Global Regularity Of Incompressible Boussinesq Equations

In this paper,we discuss the global regularity of the incompressible Boussinesq equations without heat conduction.Using a method Chae used to discuss the three dimensional incompressible Euler equations,we obtain a sufficient condition for the local smooth solution of the two or three dimensional inviscid Boussinesq equations without heat conduction to blow up along particle trajectories.Using a method that Chae used in discussing the three dimensional incompressible Euler and Navier-Stokes equations,we use quantities similar to the enstrophy to give a bolw-up condition for local smooth solutions of the two and three dimensional Boussinesq equations without heat conduction.In the two dimension case,the equations is known to be globally regular when the viscosity is greater than zero.