Generalized Ricci-Bourguignon flow

In this paper, we consider a kind of generalized Ricci-Bourguignon flow system, which is closely like the Ricci-Bourguignon flow and possesses a gradient form. We establish the existence and uniqueness of the solution to this flow on an $n$-dimensional closed Riemannian manifold. We introduce generalized Ricci-Bourguignon system soliton and give a condition to a gradient generalized Ricci-Bourguignon system soliton to be isometric to an Euclidean sphere. Then we give the evolution of some geometric structure of manifold along this flow and establish higher-derivative estimates for compact manifolds and the compactness theorem for this general Ricci-Bourguignon flow system on closed Riemannian manifolds.