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We prove the compactness of solutions to general fourth order ellipticequations which are L^1-perturbations of the Q-curvature equation on compactRiemannian 4-maniods. Consequently, we prove the global existence andconvergence of the Q-curvature flow on a generic class of Riemannian4-manifolds. As a by product, we give a positive answer to an open question byA. Malchiodi on the existence of bounded Palais-Smale sequences for theQ-curvature problem when the Paneitz operator is positive with trivial kernel.

Compactness Properties for Geometric Fourth order Elliptic equations with Application to the Q-curvature Flow