Diffeomorphisms and nonlinear heat flows

We show that the gradient flow ${\mathbf u}$ on $L^2$ generated by the energy functional $I[{\mathbf u}] := \int_U \Phi(\det D{\mathbf u}) \, dx$ for vector-valued mappings is in some sense "integrable," meaning that (i) the inverse Jacobian $\beta := (\det D{\mathbf u})^{-1}$ satisfies a scalar nonlinear diffusion equation, and (ii) we can recover ${\mathbf u}$ by solving an ODE determined by $\beta$.