## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 48, Number 2 (2019), 357-383.

### Note on asymptotic profile of solutions to the linearized compressible Navier-Stokes flow

Ruy COIMBRA CHARÃO and Ryo IKEHATA

#### Abstract

We consider the asymptotic behavior as $t \to +\infty$ of the $L^{2}$-norm of the velocity of the linearized compressible Navier-Stokes equations in ${\bf R}^{n}$ ($n \geq 2$). As an application we shall study the optimality of the decay rate for the $L^{2}$-norm of the velocity by deriving a decay estimate from below as $t \to +\infty$. To get the estimates in the zone of high frequency we use a version of the energy method in the Fourier space combined with the Haraux-Komornik inequality and this seems much different from known techniques to study compressible Navier-Stokes system.

#### Article information

**Source**

Hokkaido Math. J., Volume 48, Number 2 (2019), 357-383.

**Dates**

First available in Project Euclid: 11 July 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1562810515

**Digital Object Identifier**

doi:10.14492/hokmj/1562810515

**Mathematical Reviews number (MathSciNet)**

MR3980948

**Zentralblatt MATH identifier**

07080100

**Subjects**

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 35B40: Asymptotic behavior of solutions

Secondary: 76N99: None of the above, but in this section 35C20: Asymptotic expansions

**Keywords**

Compressible Navier-Stokes equations Cauchy problem Asymptotic profiles Weighted $L^{1}$-initial data Low and high frequencies

#### Citation

COIMBRA CHARÃO, Ruy; IKEHATA, Ryo. Note on asymptotic profile of solutions to the linearized compressible Navier-Stokes flow. Hokkaido Math. J. 48 (2019), no. 2, 357--383. doi:10.14492/hokmj/1562810515. https://projecteuclid.org/euclid.hokmj/1562810515