$ \varepsilon $-regularity, harmonic maps vs harmonic heat flow

Question

Let $ \Omega\subset\mathbb{R}^n $ be a bounded domain with smooth boundary and $ (N,h)\subset\mathbb{R}^L $ is a smooth compact Riemannian manifold. Consider the local minimizer $ u\in W^{1,2}(\Omega,N) $, i.e. $$ \int_{B_r(x)}|\nabla u|^2\mathrm{d}y\leq\int_{B_r(x)}|\nabla v|^2\mathrm{d}y $$ for any $ v\in W^{1,2}(\Omega,N) $ such that $ v=u $ on $ \partial B_r(x) $ with $ B_r(x)\subset\Omega $. The famous Sochen-Uhlenbeck theory implies that there exists $ \varepsilon_0=\varepsilon_0(N)>0 $ such that if $$ R^{2-n}\int_{B_R(x)}|\nabla u|^2\mathrm{d}y\leq\varepsilon_0, $$ then $ u $ is smooth in $ B_{R/2}(x) $. For harmonic heat flows $$ \partial_tu-\Delta u=\Gamma(u)(\nabla u,\nabla u), $$ where $ \Gamma(u) $ is the second fundamental form of $ N $ at the point $ u $, there is also a $ \varepsilon $-regularity theory. However I find that such results are obtained for smooth solutions, i.e. for those solutions such that $ u\in C^{\infty}(\Omega\times(0,T)) $ with $ T>0 $. Can I obtain similar results for weak solutions of harmonic heat flows like harmonic maps?