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On the inviscid limit of the compressible Navier-Stokes equations near Onsager’s regularity in bounded domains

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Abstract

The viscous dissipation limit of weak solutions is considered for the Navier-Stokes equations of compressible isentropic flows confined in a bounded domain. We establish a Kato-type criterion for the validity of the inviscid limit for the weak solutions of the Navier-Stokes equations in a function space with the regularity index close to Onsager’s critical threshold. In particular, we prove that under such a regularity assumption, if the viscous energy dissipation rate vanishes in a boundary layer of thickness in the order of the viscosity, then the weak solutions of the Navier-Stokes equations converge to a weak admissible solution of the Euler equations. Our approach is based on the commutator estimates and a subtle foliation technique near the boundary of the domain.

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Acknowledgements

Robin Ming Chen was supported by National Science Foundation of USA (Grant No. DMS-1907584). Zhilei Liang was supported by the Fundamental Research Funds for the Central Universities (Grant No. JBK 2202045). Dehua Wang was supported by National Science Foundation of USA (Grant Nos. DMS-1907519 and DMS-2219384). Runzhang Xu was supported by National Natural Science Foundation of China (Grant No. 12271122).

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Authors and Affiliations

  1. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Robin Ming Chen & Dehua Wang

  2. School of Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China

    Zhilei Liang

  3. College of Mathematical Sciences, Harbin Engineering University, Harbin, 150001, China

    Runzhang Xu

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  1. Robin Ming Chen
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  4. Runzhang Xu
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Correspondence to Dehua Wang.

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Chen, R.M., Liang, Z., Wang, D. et al. On the inviscid limit of the compressible Navier-Stokes equations near Onsager’s regularity in bounded domains. Sci. China Math. 67, 1–22 (2024). https://doi.org/10.1007/s11425-022-2085-3

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