  Profile decompositions for "critical" Sobolev-type embeddings are
established, allowing one to regain some compactness despite the non-compact
nature of the embeddings. Such decompositions have wide applications to the
regularity theory of nonlinear partial differential equations, and have
typically been established for spaces with Hilbert structure. Following the
method of S. Jaffard, we treat settings of spaces with only Banach structure by
use of wavelet bases. This has particular applications to the regularity theory
of the Navier-Stokes equations, where many natural settings are non-Hilbertian.
