On the smallness of mean oscillations and regularity of weak solutions to regular/degenerate strongly coupled parabolic systems

It will be established that the mean oscillation of bounded weak solutions to strongly coupled parabolic systems is small in small balls. If the systems are regular elliptic then their bounded weak solutions are Hölder continuous. Further assumptions on the systems will even prove that these solutions exist globally. Weak solutions to degenerate systems of porous media type are also studied.