Sensitivity equations for measure-valued solutions to transport equations

Azmy S. Ackleh; Nicolas Saintier; Jakub Skrzeczkowski; Azmy S. Ackleh; Nicolas Saintier; Jakub Skrzeczkowski

doi:10.3934/mbe.2020028

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 1: 514-537. doi: 10.3934/mbe.2020028

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Sensitivity equations for measure-valued solutions to transport equations

1.
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504, USA
2.
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires (1428) Pabellón I-Ciudad Universitaria-Buenos Aires, Argentina
3.
Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

Received: 28 July 2019 Accepted: 24 September 2019 Published: 17 October 2019

We consider the following transport equation in the space of bounded, nonnegative Radon measures $\mathcal{M}^+(\mathbb{R}^d)$: $ \partial_t\mu_t + \partial_x(v(x) \mu_t) = 0. $ We study the sensitivity of the solution $\mu_t$ with respect to a perturbation in the vector field, $v(x)$. In particular, we replace the vector field $v$ with a perturbation of the form $v^h = v_0(x) + h v_1(x)$ and let $\mu^h_t$ be the solution of $ \partial _t\mu^h_t + \partial_x(v^h(x)\mu^h_t) = 0. $ We derive a partial differential equation that is satisfied by the derivative of $\mu^h_t$ with respect to $h$, $\partial_h(\mu_t^h)$. We show that this equation has a unique very weak solution on the space $Z$, being the closure of $\mathcal{M}(\mathbb{R}^d)$ endowed with the dual norm $(C^{1, \alpha}(\mathbb{R}^d))^*$. We also extend the result to the nonlinear case where the vector field depends on $\mu_t$, i.e., $v = v[\mu_t](x)$.
- transport equations,
- space of Radon measures,
- differentiability of solutions,
- very weak solutions
Citation: Azmy S. Ackleh, Nicolas Saintier, Jakub Skrzeczkowski. Sensitivity equations for measure-valued solutions to transport equations[J]. Mathematical Biosciences and Engineering, 2020, 17(1): 514-537. doi: 10.3934/mbe.2020028

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Abstract

We consider the following transport equation in the space of bounded, nonnegative Radon measures $\mathcal{M}^+(\mathbb{R}^d)$: $ \partial_t\mu_t + \partial_x(v(x) \mu_t) = 0. $ We study the sensitivity of the solution $\mu_t$ with respect to a perturbation in the vector field, $v(x)$. In particular, we replace the vector field $v$ with a perturbation of the form $v^h = v_0(x) + h v_1(x)$ and let $\mu^h_t$ be the solution of $ \partial _t\mu^h_t + \partial_x(v^h(x)\mu^h_t) = 0. $ We derive a partial differential equation that is satisfied by the derivative of $\mu^h_t$ with respect to $h$, $\partial_h(\mu_t^h)$. We show that this equation has a unique very weak solution on the space $Z$, being the closure of $\mathcal{M}(\mathbb{R}^d)$ endowed with the dual norm $(C^{1, \alpha}(\mathbb{R}^d))^*$. We also extend the result to the nonlinear case where the vector field depends on $\mu_t$, i.e., $v = v[\mu_t](x)$.

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