A Blow-up Criterion For Classical Solutions To The3-D Inhomogeneous Incompressible Navier-stokes Equations

In this paper, we prove a blow-up criterion in terms of the L2(L∞)-norm of the gradient of the velocity for classical solutions to the three dimensional inhomogeneous incompressible Navier-Stokes equations. It may be helpful in investigating the global existence of classical solutions, the uniqueness of weak solutions and the manner in which classical solutions blow up if they do. For the density-dependent Navier-Stokes equations, Cho and Kim (2004) have derived a blow-up criterion for strong solutions in terms of the L∞(L2)-norm of the gradient of the velocity and the L∞(Lq)-norm of the gradient of the density for any q>3. For the equations in this paper, Kim (2006) has derived a blow-up criterion for strong solutions in terms of the Ls(LT)-norm of the velocity for any (r, s) with2/s+3/r=1and3<r≤∞. For compressible Navier-Stokes equations, Huang-Li-Xin (2011) have used energy estimates to derive a blow-up condition for strong solutions by the L1(L∞)-norm of the gradient of the velocity. Here we use similar methods to prove the main result in this paper.