Profile decompositions for critical Lebesgue and Besov space embeddings

TL;DR

This paper develops profile decompositions for critical Lebesgue and Besov space embeddings using wavelet bases, enabling analysis of non-Hilbertian spaces relevant to nonlinear PDEs like Navier-Stokes.

Contribution

It extends profile decomposition techniques to Banach spaces with wavelet bases, broadening their applicability beyond Hilbert spaces.

Findings

Established profile decompositions for critical Lebesgue and Besov embeddings.

Applied wavelet-based methods to non-Hilbertian spaces.

Facilitated analysis of PDE regularity in non-Hilbertian contexts.

Abstract

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.