Global Smooth Solutions With Exponential Growth to 2D Inviscid Boussinesq Equations Without Heat Conduction and 3D Axisymmetric Incompressible Euler Equations on Smooth Domains

The growth of smooth solutions to 2D inviscid Boussinesq equations without heat conduction and the 3D axisymmetric Euler equations was investigated, to find regions where these systems have fast growing solutions. Through appropriately choosing the initial temperature and the velocity component, the Boussinesq system was decoupled into 2 parts. From the part involving only the vorticity, the vorticity and velocity can be solved and the smooth regions determined. From the part involving the temperature, one can see that the growth of temperature derivatives depends only on the velocity component. Through choosing that component appropriately, solutions with temperature derivatives of exponential growth were constructed on certain unbound smooth regions. The same method was applied to the axisymmetric Euler equations. Through choosing the radial velocity component appropriately, the system can be decoupled and one can ultimately find a class of smooth domains, and on them smooth global solutions of exponential growth. This investigation extends the results of Chae, Constantin and Wu on the inviscid Boussinesq system without heat conduction on a 2D cone to a class of smooth domains. Their method was also applied to the 3D axisymmetric Euler equations to obtain a similar result.