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
[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. |