Sensitivity equations for measure-valued solutions to transport equations

We consider the following transport equation in the space of bounded, nonnegative Radon measures M+(R-d): partial derivative(t)mu(t )+partial derivative(x)(v(x)mu(t)) = 0. We study the sensitivity of the solution mu(1) with respect to a perturbation in the vector field, v(x). In particular, we replace the vector field v with a perturbation of the form V-h = v(0)(x) + h(v1)(x) and let mu(h)(t) be the solution of partial derivative(t)mu(h)(t)( )+partial derivative(x)(v(h)(x)mu(h)(t)) = 0. We derive a partial differential equation that is satisfied by the derivative of mu(h)(t) with respect to h,partial derivative(h)(mu(h)(t)). We show that this equation has a unique very weak solution on the space Z, being the closure of M(R-d) endowed with the dual norm (C-1,C-alpha(R-d))*. We also extend the result to the nonlinear case where the vector field depends on mu(t), i.e., v = v[mu(t)](x).