A regularity criterion to the biharmonic map heat flow in ℜ ⁴

We consider the regularity problem under the critical condition to the biharmonic map heat flow from ℜ ⁴ to a smooth compact Riemannian manifold without boundary. Using Gagliardo-Nirenberg inequalities and delicate estimates, the Serrin type regularity criterion for the smooth solutions of biharmonic map heat flow is obtained without assuming a smallness condition on the initial energy. Our result improved the results of Lamm in 5 and 6 and generalized the results of Chang, Wang, Yang 1, Strzelecki 11 and Wang 13, 14 to non-stationary case.