Existence and differentiability in parameter of the measure solution to a perturbed non-linear transport equation

Abstract

We consider a perturbation in the non-linear transport equation on measures i.e. both initial condition ${\mu }_0$ and the solution ${\mu }_t^h$ are bounded Radon measures $\mathcal{M}({\mathbb{R}}^d)$. The perturbations occur in the velocity field and also in the right-hand side scalar function. It is shown that the solution is differentiable with respect to the perturbation parameter $h$ i.e. that derivative is an element of a proper Banach space. This result extends our previous result which considered the linear transport equation. The proof exploits approximation of the non-linear problem which is based on the study of the linear equation.

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