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

TL;DR

This paper proves the existence and differentiability of measure solutions to a perturbed non-linear transport equation with respect to perturbations in the velocity and scalar functions, extending previous linear results.

Contribution

It establishes the differentiability of measure solutions to a non-linear transport equation under perturbations, expanding prior linear case results.

Findings

Solution differentiability with respect to perturbation parameter h

Extension from linear to non-linear transport equations

Solution derivative exists in a Banach space

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.