Asymptotic Profile for Diffusion Wave Terms of the Compressible Navier–Stokes–Korteweg System

Takayuki Kobayashi, Masashi Misawa and Kazuyuki Tsuda
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Takayuki Kobayashi: Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, 1-3, Machikaneyamacho, Toyonakashi 560-8531, Japan
Masashi Misawa: Department of Science, Faculty of Science, Kumamoto University, 2-39-1, Kurokami, Chuo-ku, Kumamoto 860-8555, Japan
Kazuyuki Tsuda: Faculty of Science and Engineering, Kyushu Sangyo University, 3-1, Matsukadai 2-Chome, Higashi-ku, Fukuoka 813-8503, Japan

Mathematics, 2021, vol. 9, issue 6, 1-20

Abstract: The asymptotic profile for diffusion wave terms of solutions to the compressible Navier–Stokes–Korteweg system is studied on R 2 . The diffusion wave with time-decay estimate was studied by Hoff and Zumbrun (1995, 1997), Kobayashi and Shibata (2002), and Kobayashi and Tsuda (2018) for compressible Navier–Stokes and compressible Navier–Stokes–Korteweg systems. Our main assertion in this paper is that, for some initial conditions given by the Hardy space, asymptotic behaviors in space–time L 2 of the diffusion wave parts are essentially different between density and the potential flow part of the momentum. Even though measuring by L 2 on space, decay of the potential flow part is slower than that of the Stokes flow part of the momentum. The proof is based on a modified version of Morawetz’s energy estimate, and the Fefferman–Stein inequality on the duality between the Hardy space and functions of bounded mean oscillation.

Keywords: compressible Navier–Stokes–Korteweg system; asymptotic profile; diffusion wave; hardy space (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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