Abstract
It is proved that a distributional solution u to the stationary Navier–Stokes equations in a bounded domain \(\Omega \) of \({\mathbb R}^n\) \((n>2)\) is regular, provided its norm in the weak-\(L^n(\Omega )\) space is small.
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Notes
In indicial notation \(({\mathrm{div}}\,(u\otimes u))_i=\partial _j(u_iu_j)\) so that \({\mathrm{div}}\,(u\otimes (u-\kappa ))\), for every constant vector \(\kappa \).
To a distributional [resp. weak] solution u to (1.1) in \(\Omega \) is associated in a standard way a pressure field \(p\in W^{-1,q}(\Omega )\) [resp. \(p\in L^q(\Omega )\)] in such a way that the pair (u, p) is a solution to (1.1) in the sense of distributions [3]. If u is regular, then from the equation \(\Delta p+\partial _{ij}(u_iu_j)=0\) it follows that also p is regular and (u, p) is a classical solution to (1.1) in \(\Omega \).
Here and in sequel, at least for interior regularity, we can require that the hypotheses on u are satisfied locally. The symbol c will be reserved to denote positive constants whose numerical value is not essential to our purposes.
When it will be necessary, we use the symbol \(\mathscr {B}_R(x)\) to mean that the ball or the half-ball is centered at x.
This inequality follows from the fact that the function \(\int _{B_R}|u-t|^2\) of t has the minimum at \(t=u_R\).
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Acknowledgements
The work of M.Korobkov is supported by Mathematical Center in Akademgorodok, the agreement with Ministry of Science and High Education of the Russian Federation Number 075-15-2019-1613. The research of K. Pileckas was funded by the Grant No. S-MIP-17-68 from the Research Council of Lithuania. The research of R. Russo has been supported by the funding program VALERE of Università degli Studi della Campania “Luigi Vanvitelli”, Italy.
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Korobkov, M.V., Pileckas, K. & Russo, R. A Simple Proof of Regularity of Steady-State Distributional Solutions to the Navier–Stokes Equations. J. Math. Fluid Mech. 22, 55 (2020). https://doi.org/10.1007/s00021-020-00517-3
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DOI: https://doi.org/10.1007/s00021-020-00517-3