A Contribution to the Theory of Regularity of a Weak Solution to the Navier–Stokes Equations via One Component of Velocity and Other Related Quantities

Abstract

We deal with a suitable weak solution \((\mathbf {v},p)\) to the Navier–Stokes equations in \(\Omega \times (0,T)\), where \(\Omega \) is a domain in \({\mathbb R}^3\), \(T>0\) and \(\mathbf {v}=(v_1,v_2,v_3)\). We show that the regularity of \((\mathbf {v},p)\) at a point \((\mathbf {x}_0, t_0)\in \Omega \times (0,T)\) is essentially determined by the Serrin-type integrability of the positive part of a certain linear combination of \(v_1^2,\, v_2^2,\, v_3^2\) and p in a backward neighborhood of \((\mathbf {x}_0,t_0)\). An appropriate choice of the coefficients in the linear combination leads to the Serrin-type condition on the positive part of the Bernoulli pressure \(\frac{1}{2}|\mathbf {v}|^2+p\) or the negative part of p (Theorem 1), or one component of \(\mathbf {v}\) (Theorem 2), etc.

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Correspondence to Jiří Neustupa.

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Communicated by G. P. Galdi

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Neustupa, J. A Contribution to the Theory of Regularity of a Weak Solution to the Navier–Stokes Equations via One Component of Velocity and Other Related Quantities. J. Math. Fluid Mech. 20, 1249–1267 (2018). https://doi.org/10.1007/s00021-018-0365-6

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Keywords

  • Navier–Stokes equations
  • weak solution
  • regularity

Mathematics Subject Classification

  • Primary 35Q30
  • Secondary 76D03
  • 76D05