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arXiv · 2007.01631

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

Piotr Gwiazda·Sander C. Hille·Kamila Łyczek
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Abstract

We consider a perturbation in the non-linear transport equation on measures i.e. both initial condition $μ_0$ and the solution $μ_t^h$ 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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Piotr Gwiazda, Sander C. Hille, Kamila Łyczek. 2020-07-03. Existence and differentiability in parameter of the measure solution to a perturbed non-linear transport equation. https://arxiv.org/abs/2007.01631

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