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A Pressure Associated with a Weak Solution to the Navier–Stokes Equations with Navier’s Boundary Conditions

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Abstract

We show that if \(\mathbf {u}\) is a weak solution to the Navier–Stokes initial–boundary value problem with Navier’s slip boundary conditions in \(Q_T:=\Omega \times (0,T)\), where \(\Omega \) is a domain in \({{\mathbb {R}}}^3\), then an associated pressure p exists as a distribution with a certain structure. Furthermore, we also show that if \(\Omega \) is a “smooth” domain in \({{\mathbb {R}}}^3\) then the pressure is represented by a function in \(Q_T\) with a certain rate of integrability. Finally, we study the regularity of the pressure in sub-domains of \(Q_T\), where \(\mathbf {u}\) satisfies Serrin’s integrability conditions.

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Acknowledgements

The authors have been supported by the Academy of Sciences of the Czech Republic (RVO 67985840) and by the Grant Agency of the Czech Republic, Grant No. 17-01747S.

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Authors and Affiliations

  1. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67, Prgaue 1, Czech Republic

    Jiří Neustupa, Šárka Nečasová & Petr Kučera

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  1. Jiří Neustupa
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  2. Šárka Nečasová
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  3. Petr Kučera
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Correspondence to Jiří Neustupa.

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Neustupa, J., Nečasová, Š. & Kučera, P. A Pressure Associated with a Weak Solution to the Navier–Stokes Equations with Navier’s Boundary Conditions. J. Math. Fluid Mech. 22, 37 (2020). https://doi.org/10.1007/s00021-020-00500-y

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