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Optimal decay estimates in the critical \(L^{p}\) framework for flows of compressible viscous and heat-conductive gases

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Abstract

The global existence issue in critical regularity spaces for the full Navier–Stokes equations satisfied by compressible viscous and heat-conductive gases has been first addressed in Danchin (Arch Ration Mech Anal 160:1–39, 2001), then recently extended to the general \(L^p\) framework in Danchin and He (Math Ann 64:1–38, 2016). In the present work, we establish decay estimates for the global solutions constructed in [13], under an additional mild integrability assumption that is satisfied if the low frequencies of the initial data are in \(L^{r}(\mathbb {R}^d)\) with \(\frac{p}{2}\le r \le \min \{2,\frac{d}{2}\}\). As a by-product in the case \(d=3,\) we recover the classical decay rate \(t^{-\frac{3}{4}}\) for \(t\rightarrow +\infty \) that has been observed by Matsumura and Nishida (J Math Kyoto Univ 20:67–104, 1980) for solutions with high Sobolev regularity. Compared to a recent paper of us (Danchin and Xu in Arch Ration Mech Anal 224:53–90, 2017) dedicated to the barotropic case, not only we are able to treat the full system, but we also weaken the low frequency assumption and improve the decay exponents for the high frequencies of the solution.

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Authors and Affiliations

  1. LAMA (UMR 8050), UPEMLV, UPEC, CNRS, Université Paris-Est, 61 avenue du Général de Gaulle, 94010, Créteil Cedex 10, France

    Raphaël Danchin

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

    Jiang Xu

Authors
  1. Raphaël Danchin
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  2. Jiang Xu
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Correspondence to Raphaël Danchin.

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The authors declare that they have no conflict of interest.

Additional information

Communicated by M. Hieber

The first author has been partially supported by ANR-15-CE40-0011 and Institut Universitaire de France.

The second author is partially supported by the National Natural Science Foundation of China (11471158) and the Fundamental Research Funds for the Central Universities (NE2015005). He would like to thank Professor A. Matsumura for introducing him to the decay problem for partially parabolic equations when he visited Osaka University. He is also grateful to Professor R. Danchin for the kind hospitality when visiting the LAMA in UPEC.

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Danchin, R., Xu, J. Optimal decay estimates in the critical \(L^{p}\) framework for flows of compressible viscous and heat-conductive gases. J. Math. Fluid Mech. 20, 1641–1665 (2018). https://doi.org/10.1007/s00021-018-0381-6

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  • DOI: https://doi.org/10.1007/s00021-018-0381-6

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