Well-posedness of the equation for the three-form field in the eleven dimensional supergravity

We analyze the equations for the three-form field - a system of semi-linear gauge-invariant wave equations which arises in the theory of eleven dimensional supergravity. We prove that the Cauchy problem is well-posed globally in time for the fixed-gauge version of the equation for a small compactly supported smooth data. We employ the method of Klainerman vector fields along with a finer analysis of the nonlinearity to establish an integrable decay in the energy estimate.