Journal of the Mathematical Society of Japan
Online ISSN : 1881-1167
Print ISSN : 0025-5645
ISSN-L : 0025-5645
Time periodic problem for the compressible Navier–Stokes equation on ℝ2 with antisymmetry
Kazuyuki Tsuda
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2018 年 70 巻 1 号 p. 243-281

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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 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 ℝ2.

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