October  2019, 12(5): 1093-1108. doi: 10.3934/krm.2019041

Differentiability in perturbation parameter of measure solutions to perturbed transport equation

1. 

Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland

2. 

Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, Netherlands

3. 

Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Received  November 2018 Revised  April 2019 Published  July 2019

We consider a linear perturbation in the velocity field of the transport equation. We investigate solutions in the space of bounded Radon measures and show that they are differentiable with respect to the perturbation parameter in a proper Banach space, which is predual to the Hölder space $ \mathcal{C}^{1+\alpha}( {\mathbb{R}^d}) $. This result on differentiability is necessary for application in optimal control theory, which we also discuss.

Citation: Piotr Gwiazda, Sander C. Hille, Kamila Łyczek, Agnieszka Świerczewska-Gwiazda. Differentiability in perturbation parameter of measure solutions to perturbed transport equation. Kinetic and Related Models, 2019, 12 (5) : 1093-1108. doi: 10.3934/krm.2019041
References:
[1]

G. AlbiY.-P. ChoiM. Fornasier and D. Kalise, Mean field control hierarchy, Applied Mathematics and Optimization, 76 (2017), 93-135.  doi: 10.1007/s00245-017-9429-x.

[2]

G. AlbiM. Herty and L. Pareschi, Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems, Applied Mathematics and Computation, 187 (2019), 460-477.  doi: 10.1016/j.amc.2019.02.021.

[3]

H. Amann and J. Escher., Analysis II, Birkhäuser Verlag, Basel, 2008.

[4]

L. Ambrosio, M. Fornasier and M. Morandotti, Spatially inhomogeneous evolutionary games, arXiv: 1805.04027v1.

[5]

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, 2008.

[6]

N. Bellomo, P. Degond and E. Tadmor, Active Particles, Volume 1: Advances in Theory, Models, and Applications, Springer, Birkhäuser, 2017.

[7]

F. BerthelinP. DegondM. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Archive for Rational Mechanics and Analysis, 187 (2008), 185-220.  doi: 10.1007/s00205-007-0061-9.

[8]

G. A. BonaschiJ. A. CarrilloM. Di Francesco and M. A. Peletier, Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1{D}, ESAIM Control Optim. Calc. Var., 21 (2015), 414-441.  doi: 10.1051/cocv/2014032.

[9]

M. BonginiM. FornasierF. Rossi and F. Solombrino, Mean-field Pontryagin maximum principle, Journal of Optimization Theory and Applications, 175 (2017), 1-38.  doi: 10.1007/s10957-017-1149-5.

[10]

B. Bonnet and F. Rossi, The Pontryagin maximum principle in the Wasserstein space, Calculus of Variations and Partial Differential Equations, 58 (2019), Art. 11, 36pp. doi: 10.1007/s00526-018-1447-2.

[11]

Y. BrenierW. GangboG. Savaré and M. Westdickenberg, Sticky particle dynamics with interactions, Journal de Mathématiques Pures et Appliquées. Neuvième Série, 99 (2013), 577-617.  doi: 10.1016/j.matpur.2012.09.013.

[12]

J. A. CañizoJ. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics, Acta Applicandae Mathematicae, 123 (2013), 141-156.  doi: 10.1007/s10440-012-9758-3.

[13]

J. A. CañizoJ. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Mathematical Models and Methods in Applied Sciences, 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.

[14]

J. A. CarrilloR. M. ColomboP. Gwiazda and A. Ulikowska, Structured populations, cell growth and measure valued balance laws, Journal of Differential Equation, 252 (2012), 3245-3277.  doi: 10.1016/j.jde.2011.11.003.

[15]

R. M. ColomboP. Gwiazda and M. Rosińska, Optimization in structure population models through the Escalator Boxcar Train, ESAIM: Control, Optimisation and Calculus of Variations, 24 (2018), 377-399.  doi: 10.1051/cocv/2017003.

[16]

P. DegondM. Herty and J.-G. Liu, Flow on sweeping networks, Multiscale Model. Simul., 12 (2014), 538-565.  doi: 10.1137/130927061.

[17]

J. H. M. EversS. C. Hille and A. Muntean, Measure-valued mass evolution problems with flux boundary conditions and solution-dependent velocities, SIAM Journal on Mathematical Analysis, 48 (2016), 1929-1953.  doi: 10.1137/15M1031655.

[18]

M. Fornasier, L. Lisini, C. Orrieri and G. Savaré, Mean-field optimal control as gamma-limit of finite agent controls, arXiv: 1803.04689v1.

[19]

P. Goatin and F. Rossi, A traffic flow model with non-smooth metric interaction: Well-posedness and micro-macro limit, Communications in Mathematical Sciences, 15 (2017), 261-287.  doi: 10.4310/CMS.2017.v15.n1.a12.

[20]

P. Goatin and S. Scialanga, Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity, Networks and Heterogeneous Media, 11 (2016), 107-121.  doi: 10.3934/nhm.2016.11.107.

[21]

P. Gwiazda, S. C. Hille and K. Łyczek, Differentiability in perturbation parameter of measure solution to non-linear perturbed transport equation, In preparation.

[22]

P. GwiazdaJ. JabłońskiA. Marciniak-Czochra and A. Ulikowska, Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded Lipschitz distance, Numerical Methods for Partial Differential Equations, 30 (2014), 1797-1820.  doi: 10.1002/num.21879.

[23]

P. GwiazdaT. Lorenz and A. Marciniak-Czochra, A non-linear structured population model: Lipschitz continuity of measure- valued solutions with respect to model ingredients, Journal of Differential Equations, 248 (2010), 2703-2735.  doi: 10.1016/j.jde.2010.02.010.

[24]

P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces, Journal of Hyperbolic Differential Equations, 7 (2010), 733-773.  doi: 10.1142/S021989161000227X.

[25]

P. Hartman, Ordinary Differential Equations, Society for Industrial and Applied Mathematics, 2002. doi: 10.1137/1.9780898719222.

[26]

S. C. Hille and D. H. T. Worm, Embedding of semigroups of Lipschitz maps into positive linear semigroups on ordered Banach spaces generated by measures, Integral Equations and Operator Theory, 63 (2009), 351-371.  doi: 10.1007/s00020-008-1652-z.

[27]

D. Kinderlehrer and P. Pedregal, Weak convergence of integrands and the Young measure representation, SIAM Journal on Mathematical Analysis, 23 (1992), 1-19.  doi: 10.1137/0523001.

[28]

S. Maniglia, Probabilistic representation and uniqueness results for measure-valued solutions of transport equations, Journal de Math matiques Pures et Appliqu es, 87 (2007), 601-626.  doi: 10.1016/j.matpur.2007.04.001.

[29]

B. Piccoli, Measure differential equations, arXiv: 1708.09738v1.

[30]

B. Piccoli and F. Rossi, Generalized Wasserstein distance and its application to transport equations with source, Archive for Rational Mechanics and Analysis, 211 (2014), 335-358.  doi: 10.1007/s00205-013-0669-x.

[31]

B. Piccoli and F. Rossi, On properties of the generalized Wasserstein distance, Archive for Rational Mechanics and Analysis, 222 (2016), 1339-1365.  doi: 10.1007/s00205-016-1026-7.

[32]

J. Skrzeczkowski, Measure solutions to perturbed structured population models – differentiability with respect to perturbation parameter, arXiv: 1812.01747v3.

[33] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, 2003. 
[34]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, M. Dekker New York, 1985.

show all references

References:
[1]

G. AlbiY.-P. ChoiM. Fornasier and D. Kalise, Mean field control hierarchy, Applied Mathematics and Optimization, 76 (2017), 93-135.  doi: 10.1007/s00245-017-9429-x.

[2]

G. AlbiM. Herty and L. Pareschi, Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems, Applied Mathematics and Computation, 187 (2019), 460-477.  doi: 10.1016/j.amc.2019.02.021.

[3]

H. Amann and J. Escher., Analysis II, Birkhäuser Verlag, Basel, 2008.

[4]

L. Ambrosio, M. Fornasier and M. Morandotti, Spatially inhomogeneous evolutionary games, arXiv: 1805.04027v1.

[5]

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, 2008.

[6]

N. Bellomo, P. Degond and E. Tadmor, Active Particles, Volume 1: Advances in Theory, Models, and Applications, Springer, Birkhäuser, 2017.

[7]

F. BerthelinP. DegondM. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Archive for Rational Mechanics and Analysis, 187 (2008), 185-220.  doi: 10.1007/s00205-007-0061-9.

[8]

G. A. BonaschiJ. A. CarrilloM. Di Francesco and M. A. Peletier, Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1{D}, ESAIM Control Optim. Calc. Var., 21 (2015), 414-441.  doi: 10.1051/cocv/2014032.

[9]

M. BonginiM. FornasierF. Rossi and F. Solombrino, Mean-field Pontryagin maximum principle, Journal of Optimization Theory and Applications, 175 (2017), 1-38.  doi: 10.1007/s10957-017-1149-5.

[10]

B. Bonnet and F. Rossi, The Pontryagin maximum principle in the Wasserstein space, Calculus of Variations and Partial Differential Equations, 58 (2019), Art. 11, 36pp. doi: 10.1007/s00526-018-1447-2.

[11]

Y. BrenierW. GangboG. Savaré and M. Westdickenberg, Sticky particle dynamics with interactions, Journal de Mathématiques Pures et Appliquées. Neuvième Série, 99 (2013), 577-617.  doi: 10.1016/j.matpur.2012.09.013.

[12]

J. A. CañizoJ. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics, Acta Applicandae Mathematicae, 123 (2013), 141-156.  doi: 10.1007/s10440-012-9758-3.

[13]

J. A. CañizoJ. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Mathematical Models and Methods in Applied Sciences, 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.

[14]

J. A. CarrilloR. M. ColomboP. Gwiazda and A. Ulikowska, Structured populations, cell growth and measure valued balance laws, Journal of Differential Equation, 252 (2012), 3245-3277.  doi: 10.1016/j.jde.2011.11.003.

[15]

R. M. ColomboP. Gwiazda and M. Rosińska, Optimization in structure population models through the Escalator Boxcar Train, ESAIM: Control, Optimisation and Calculus of Variations, 24 (2018), 377-399.  doi: 10.1051/cocv/2017003.

[16]

P. DegondM. Herty and J.-G. Liu, Flow on sweeping networks, Multiscale Model. Simul., 12 (2014), 538-565.  doi: 10.1137/130927061.

[17]

J. H. M. EversS. C. Hille and A. Muntean, Measure-valued mass evolution problems with flux boundary conditions and solution-dependent velocities, SIAM Journal on Mathematical Analysis, 48 (2016), 1929-1953.  doi: 10.1137/15M1031655.

[18]

M. Fornasier, L. Lisini, C. Orrieri and G. Savaré, Mean-field optimal control as gamma-limit of finite agent controls, arXiv: 1803.04689v1.

[19]

P. Goatin and F. Rossi, A traffic flow model with non-smooth metric interaction: Well-posedness and micro-macro limit, Communications in Mathematical Sciences, 15 (2017), 261-287.  doi: 10.4310/CMS.2017.v15.n1.a12.

[20]

P. Goatin and S. Scialanga, Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity, Networks and Heterogeneous Media, 11 (2016), 107-121.  doi: 10.3934/nhm.2016.11.107.

[21]

P. Gwiazda, S. C. Hille and K. Łyczek, Differentiability in perturbation parameter of measure solution to non-linear perturbed transport equation, In preparation.

[22]

P. GwiazdaJ. JabłońskiA. Marciniak-Czochra and A. Ulikowska, Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded Lipschitz distance, Numerical Methods for Partial Differential Equations, 30 (2014), 1797-1820.  doi: 10.1002/num.21879.

[23]

P. GwiazdaT. Lorenz and A. Marciniak-Czochra, A non-linear structured population model: Lipschitz continuity of measure- valued solutions with respect to model ingredients, Journal of Differential Equations, 248 (2010), 2703-2735.  doi: 10.1016/j.jde.2010.02.010.

[24]

P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces, Journal of Hyperbolic Differential Equations, 7 (2010), 733-773.  doi: 10.1142/S021989161000227X.

[25]

P. Hartman, Ordinary Differential Equations, Society for Industrial and Applied Mathematics, 2002. doi: 10.1137/1.9780898719222.

[26]

S. C. Hille and D. H. T. Worm, Embedding of semigroups of Lipschitz maps into positive linear semigroups on ordered Banach spaces generated by measures, Integral Equations and Operator Theory, 63 (2009), 351-371.  doi: 10.1007/s00020-008-1652-z.

[27]

D. Kinderlehrer and P. Pedregal, Weak convergence of integrands and the Young measure representation, SIAM Journal on Mathematical Analysis, 23 (1992), 1-19.  doi: 10.1137/0523001.

[28]

S. Maniglia, Probabilistic representation and uniqueness results for measure-valued solutions of transport equations, Journal de Math matiques Pures et Appliqu es, 87 (2007), 601-626.  doi: 10.1016/j.matpur.2007.04.001.

[29]

B. Piccoli, Measure differential equations, arXiv: 1708.09738v1.

[30]

B. Piccoli and F. Rossi, Generalized Wasserstein distance and its application to transport equations with source, Archive for Rational Mechanics and Analysis, 211 (2014), 335-358.  doi: 10.1007/s00205-013-0669-x.

[31]

B. Piccoli and F. Rossi, On properties of the generalized Wasserstein distance, Archive for Rational Mechanics and Analysis, 222 (2016), 1339-1365.  doi: 10.1007/s00205-016-1026-7.

[32]

J. Skrzeczkowski, Measure solutions to perturbed structured population models – differentiability with respect to perturbation parameter, arXiv: 1812.01747v3.

[33] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, 2003. 
[34]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, M. Dekker New York, 1985.

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