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A Low-Frequency Assumption for Optimal Time-Decay Estimates to the Compressible Navier–Stokes Equations

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Abstract

The global existence issue in critical spaces for compressible Navier–Stokes equations, was addressed by Danchin (Invent Math 141:579–614, 2000) and then developed by Charve and Danchin (Arch Rational Mech Anal 198:233–271, 2010), Chen et al. (Commun Pure Appl Math 63:1173–1224, 2010) and Haspot (Arch Rational Mech Anal 202:427–460, 2011) in more general Lp setting. The main aim of this paper is to exhibit (more precisely) time-decay estimates of solutions constructed in the critical regularity framework. To the best of our knowledge, the low-frequency assumption usually plays a key role in the large-time asymptotics of solutions, which was firstly observed by Matsumura and Nishida (J Math Kyoto Univ 20:67–104, 1980) in the \({L^1(\mathbb{R}^d)}\) space. We now claim a new low-frequency assumption for barotropic compressible Navier–Stokes equations, which may be of interest in the mathematical analysis of viscous fluids. Precisely, if the initial density and velocity additionally belong to some Besov space \({\dot{B}^{-\sigma_1}_{2,\infty}(\mathbb{R}^d)}\) with the regularity \({\sigma_1\in (1-d/2, 2d/p-d/2]}\), then a sharp time-weighted inequality including enough time-decay information can be available, where optimal decay exponents for the high frequencies are exhibited. The proof mainly depends on some non standard Besov product estimates. As a by-product, those optimal time-decay rates of Lq–Lr type are also captured in the critical framework.

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Acknowledgements

The author (J. Xu) is partially supported by the National Natural Science Foundation of China (11471158, 11871274) and the Fundamental Research Funds for the Central Universities (NE2015005).

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  1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, People’s Republic of China

    Jiang Xu

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Correspondence to Jiang Xu.

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Communicated by C. De Lellis

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The author would like to thank Professor A. Matsumura for introducing him to the decay problem for partially parabolic equations when he visited Osaka University. Also, he is grateful to Professor R. Danchin for addressing the conjecture on the regularity of low frequencies when visiting the LAMA in UPEC.

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Xu, J. A Low-Frequency Assumption for Optimal Time-Decay Estimates to the Compressible Navier–Stokes Equations. Commun. Math. Phys. 371, 525–560 (2019). https://doi.org/10.1007/s00220-019-03415-6

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