Blow-up criterion for the incompressible viscoelastic flows

In this paper, we are concerned with the incompressible viscoelastic flows in the periodic domain. We establish a Serrin-type blow-up criterion for 3-D periodic initial boundary problem, which states a strong solution exists globally, provided that the velocity satisfies Serrin's condition and the Lt∞Lx∞-norm of the deformation gradient are bounded. We also establish blow-up criterion in terms of the upper bound of the deformation gradient for 2-D periodic initial boundary problem. The main ingredient of the proof is a priori estimate for an important quantity under the assumption that the deformation gradient is upper bounded, whose divergence can be viewed as the effective viscous flux.