Differentiability in perturbation parameter of measure solutions to perturbed transport equation

Piotr Gwiazda; Sander C. Hille; Kamila Łyczek; Agnieszka Świerczewska-Gwiazda

doi:10.3934/krm.2019041

Article Contents

2019, Volume 12, Issue 5: 1093-1108. Doi: 10.3934/krm.2019041

This issue Previous Article The discrete unbounded coagulation-fragmentation equation with growth, decay and sedimentation Next Article Kinetic methods for inverse problems

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

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Abstract

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.

Keywords:

Transport equation in the space of Radon measures,
bounded Radon measures,
differentiability with respect to the parameter.

Mathematics Subject Classification: Primary: 35Q93; Secondary: 28A33.

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References

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