The sharp time decay rate of the isentropic Navier-Stokes system in $ {\mathop{\mathbb R\kern 0pt}\nolimits}^3 $

  • Received: 01 November 2019 Revised: 01 July 2020 Published: 23 September 2020
  • 35A01, 35B45, 35Q35, 76A05, 76D03

  • We investigate the sharp time decay rates of the solution $ U $ for the compressible Navier-Stokes system (1.1) in $ {\mathop{\mathbb R\kern 0pt}\nolimits}^3 $ to the constant equilibrium $ (\bar\rho>0, 0) $ when the initial data is a small smooth perturbation of $ (\bar\rho,0) $. Let $ \widetilde U $ be the solution to the corresponding linearized equations with the same initial data. Under a mild non-degenerate condition on initial perturbations, we show that $ \|U-\widetilde U\|_{L^2} $ decays at least at the rate of $ (1+t)^{-\frac54} $, which is faster than the rate $ (1+t)^{-\frac34} $ for the $ \widetilde U $ to its equilibrium $ (\bar\rho ,0) $. Our method is based on a combination of the linear sharp decay rate obtained from the spectral analysis and the energy estimates.

    Citation: Yuhui Chen, Ronghua Pan, Leilei Tong. The sharp time decay rate of the isentropic Navier-Stokes system in $ {\mathop{\mathbb R\kern 0pt}\nolimits}^3 $[J]. Electronic Research Archive, 2021, 29(2): 1945-1967. doi: 10.3934/era.2020099

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  • We investigate the sharp time decay rates of the solution $ U $ for the compressible Navier-Stokes system (1.1) in $ {\mathop{\mathbb R\kern 0pt}\nolimits}^3 $ to the constant equilibrium $ (\bar\rho>0, 0) $ when the initial data is a small smooth perturbation of $ (\bar\rho,0) $. Let $ \widetilde U $ be the solution to the corresponding linearized equations with the same initial data. Under a mild non-degenerate condition on initial perturbations, we show that $ \|U-\widetilde U\|_{L^2} $ decays at least at the rate of $ (1+t)^{-\frac54} $, which is faster than the rate $ (1+t)^{-\frac34} $ for the $ \widetilde U $ to its equilibrium $ (\bar\rho ,0) $. Our method is based on a combination of the linear sharp decay rate obtained from the spectral analysis and the energy estimates.



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