Gabriel Koch (Sussex)
Blow-up of critical spatial norms at any Navier-Stokes singular time
Abstract: Consider a Besov space X=X(R^3) which is invariant under the spatial
scaling of the Navier-Stokes equations and for which a local existence
theory for any initial datum in X has been developed. We show that for
any local Navier-Stokes solution u starting from an initial datum u_0 in X
and which (hypothetically) has a singularity at time T>0, the (spatial) X
norm of u(.,t) must become unbounded as t approaches T. This extends a
famous result of Escauriaza-Seregin-Sverak (2003) which establishes the
same result for X=L^3(R^3), which is continuously embedded in all of the
Besov spaces mentioned above. Our proof uses profile decompositions for
bounded sequences in Banach spaces as well as the related "critical
element" theory developed for and implemented in the setting of critical
dispersive and hyperbolic equations by C. Kenig and F. Merle. The program
for Navier-Stokes (the first parabolic setting in which the critical
element method was successfully implemented) was started in collaboration
with Carlos Kenig (Chicago) and completed in collaboration with Isabelle
Gallagher (Paris) and Fabrice Planchon (Nice). As a by-product of the
tools developed for this program, we were able to easily extend a recent
result of Rusin-Sverak regarding "minimal blow-up data" for Navier-Stokes
in critical spaces to all of the spaces mentioned above.