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A Kato-Type Criterion for Vanishing Viscosity Near Onsager’s Critical Regularity

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Abstract

We consider a vanishing viscosity sequence of weak solutions for the three-dimensional Navier–Stokes equations of incompressible fluids in a bounded domain. In a seminal paper (Kato in Seminar on nonlinear partial differential equations, Springer, New York, 1983), Kato showed that for sufficiently regular solutions, the vanishing viscosity limit is equivalent to having vanishing viscous dissipation in a boundary layer of width proportional to the viscosity. We prove that Kato’s criterion holds for the Hölder continuous solutions with the regularity index arbitrarily close to Onsager’s critical exponent through a new boundary layer foliation and a global mollification technique.

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Acknowledgements

The work of R. M. Chen is partially supported by National Science Foundation under grants DMS-1907584 and DMS-2205910. The work of Z. Liang is partially supported by the fundamental research funds for central universities (JBK 2202045). The work of D. Wang is partially supported by the National Science Foundation under grants DMS-1907519 and DMS-2219384. The authors are very grateful to the anonymous referee for the valuable comments and suggestions, and for pointing out the references [37, 38].

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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

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  1. Robin Ming Chen
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  2. Zhilei Liang
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  3. Dehua Wang
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Correspondence to Robin Ming Chen.

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Chen, R.M., Liang, Z. & Wang, D. A Kato-Type Criterion for Vanishing Viscosity Near Onsager’s Critical Regularity. Arch Rational Mech Anal 246, 535–559 (2022). https://doi.org/10.1007/s00205-022-01822-z

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