  Ansatze are constructed under which the solutions of 11D supergravity must be
stationary points of a parabolic flow on a Riemannian manifold $M^{10-p}$. This
parabolic flow turns out to be the Ricci flow coupled to a scalar field, a
$(3-p)$-form, and a $4$-form. This allows the introduction of techniques from
parabolic partial differential equations to the search of solutions to 11D
supergravity. As a first step, Shi-type estimates and criteria for the
long-time existence of the flow are established.
