Abstract：

In two dimensional incompressible flows, the vorticity is being transported by the flow conserving all the L^p norms. Still, the support of the vorticity could stretch in time and we review known results on maximal stretching rates. We consider similar problems for the three dimensional incompressible fluids, under the axisymmetric and swirl free assumptions. While the vorticity equation looks similar as in 2D, the vorticity can exhibit very different dynamical behavior, and the vorticity support confinement problem is linked with vorticity growth. We present our recent result which gives the optimal growth upper bound of t^4/3 for the vorticity maximum, which confirms a prediction of Childress in 2009. This is joint work with Deokwoo Lim.