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In this paper, we obtain interior Hölder continuity for solutions of the fourth‐order elliptic system Δ2u=Δ(V·∇u)+div(w∇u)+W·∇uformulated by Lamm and Rivière [Comm. Partial Differential Equations 33 (2008) 245–262]. Boundary continuity is also obtained under a standard Dirichlet or Navier boundary condition. We also use conservation law to establish a weak compactness result which generalizes a result of Rivière for the second‐order problem. [ABSTRACT FROM AUTHOR]