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Weak Solutions for Compressible Isentropic Navier–Stokes Equations in Dimensions Three

Abstract

Except a closed set \({\mathfrak {S}}^c\) with zero parabolic Hausdorff measure, the weak limit \((\rho ,\mathbf{u})\) of approximate solutions is a renormalized weak solution with finite energy of three dimensional compressible Navier–Stokes equations for \(\gamma \in (6/5, 3/2]\) as constructed by Lions and Feireisl et al. in the Leray sense. The key novelty of the paper is the improved integrability of pressure by localization, which is based on the faster decay of the gradient of velocity and the higher integrability of the Riesz potentials of both density and momentum.

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Notes

  1. 1.

    In dimensions three, the standard values of the adiabatic exponent \(\gamma \) are 5/3 for the monoatomic gas, between 11/9 and 7/5 for the diatomic gas, and between 1 and 4/3 for the polyatomic gas.

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Acknowledgements

This work was partially supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11332216, 11300417 and 11301919). The author would like to thank the anonymous referees for suggestions which improve the quality of redaction.

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Correspondence to Xianpeng Hu.

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Communicated by P.-L. Lions.

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Hu, X. Weak Solutions for Compressible Isentropic Navier–Stokes Equations in Dimensions Three. Arch Rational Mech Anal 242, 1907–1945 (2021). https://doi.org/10.1007/s00205-021-01716-6

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