Several topics on the Navier-Stokes problem

Whether a 3D incompressible flow can spontaneously form a singularity, i.e. the global-in-time existence of the classical solution of the 3D Navier-Stokes equation, is still an open problem. This thesis provides several possible ways to explore regularity or criticality of the equation, with the notions of spatial intermittency and sparseness at scales as well as the harmonic analysis techniques regarding pointwise multipliers, singular integral operators, heat and Stokes semigroups in oscillation function spaces.