Global Smooth Solutions With Exponential Growth Of The 2D Inviscid Boussinesq Equations Without Heat Conduction And The 3D Axisymmetric Incompressible Euler Equations On Smooth Domains

For the two dimensional inviscid Boussinesq equations without heat conduc-tion,we find solutions with derivatives of exponential growth on a class of smooth unbounded regions.Along the way,stationary solutions of the two dimensional incompressible Euler equations on various smooth unbounded domains are discov-ered.Exploiting the similarities between the two dimensional Boussinesq system and the three dimensional Euler system,solutions of the latter with derivatives of exponential growth on a class of smooth unbounded axisymmetric domains are obtained.It is well known that the global regularity of the two dimensional inviscid Boussinesq system without heat conduction and the three dimensional incom-pressible Euler system are open problems.That is,it is still unknown whether their local smooth solutions can definitly be extended to be global solutions.The growth of smooth solutions can be regarded as an extension to this problem.If a solution grows extremely fast and attains infinity in finite time,then the system is not globally regular.In the literature,Chae,Constantin and Wu have obtained solutions of exponential growth on a planar domain with corner for the two di-mensional inviscid Boussinesq system without heat conduction.For the three dimensional Euler system,there are also smooth solutions of exponential growth on R~3.In this thesis we find smooth solutions of exponential growth on various smooth unbounded domains for both systems.The stationary solutions to the two dimensional Euler system on various smooth domains are also of interest,as there are not many solutions with explicit formulas,and they are usually on nice symmetric domains.