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Note on asymptotic profile of solutions to the linearized compressible Navier-Stokes flow. (English) Zbl 1426.35177
A linearized compressible Navier-Stokes system in $$\mathbb{R}^n$$ is considered in this paper. The main point is to give the exact behavior of the leading part of the velocity, for large times values. Some results given in [the second author and M. Onodera [Differ. Integral Equ. 30, No. 7–8, 505–520 (2017; Zbl 1424.35246)], where an explicit profile of the density for $$t\rightarrow \infty$$ is given (related with the viscoelastic wave equations) are used, extended and improved. The new element of the paper is the use of the Fourier transformation. The initial problem is reduced to a system of ODF and a decomposition of the initial data is performed. The useful expression (2.15) for the Fourier transform of the velocity is obtained and each term is carefully analyzed, in the low frequency zone. In the high frequency zone, the Harnaux-Kormonik inequality is used (which is concerning the energy method in the Fourier space). The 3D Gauss kernel and the Plancherel theorem are also used. A large list of references is given in the last part.
##### MSC:
 35Q30 Navier-Stokes equations 35B40 Asymptotic behavior of solutions to PDEs 76N99 Compressible fluids and gas dynamics, general 35C20 Asymptotic expansions of solutions to PDEs 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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