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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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Keywords 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. Mathematical Biosciences and Engineering, 2020, 17(1): 514-537. doi: 10.3934/mbe.2020028


  • 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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