Sensitivity equations for measure-valued solutions to transport equations

Abstract

We consider the following transport equation in the space of bounded, nonnegative Radon measures M+(Rd): θtμt + θx(v(x)μt) = 0: We study the sensitivity of the solution μt 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 vh = v0(x) + hv1(x) and let μh t be the solution of θtμh t + θx(vh(x)μh t) = 0: We derive a partial differential equation that is satisfied by the derivative of μh t with respect to h, θh(μh t). We show that this equation has a unique very weak solution on the space Z, being the closure of M(Rd) endowed with the dual norm (C1,α(Rd))*. We also extend the result to the nonlinear case where the vector field depends on μt, i.e., v = v[μt](x).