We study a Cauchy problem for the 3-d compressible isentropic Navier-Stokes equations, in which the initial data is a 3-d periodic perturbation around a planar rarefaction wave. We prove that the solution of the Cauchy problem exists globally in time and tends to the background rarefaction wave in the $ L^\infty(R^3) $ space as $ t\to +\infty. $ The result reveals that even though the initial perturbation has infinite oscillations at the far field and is not integrable along any direction of space, the planar rarefaction wave is nonlinearly stable for the 3-d N-S equations.
Publication: Advances in Mathematics 404, Paper No. 108452, 27 pp. (2022)
Authors:
Feimin Huang
Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, China, and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Email: fhuang@amt.ac.cn
Lingda Xu
Department of Mathematics, Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China, and Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China
Email: xulingda@tsinghua.edu.cn
Qian Yuan
Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, China
Email: qyuan@amss.ac.cn
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