| Abstract: |
| We give an overview on the sufficient condition for the regularity of solutions to the evolution
Navier-Stokes equations known in the literature as Prodi-Serrin condition.
H.-O.~Bae and H.J.~Choe proved in 1997 that, in the whole space $I\!\!R^3,$ it is sufficient
that two components of the velocity satisfy the above condition in order to
guarantee the regularity of solutions. In 2017, Beir\~ao~da~Veiga extended this result to the
half-space case $I\!\!R^n_+$ under slip boundary conditions by assuming that the velocity components
\emph{parallel} to the boundary enjoy the above condition. It remained open whether the flat
boundary geometry is essential. In this talk, we elaborate on the proof that, under physical slip boundary
conditions imposed in cylindrical boundaries, the result still holds. This proof has been subject
of the 2018/19 joint work\newline
{\textit{\phantom{somespace}On a Two Components Condition\newline
\phantom{somespace}for Regularity of the 3D Navier-Stokes Equations\newline
\phantom{somespace}under Physical Slip Boundary Conditions\newline
\phantom{somespace}on Non-Flat Boundaries}}\newline
with H.~Beir\~ao~da~Veiga and J.~Bemelmans. |
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