A Blow-up Criterion For Strong Solutions Of The Density Dependent Incompressible Navier-stokes Equations

We consider the density dependent incompressible Navier-Stokes equations in bounded regions in Rn(c n=2,3), and obtain the following blow up criterion for a strong solution. If T*is the maximal existence time of the local strong solution, then‖d(u)‖L2(O,T*,L∞)=∞as t(?)T*, where d(u):=2/▽u+▽Tu is the deformation tensor, u is the velocity. The blow up criterion may be helpful in investigating the global existence of a strong solution, the uniqueness of weak solutions, and the manner in which strong solutions blow up if they do.Cho and Kim have derived the blow up criterion,‖▽ρ(t)‖Lq(O,T*,L∞)+‖▽-u(t)‖L2(O,T*,L∞)=∞as t/T*, where p is the density.We use energy estimates to prove the result, similar to the method Huang, Li and Xin used in proving the Beale-Kato-Majda blow up criterion for the strong solution to the compressible Navier-Stokes equations.