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A remark on the uniqueness of Kozono–Nakao’s mild \(L^3\)-solutions on the whole time axis to the Navier–Stokes equations in unbounded domains

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Abstract

This paper is concerned with the uniqueness of Kozono–Nakao’s bounded continuous \(L^{3}\)-solutions on the whole time axis to the Navier–Stokes equations in 3-dimensional unbounded domains. When \(\Omega \) is an unbounded domain, it is known that a small solution in \(BC({\mathbb {R}};L^{3,\infty })\) is unique within the class of solutions which have sufficiently small \(L^{\infty }({\mathbb {R}}; L^{3,\infty })\)-norm. There is another type of uniqueness theorem. Farwig, Nakatsuka and the author (2015) showed that if two solutions exist for the same force f, one is small and if other one satisfies the precompact range condition (PRC), then the two solutions coincide. Since time-periodic solutions satisfy (PRC), this uniqueness theorem is applicable to time-periodic solutions. On the other hand, there exist many solutions which do not satisfy (PRC). In this paper, by assuming the boundedness of the \(L^r\)-norm for some \(1<r<3\), we show a modified version of the above-mentioned uniqueness theorem without (PRC).

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Acknowledgements

The author would like to thank the anonymous referees for their valuable comments. This work was supported by JSPS KAKENHI Grant Number No.16K05228.

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  1. Department of Mathematical Sciences Shinshu University, Matsumoto, 390-8621, Japan

    Yasushi Taniuchi

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  1. Yasushi Taniuchi
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Correspondence to Yasushi Taniuchi.

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Dedicated to Professor Hideo Kozono on his 60th birthday.

Mathematical Fluid Mechanics and Related Topics: In Honor of Professor Hideo Kozono’s 60th Birthday. This article is part of the topical collection dedicated to Prof. Hideo Kozono on the occasion of his 60th birthday, edited by Kazuhiro Ishige, Tohru Ozawa, Senjo Shimizu, and Yasushi Taniuchi.

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Taniuchi, Y. A remark on the uniqueness of Kozono–Nakao’s mild \(L^3\)-solutions on the whole time axis to the Navier–Stokes equations in unbounded domains. Partial Differ. Equ. Appl. 2, 68 (2021). https://doi.org/10.1007/s42985-021-00121-8

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