论文标题
湿性低纤维化波和schr {Ö} dinger方程的对数衰减
Logarithmic decay for damped hypoelliptic wave and Schr{ö}dinger equations
论文作者
论文摘要
我们考虑阻尼波（schr {Ö} dinger and plate）方程，该方程是由紧凑型歧管和阻尼函数B（x）驱动的。我们假设Chow-rashevski-h {Ö} rmander条件在等级K（最多需要k lie括号跨越切线空间需要）以及M的分析性以及L的分析性。我们以速率$ log（t）^{-1/k} $（t）^$ log（t）^$ log（t）^$ log（t）^$ log（t）^^{-2/k} $ for for for的能量衰减。我们表明，这种衰变是Grushin型操作员家族的最佳选择。该结果来自一个扰动参数（独立利益的），在一般的抽象环境中，波浪型方程的定量近似可观察力/可控性结果意味着相关阻尼波，Schr {Ö} dinger和板方程的先验衰减率。作者在[ll19，ll17]中获得了适用的定量近似可观测性/可控性定理。
We consider damped wave (resp. Schr{ö}dinger and plate) equations driven by a hypoelliptic "sum of squares" operator L on a compact manifold and a damping function b(x). We assume the Chow-Rashevski-H{ö}rmander condition at rank k (at most k Lie brackets needed to span the tangent space) together with analyticity of M and the coefficients of L. We prove decay of the energy at rate $log(t)^{-1/k}$ (resp. $log(t)^{-2/k}$ ) for data in the domain of the generator of the associated group. We show that this decay is optimal on a family of Grushin-type operators. This result follows from a perturbative argument (of independent interest) showing, in a general abstract setting, that quantitative approximate observability/controllability results for wave-type equations imply a priori decay rates for associated damped wave, Schr{ö}dinger and plate equations. The adapted quantitative approximate observability/controllability theorem for hypoelliptic waves is obtained by the authors in [LL19, LL17].
