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We prove the compactness of solutions of general fourth order elliptic equations which areL  1L^{1}   -perturbations of the  Q -curvature equation on compact Riemannian 4-manifolds. Consequently, we prove the global existence and convergence of the  Q -curvature flow on a generic class of Riemannian 4-manifolds. As a by-product, we give a positive answer to an open question by A. Malchiodi [J. reine angew. Math. 594 (2006), 137–174] on the existence of bounded Palais–Smale sequences for the  Q -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