Optimal Exponential Decay for the Linearized Ellipsoidal BGK Model in Weighted Sobolev Spaces

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-λt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{-\lambda t}$$\end{document} is optimal in the sense that λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda >0$$\end{document} 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.