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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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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DOI: https://doi.org/10.1007/s11425-022-2085-3
Keywords
- inviscid limit
- Navier-Stokes equations
- Euler equations
- weak solutions
- bounded domain
- Kato-type criterion
- Onsager’s regularity