REMARKS ON THE A PRIORI BOUND FOR THE VORTICITY OF THE AXISYMMETRIC NAVIER-STOKES EQUATIONS

We study the axisymmetric Navier-Stokes equations. In 2010, Loftus-Zhang used a refined test function and re-scaling scheme, and showed that vertical bar w(r) (x, t)vertical bar + |w(z) (r, t)vertical bar <= C/r(10), 0 < r <= 1/2. By employing the dimension reduction technique by Lei-Navas-Zhang, and analyzing w(r), w(z) and w(theta)/r on different hollow cylinders, we are able to improve it and obtain vertical bar w(r) (x, t)vertical bar + |w(z) (r, t)vertical bar <= C vertical bar lnr vertical bar/r(10), 0 < r <= 1/2.

Automated summary

Studying the axisymmetric Navier-Stokes equations led to the formulation of a new inequality through refined test functions and re-scaling schemes. Further improvement was achieved by employing dimension reduction techniques.