Research article Special Issues

Sensitivity equations for measure-valued solutions to transport equations

  • 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., $v = v[\mu_t](x)$.

    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

    Related Papers:

  • 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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    [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. doi: 10.1002/num.21879
    [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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