Abstract
In this note we improve the standard regularity of the dynamic part of the pressure in the Navier–Stokes system. Using the theory of elliptic equations with \(L^1\) right-hand side we prove that, in addition to be in \(L^2\), the dynamic pressure belongs to \(W^{1,\alpha }_{loc} \) with \(1<\alpha <\frac{n}{n-1}\), in case of Dirichlet boundary condition. For pressure boundary condition the dynamic pressure is proved to be in \(W^{1,\alpha } \). As a consequence, for the force \(\mathbf{f} \in L^q (\Omega )^n \) and \(q>n /2 \) the pressure turns out to be continuous.
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Friganović, S. A note on regularity of the pressure in the Navier–Stokes system. Ann Univ Ferrara 64, 361–370 (2018). https://doi.org/10.1007/s11565-018-0302-x
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DOI: https://doi.org/10.1007/s11565-018-0302-x