Differentiability of measure solutions to the nonlinear transport equation

We consider the nonlinear transport equation in the space of bounded Radon measures. Previous results concerning this type of equation include well-posedness and Lipschitz dependence of the solution (on the initial condition and model ingredients). However, these results do not allow to analyze the differentiability of solutions with respect to a perturbation of the system. Thus, a new setting is necessary. We have shown that solutions are differentiable with respect to the perturbing parameter in a proper space, which is predual to (C^{1+\alpha}). The knowledge of differentiability is necessary for various applications, e.g. in the optimal control theory.