Profile decompositions for critical Lebesgue and Besov space embeddings

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. In this article, we characterize the causes for the lack of compactness in certain embeddings of Banach spaces. In particular, one can re-write a bounded sequence in such a way that it is obvious why there fails to be a convergent subsequence (even in the weaker space). Once the defects of compactness are thus identified, it is often possible to isolate them and regain some aspects of compactness. This useful tool for re-writing bounded sequences is known as a “decomposition into profiles” (typically after passing to a subsequence), the “profiles” being the typical obstacle to compactness – specifically, norm-invariant transformations of fixed non-zero elements of the space. Since these elements are fixed (for the whole sequence) one can think of them as “limits” which can replace the need for an actual convergent subsequence. This is particularly useful in the study of nonlinear partial differential equations, where the natural setting is often that of such Banach spaces, and the convergence of certain sequences corresponds to establishing the existence of certain solutions to the equation being considered. In fact, it is primarily for this purpose that such decompositions have historically been established.