Finite-time Blow-up Of Local Smooth Solutions Of The Boussinesq System On A Two-dimensional Domain With Cusps

For the Boussinesq system without viscosity and heat conduction on a two dimensional domain with cusps,we construct a class of local smooth solutions that blow up in finite time.For the 2D incompressible Euler equation,it is well-known that it is globally regular on R2 and smooth domains.However,on a domain with cusps,Kiselev and Zlatos construct local smooth solutions that blow up in finite time.For the Boussinesq system without viscosity and heat conductivity,the global regularity problem on these three kinds of domains are open.In this article,using the method of Kiselev and Zlatos,and in addition controlling the size of the region where the vorticity can change with respect to time,we construct local smooth solutions that blow up in finite time on a domain with cusps.