Chae and Wolf recently constructed discretely self-similar solutions
to the Navier–Stokes equations for any discretely self-similar data in
.
Their solutions are in the class of local Leray solutions with projected pressure and
satisfy the “local energy inequality with projected pressure”. In this note, for the
same class of initial data, we construct discretely self-similar suitable weak
solutions to the Navier–Stokes equations that satisfy the classical local energy
inequality of Scheffer and Caffarelli–Kohn–Nirenberg. We also obtain an explicit
formula for the pressure in terms of the velocity. Our argument involves a new
purely local energy estimate for discretely self-similar solutions with data in
and
an approximation of divergence-free, discretely self-similar vector fields in
by divergence-free, discretely self-similar elements of
.
