Characterization of blowup for the Navier-Stokes equations using vector potentials

• Main Author: Koji Ohkitani
• Format: Article
• Language: English
• Published: AIP Publishing LLC 2017-01-01
• Series: AIP Advances
• Online Access: http://dx.doi.org/10.1063/1.4975406

We characterize a possible blowup for the 3D Navier-Stokes on the basis of dynamical equations for vector potentials 𝑨. This is motivated by a known interpolation ∥𝑨∥BMO≤∥𝒖∥L3, together with recent mathematical results. First, by working out an inversion formula for singular integrals that appear in the governing equations, we derive a criterion using the nonlinear term of 𝑨 as ∫0t∗∥∂𝑨∂t−ν△𝑨∥L∞dt=∞ for a blowup at t∗. Second, for a particular form of a scale-invariant singularity of the nonlinear term we show that the vector potential becomes unbounded in its L∞ and BMO norms. Using the stream function, we also consider the 2D Navier-Stokes equations to seek an alternative proof of their known global regularity. It is not yet proven that the BMO norm of vector potentials in 3D (or, the stream function in 2D) serve as a blow up criterion in more general cases.