A conditional regularity result for p-harmonic flows

We prove an $\epsilon$-regularity result for a wide class of parabolic systems $$u_t-\text{div}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u)=B(u,\mathrm{\nabla }u)$$ with the right hand side $B$ growing like $\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p$. It is assumed that the solution $u(t,\cdot )$ is uniformly small in the space of functions of bounded mean oscillation. The crucial tool is provided by a sharp nonlinear version of the Gagliardo-Nirenberg inequality which has been used earlier in an elliptic context by T. Rivi`ere and the last named author.