We will consider regularity criteria for the Navier–Stokes equation in mixed Lebesgue
sum spaces. In particular, we will prove regularity criteria that only require control of
the velocity, vorticity, or the positive part of the second eigenvalue of the strain
matrix, in the sum space of two scale-critical spaces. This represents a significant step
forward, because each sum-space regularity criterion covers a whole family of
scale-critical regularity criteria in a single estimate. In order to show this, we
will also prove a new inclusion and inequality for sum spaces in families of
mixed Lebesgue spaces with a scale invariance that is also of independent
interest.
