### Electronic Research Archive

2021, Issue 2: 1945-1967. doi: 10.3934/era.2020099

# 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

### Related Papers:

• 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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