SMALL SCALE FORMATION FOR THE 2-DIMENSIONAL BOUSSINESQ EQUATION

Abstract

We study the 2-dimensional incompressible Boussinesq equations without thermal diffusion, and aim to construct rigorous examples of small scale formations as time goes to infinity. In the viscous case, we construct examples of global smooth solutions satisfying (Formula presented.) for some α > 0. For the inviscid equation in the strip, we construct examples satisfying (Formula presented.) and (Formula presented.) during the existence of a smooth solution. These growth results hold for a broad class of initial data, where we only require certain symmetry and sign conditions. As an application, we also construct solutions to the 3-dimensional axisymmetric Euler equation whose velocity has infinite-in-time growth.