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January, 2018 Time periodic problem for the compressible Navier–Stokes equation on $\mathbb{R}^2$ with antisymmetry
Kazuyuki TSUDA
J. Math. Soc. Japan 70(1): 243-281 (January, 2018). DOI: 10.2969/jmsj/07017524


The compressible Navier–Stokes equation is considered on the two dimensional whole space when the external force is periodic in the time variable. The existence of a time periodic solution is proved for sufficiently small time periodic external force with antisymmetry condition. The proof is based on using the time-$T$-map associated with the linearized problem around the motionless state with constant density. In some weighted $L^\infty$ and Sobolev spaces the spectral properties of the time-$T$-map are investigated by a potential theoretic method and an energy method. The existence of a stationary solution to the stationary problem is also shown for sufficiently small time-independent external force with antisymmetry condition on $\mathbb{R}^2$.


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Kazuyuki TSUDA. "Time periodic problem for the compressible Navier–Stokes equation on $\mathbb{R}^2$ with antisymmetry." J. Math. Soc. Japan 70 (1) 243 - 281, January, 2018.


Published: January, 2018
First available in Project Euclid: 26 January 2018

zbMATH: 06859852
MathSciNet: MR3750276
Digital Object Identifier: 10.2969/jmsj/07017524

Primary: 35B10
Secondary: 76N10

Keywords: compressible Navier–Stokes equation , stationary solution , time periodic solution , two dimensional case

Rights: Copyright © 2018 Mathematical Society of Japan


Vol.70 • No. 1 • January, 2018
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