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On Mixed Pressure-Velocity Regularity Criteria to the Navier-Stokes Equations in Lorentz Spaces

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Abstract

In this paper the authors derive regular criteria in Lorentz spaces for Leray-Hopf weak solutions υ of the three-dimensional Navier-Stokes equations based on the formal equivalence relation π ≅ ∣υ∣2, where π denotes the fluid pressure and υ denotes the fluid velocity. It is called the mixed pressure-velocity problem (the P-V problem for short). It is shown that if \({\pi \over {{{({e^{ - {{\left| x \right|}^2}}} + \left| v \right|)}^\theta }}} \in {L^p}\left( {0,T;{L^{q,\infty }}} \right)\), where 0 ≤ θ ≤ 1 and \({2 \over p} + {3 \over q} = 2 - \theta \), then {itυ} is regular on (0, {itT}]. Note that, if Ω is periodic, \({{e^{ - {{\left| x \right|}^2}}}}\) may be replaced by a positive constant. This result improves a 2018 statement obtained by one of the authors. Furthermore, as an integral part of the contribution, the authors give an overview on the known results on the P-V problem, and also on two main techniques used by many authors to establish sufficient conditions for regularity of the so-called Ladyzhenskaya-Prodi-Serrin (L-P-S for short) type.

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Authors and Affiliations

  1. Department of Mathematics, Pisa University, Pisa, Italy

    Hugo Beirão da Veiga

  2. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, China

    Jiaqi Yang

Authors
  1. Hugo Beirão da Veiga
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  2. Jiaqi Yang
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Corresponding authors

Correspondence to Hugo Beirão da Veiga or Jiaqi Yang.

Additional information

This work was supported by FCT (Portugal) under the project: UIDB/MAT/04561/2020 and the Fundamental Research Funds for the Central Universities under grant: G2019KY05114.

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Beirão da Veiga, H., Yang, J. On Mixed Pressure-Velocity Regularity Criteria to the Navier-Stokes Equations in Lorentz Spaces. Chin. Ann. Math. Ser. B 42, 1–16 (2021). https://doi.org/10.1007/s11401-021-0242-0

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  • DOI: https://doi.org/10.1007/s11401-021-0242-0

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