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 prob-lem,which states a strong solution exists globally,provided that the velocity satisfies serrin's condition and the L_t~?L_x~?-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 im-portant quantity under the assumption that the deformation gradient is upper bounded,whose divergence can be viewed as the effective viscous flux.