I consider a riemannian manifold (M,g) with a fiber bundle E equipped by one parameter family of connections ∇t. The curvature of ∇t is R(∇t) and the codifferential is δ∇t=∗∘d∇t∘∗, with ∗ the Hodge operator and d∇t the differential of the connection on the forms Λ∗(M)⨂End(E). The Yang-Mills flow is then defined by the formula:
∂∂t(∇t)=δ∇t(R(∇t))
I have in each term of the equality a 1-form with values in the endomorphisms of E. The fixed points of the flow are the connections with harmonic curvature (as the curvature is closed). They are the instantons in physics for SU(2)-connections over the space-time manifold.
Is such a flow well-defined for a short interval of the time near the initial connection? When is the flow convergent?