On the Hölder continuity of weak solutions to nonlinear parabolic systems in two space dimensions

summary:We prove the interior Hölder continuity of weak solutions to parabolic systems $$ \frac{\partial u^j}{\partial t}-D_\alpha a_j^\alpha(x,t,u,\nabla u)=0 \text{ in } Q \quad (j=1,\ldots,N) $$ ($Q=\Omega\times(0,T),\Omega\subset\Bbb R^2$), where the coefficients $a_j^\alpha(x,t,u,\xi)$ are measurable in $x$, Hölder continuous in $t$ and Lipschitz continuous in $u$ and $\xi$