# American Institute of Mathematical Sciences

October  2019, 12(5): 1093-1108. doi: 10.3934/krm.2019041

## Differentiability in perturbation parameter of measure solutions to perturbed transport equation

 1 Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland 2 Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, Netherlands 3 Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Received  November 2018 Revised  April 2019 Published  July 2019

We consider a linear perturbation in the velocity field of the transport equation. We investigate solutions in the space of bounded Radon measures and show that they are differentiable with respect to the perturbation parameter in a proper Banach space, which is predual to the Hölder space $\mathcal{C}^{1+\alpha}( {\mathbb{R}^d})$. This result on differentiability is necessary for application in optimal control theory, which we also discuss.

Citation: Piotr Gwiazda, Sander C. Hille, Kamila Łyczek, Agnieszka Świerczewska-Gwiazda. Differentiability in perturbation parameter of measure solutions to perturbed transport equation. Kinetic & Related Models, 2019, 12 (5) : 1093-1108. doi: 10.3934/krm.2019041
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