AI 中文总结
本文研究一维环面上线性化周期Whitham-Boussinesq系统的稳定化，建立其适定性，证明阻尼半群强稳定性，得到最优多项式衰减率。
AI 中文摘要
我们研究一维环面上线性化周期Whitham-Boussinesq系统的稳定化问题。我们在自然能量空间中建立了保守动力学和阻尼动力学的适定性，并描述了保守生成元的谱结构，其频率在高频处呈现|k|^{1/2}阶的次线性增长。随后我们证明了阻尼半群的强稳定性，并得到了谱参数线性增长的高频预解估计。在真正局域化阻尼的条件下，一族高频拟模式给出了匹配的下界，表明该预解增长是最优的。根据Borichev-Tomilov定理[5]，我们推导出生成元定义域上半群的t^{-1}衰减率，对应能量的t^{-2}衰减率。在相同局域化假设下，这些多项式衰减率是最优的。
英文摘要
We study the stabilization of the linearized periodic Whitham--Boussinesq system on the one-dimensional torus. We establish the well-posedness of the conservative and damped dynamics in the natural energy space and describe the spectral structure of the conservative generator, whose frequencies exhibit sublinear growth of order $|k|^{1/2}$ at high frequency. We then prove strong stability of the damped semigroup and obtain a high-frequency resolvent estimate with linear growth in the spectral parameter. Under genuinely localized damping, a family of high-frequency quasimodes provides the matching lower bound and shows that this resolvent growth is optimal. By the Borichev--Tomilov theorem [5], we deduce a $t^{-1}$ decay rate for the semigroup on the domain of the generator, corresponding to a $t^{-2}$ decay rate for the energy. Under the same localization assumption, these polynomial decay rates are optimal.
Comments38 pages. Comments are welcome
