We consider a class of parabolic systems and equations in divergence form modeled by the evolutionary -Laplacean system
and provide -bounds for the spatial gradient of solutions via nonlinear potentials of the right hand side datum . Such estimates are related to those obtained by Kilpeläinen and Malý  in the elliptic case. In turn, the potential estimates found imply optimal conditions for the boundedness of in terms of borderline rearrangement invariant function spaces of Lorentz type. In particular, we prove that if then , where is the space dimension, and this gives the borderline case of a result of DiBenedetto ; a significant point is that the condition is independent of . Moreover, we find explicit forms of local a priori estimates extending those from  valid for the homogeneous case .
Cite this article
Tuomo Kuusi, Giuseppe Mingione, Potential estimates and gradient boundedness for nonlinear parabolic systems. Rev. Mat. Iberoam. 28 (2012), no. 2, pp. 535–576DOI 10.4171/RMI/684