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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

Special Issues: Mathematical Modeling with Measures

We consider the following transport equation in the space of bounded, nonnegative Radon measures $\mathcal{M}^+(\mathbb{R}^d)$:$$ ∂_t\mu_t + ∂_x(v(x) \mu_t) = 0.$$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 $$ ∂_t\mu^h_t + ∂_x(v^h(x)\mu^h_t) = 0.$$We derive a partial differential equation that is satisfied by the derivative of $\mu^h_t$ with respect to $h$, $∂artial_h(\mu_t^h)$. 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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References

1. J. Smoller, Shock waves and reaction diffusion equations, volume 258. Springer Science & Business Media, 2012.

2. B. Perthame, Transport equations in biology, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007.

3. L. Pareschi and G. Toscani, Interacting multiagent systems: kinetic equations and Monte Carlo methods, OUP Oxford, 2013.

4. M. Pérez-Llanos, J. P. Pinasco, N. Saintier, et al., Opinion formation models with heterogeneous persuasion and zealotry, SIAM J. Math. Anal., 50 (2018), 4812-4837.

5. L. Pedraza, J. P. Pinasco and Saintier, Measure-valued opinion dynamics, submitted, 2019.

6. F. Camilli, R. De Maio and A. Tosin, Transport of measures on networks, Netw. Heterog. Media, 12 (2017), 191-215.

7. F. Camilli, R. De Maio and A. Tosin, Measure-valued solutions to nonlocal transport equations on networks, J. Differ. Equations, 264 (12), 7213-7241.

8. S. Cacace, F. Camilli, R. De Maio, et al., A measure theoretic approach to traffic flow optimisation on networks, Eur. J. Appl. Math., (2018), 1-23.

9. J. A. Cañizo, J. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics, Acta Appl. Math., 123 (2013), 141-156.

10. M. Di Francesco and S. Fagioli, Measure solutions for non-local interaction pdes with two species, Nonlinearity, 26 (2013), 2777.

11. J. A. Carrillo, R. M. Colombo, P. Gwiazda, et al., Structured populations, cell growth and measure valued balance laws, J. Differ. Equations, 252 (2012), 3245-3277.

12. J. H. M. Evers, S. C. Hille and A. Muntean, Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, J. Differ. Equations, 259 (2015), 1068-1097.

13. K. Adoteye, H. T. Banks and K. B. Flores, Optimal design of non-equilibrium experiments for genetic network interrogation, Appl. Math. Lett., 40 (2015), 84-89.

14. M. Burger, Infinite-dimensional optimization and optimal design, 2003.

15. H. T. Banks and K. Kunisch, Estimation techniques for distributed parameter systems, Birkhäuser Verlag, Basel, 1989.

16. A. S. Ackleh, J. Carter, K. Deng, et al., Fitting a structured juvenile-adult model for green tree frogs to population estimates from capture-mark-recapture field data, Bull. Math. Biol., 74 (2012), 641-665.

17. M. T. Wentworth, R. C. Smith and H. T. Banks, Parameter selection and verification techniques based on global sensitivity analysis illustrated for an hiv model, SIAM-ASA J. Uncertain., 4 (2016), 266-297.

18. A. S. Ackleh, X. Li and B. Ma, Parameter estimation in a size-structured population model with distributed states-at-birth, In IFIP Conference on System Modeling and Optimization, pages 43-57. Springer, 2015.

19. A. S. Ackleh and R. L. Miller, A model for the interaction of phytoplankton aggregates and the environment: approximation and parameter estimation, Inverse Probl. Sci. En., 26 (2018), 152-182.

20. J. A. Canizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Mod. Meth. Appl. S., 21 (2011), 515-539.

21. S. Maniglia, Probabilistic representation and uniqueness results for measure-valued solutions of transport equations, J. Math. Pures Appl., 87 (2007), 601-626.

22. P. Gwiazda, T. Lorenz and A. Marciniak-Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, J. Differ. Equations, 248 (2010), 2703-2735.

23. P. Gwiazda, S. C. Hille, K. Łyczek, et al., Differentiability in perturbation parameter of measure solutions to perturbed transport equation, arXiv preprint arXiv:1806.00357, 2018.

24. J. Skrzeczkowski, Measure solutions to perturbed structured population models-differentiability with respect to perturbation parameter, arXiv preprint arXiv:1812.01747, 2018.

25. C. Villani, Topics in optimal transportation, Springer Texts in Statistics. Springer, New York, 2006.

26. K. B. Athreya and S. N. Lahiri, Measure theory and probability theory, Springer Texts in Statistics. Springer, New York, 2006.

27. L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, second edition, 2008.

28. H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext. Springer, New York, 2011.

29. L. Székelyhidi, Jr. From isometric embeddings to turbulence, In HCDTE lecture notes. Part Ⅱ. Nonlinear hyperbolic PDEs, dispersive and transport equations, volume 7 of AIMS Ser. Appl. Math., page 63. Am. Inst. Math. Sci. (AIMS), Springfield, MO, 2013.

30. L. C. Evans, Partial differential equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second edition, 2010.

31. P. Gwiazda, J. Jabłoński, A. Marciniak-Czochra, et al., Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded lipschitz distance, Numer. Meth. Part. D. E., 30 (2014), 1797-1820.    

32. J. A. Carrillo, P. Gwiazda and A. Ulikowska, Splitting-particle methods for structured population models: convergence and applications, Math. Mod. Meth. Appl. S., 24 (2014), 2171-2197.

33. R. M. Dudley, Convergence of Baire measures, Studia Math., 27 (1966), 251-268.

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