  We establish existence of solutions in a scale of classes weaker than the
finite energy Leray class and stronger than the infinite energy
Lemari\'e-Rieusset class. The new classes are based on the $L^2$ Wiener amalgam
spaces. Solutions in the classes closer to the Leray class are shown to satisfy
some properties known in the Leray class but not the Lemari\'e-Rieusset class,
namely eventual regularity and long time estimates on the growth of the local
energy. In this sense, these solutions bridge the gap between Leray's original
solutions and Lemari\'e-Rieusset's solutions and help identify scalings at
which certain properties may break down.
