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.
Highlights
Analysis of perturbations in partial differential equation systems is an important issue
Previous considerations concerning the transport equation in the space of measures did not allow to analyse the differentiability of solutions with respect to a perturbation of the system [AGS08, Thi[03], PF14]
In this paper we consider solutions to a perturbed transport equation in the space of bounded Radon measures, denoted by M(Rd), where the perturbation is linear in the velocity field
Summary
Analysis of perturbations in partial differential equation systems is an important issue. Previous considerations concerning the transport equation in the space of measures did not allow to analyse the differentiability of solutions with respect to a perturbation of the system [AGS08, Thi[03], PF14]. In this paper we consider solutions to a perturbed transport equation in the space of bounded Radon measures, denoted by M(Rd), where the perturbation is linear in the velocity field. Let μht be the weak solution to problem (1.4) with velocity field defined by (1.3). We would like to argue why this result cannot be obtained in the space W 1,∞ with the flat metric (called bounded Lipschitz distance) – what is a natural setting to consider transport equation in the space of bounded Radon measures [PR16, PFM, GJMU14, CLM13, GM10].
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