论文标题
在消失的粘度极限中，一维传输方程的均匀可控性
On uniform controllability of 1D transport equations in the vanishing viscosity limit
论文作者
论文摘要
我们考虑一个具有不同矢量场和较小的粘度系数的一维运输方程，由间隔的一个端点控制。我们在控制粘度极限下均匀地控制零所需的最小时间和下限。我们假设向量场在整个间隔内有所不同。我们获得的上/下估计取决于几何量，例如Agmon距离和相关的半经典Schr {Ö} dinger操作员的光谱间隙。在这种特殊情况下，它们在同伴论文[LL21]中获得的结果改善。这些证据依靠对问题的重新制定为半经典热方程的统一可观察性问题，以及对在半经典允许和禁止区域中特征函数的定位进行精细分析[LL22]，以及频谱间隙的估计[HS84，All98]。沿着证据，我们为具有明确界限的生物三相统治家庭提供了建造，我们认为这具有独立的兴趣。
We consider a one dimensional transport equation with varying vector field and a small viscosity coefficient, controlled by one endpoint of the interval. We give upper and lower bounds on the minimal time needed to control to zero, uniformly in the vanishing viscosity limit. We assume that the vector field varies on the whole interval except at one point. The upper/lower estimates we obtain depend on geometric quantities such as an Agmon distance and the spectral gap of an associated semiclassical Schr{ö}dinger operator. They improve, in this particular situation, the results obtained in the companion paper [LL21]. The proofs rely on a reformulation of the problem as a uniform observability question for the semiclassical heat equation together with a fine analysis of localization of eigenfunctions both in the semiclassically allowed and forbidden regions [LL22], together with estimates on the spectral gap [HS84, All98]. Along the proofs, we provide with a construction of biorthogonal families with fine explicit bounds, which we believe is of independent interest.
