的界推广到我们的情形。
英文摘要
We consider a general smooth bounded strictly convex domain $Ω\subset\mathbb{R}^d$ of dimension $d\geq2$ and describe dispersion for the semiclassical Schrödinger equation with Dirichlet boundary condition. Our results hold more generally on compact smooth Riemannian manifolds with smooth strictly convex boundary. More specifically, we construct a sharp local in (semiclassical) time parametrix and then proceed to obtain dispersion estimates: our fixed-time decay rate for the Green function exhibits a loss of $1/4$ in the exponent of $h/t$ with respect to the boundaryless case. The loss is sharp for dispersion, as shown earlier by the first author in the case of a model convex domain. On compact three-dimensional manifolds with smooth strictly convex boundary, the resulting spectrally localized Strichartz estimates yield global well-posedness in the energy space for the defocusing cubic nonlinear Schrödinger equation, thus matching the corresponding result for boundaryless manifolds due to Burq-Gérard-Tzvetkov. Moreover, we extend bounds on the time growth of higher Sobolev norms from Planchon-Visciglia-Tzvetkov to our setting.
发表机构
- Sorbonne Université（索邦大学）
- CNRS（法国国家科学研究中心）
- Institut universitaire de France (IUF)（法国高等研究院）
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