Profile decompositions for critical Lebesgue and Besov space embeddings

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.

30 pages

References in corpus (4)

• Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data
• Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system
• Scattering for H^1/2 bounded solutions to the cubic, defocusing NLS in 3 dimensions
• Minimal initial data for potential Navier-Stokes singularities

Cited by in corpus (11)

• Blow-up of critical Besov norms at a potential Navier-Stokes singularity
• A profile decomposition approach to the $L_t^{\infty }(L_x^3)$ Navier-Stokes regularity criterion
• Blow-up criteria for the Navier-Stokes equations in non-endpoint critical Besov spaces
• Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces
• About the possibility of minimal blow up for Navier-Stokes solutions with data in ${\dot{H}}^s(R^3)$
• On the stability in weak topology of the set of global solutions to the Navier-Stokes equations
• Dynamical Behavior for the Solutions of the Navier-Stokes Equation
• Blow-up criterion and examples of global solutions of forced Navier-Stokes equations
• On the lack of compactness on stratified Lie groups
• The lack of compactness in the Sobolev-Strichartz inequalities
• On interpolation of cocompact imbeddnings