This paper deals with the asymptotic behavior of solution to the linearized ellipsoidal BGK model in torus. We prove that the solution converges exponentially to the equilibrium in the weighted Sobolev spaces with polynomial weight. Our exponential decay rate $$e^{-\lambda t}$$ e - λ t is optimal in the sense that $$\lambda >0$$ λ > 0 equals to the spectral gap of the linearized operator in the standard Hilbert space. Our strategy is taking advantage of the quantitative spectral gap estimates in a smaller reference Hilbert space, the factorization method, and the enlargement of the functional space for the associated semigroup.

中文翻译：

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加权 Sobolev 空间中线性化椭球 BGK 模型的最优指数衰减

本文讨论了圆环中线性化椭球 BGK 模型解的渐近行为。我们证明了该解在多项式权重的加权 Sobolev 空间中以指数方式收敛于均衡。我们的指数衰减率 $$e^{-\lambda t}$$ e - λ t 是最优的，因为 $$\lambda >0$$ λ > 0 等于标准希尔伯特中线性化算子的光谱间隙空间。我们的策略是利用较小参考希尔伯特空间中的定量谱间隙估计、分解方法以及相关半群的功能空间的扩大。