  Motived by the heat flow and bubble analysis of biharmonic mappings, we study
further regularity issues of the fourth order Lamm-Riviere system
$$\Delta^{2}u=\Delta(V\cdot\nabla u)+{\rm div}(w\nabla
u)+(\nabla\omega+F)\cdot\nabla u+f$$ in dimension four, with an inhomogeneous
term $f$ which belongs to some natural function space. We obtain optimal higher
order regularity and sharp Holder continuity of weak solutions. Among several
applications, we derive weak compactness for sequences of weak solutions with
uniformly bounded energy, which generalizes the weak convergence theory of
approximate biharmonic mappings.
