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 & 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. Google Scholar

[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. Google Scholar

[3]

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

[4]

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

[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. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[18]

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

[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. Google Scholar

[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. Google Scholar

[21]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

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

[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. Google Scholar

[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. Google Scholar

[32]

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

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

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

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. Google Scholar

[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. Google Scholar

[3]

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

[4]

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

[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. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[18]

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

[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. Google Scholar

[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. Google Scholar

[21]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

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

[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. Google Scholar

[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. Google Scholar

[32]

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

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

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

[1]

Agnieszka Ulikowska. An age-structured two-sex model in the space of radon measures: Well posedness. Kinetic & Related Models, 2012, 5 (4) : 873-900. doi: 10.3934/krm.2012.5.873

[2]

Fabio Camilli, Raul De Maio, Andrea Tosin. Transport of measures on networks. Networks & Heterogeneous Media, 2017, 12 (2) : 191-215. doi: 10.3934/nhm.2017008

[3]

Vladimir I. Bogachev, Stanislav V. Shaposhnikov, Alexander Yu. Veretennikov. Differentiability of solutions of stationary Fokker--Planck--Kolmogorov equations with respect to a parameter. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3519-3543. doi: 10.3934/dcds.2016.36.3519

[4]

Michael Krause, Jan Marcel Hausherr, Walter Krenkel. Computing the fibre orientation from Radon data using local Radon transform. Inverse Problems & Imaging, 2011, 5 (4) : 879-891. doi: 10.3934/ipi.2011.5.879

[5]

Simon Gindikin. A remark on the weighted Radon transform on the plane. Inverse Problems & Imaging, 2010, 4 (4) : 649-653. doi: 10.3934/ipi.2010.4.649

[6]

Todd Young. Asymptotic measures and distributions of Birkhoff averages with respect to Lebesgue measure. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 359-378. doi: 10.3934/dcds.2003.9.359

[7]

Azmy S. Ackleh, Ben G. Fitzpatrick, Horst R. Thieme. Rate distributions and survival of the fittest: a formulation on the space of measures. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 917-928. doi: 10.3934/dcdsb.2005.5.917

[8]

Mrinal Kanti Roychowdhury. Quantization coefficients for ergodic measures on infinite symbolic space. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2829-2846. doi: 10.3934/dcds.2014.34.2829

[9]

Jan Boman. A local uniqueness theorem for weighted Radon transforms. Inverse Problems & Imaging, 2010, 4 (4) : 631-637. doi: 10.3934/ipi.2010.4.631

[10]

Sunghwan Moon. Inversion of the spherical Radon transform on spheres through the origin using the regular Radon transform. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1029-1039. doi: 10.3934/cpaa.2016.15.1029

[11]

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017

[12]

Giulia Cavagnari. Regularity results for a time-optimal control problem in the space of probability measures. Mathematical Control & Related Fields, 2017, 7 (2) : 213-233. doi: 10.3934/mcrf.2017007

[13]

Kathryn Dabbs, Michael Kelly, Han Li. Effective equidistribution of translates of maximal horospherical measures in the space of lattices. Journal of Modern Dynamics, 2016, 10: 229-254. doi: 10.3934/jmd.2016.10.229

[14]

Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems & Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023

[15]

Hans Rullgård, Eric Todd Quinto. Local Sobolev estimates of a function by means of its Radon transform. Inverse Problems & Imaging, 2010, 4 (4) : 721-734. doi: 10.3934/ipi.2010.4.721

[16]

Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Bounded solutions of the Boltzmann equation in the whole space. Kinetic & Related Models, 2011, 4 (1) : 17-40. doi: 10.3934/krm.2011.4.17

[17]

Anne-Sophie de Suzzoni. Continuity of the flow of the Benjamin-Bona-Mahony equation on probability measures. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2905-2920. doi: 10.3934/dcds.2015.35.2905

[18]

Samia Challal, Abdeslem Lyaghfouri. Hölder continuity of solutions to the $A$-Laplace equation involving measures. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1577-1583. doi: 10.3934/cpaa.2009.8.1577

[19]

Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185

[20]

Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (18)
  • HTML views (98)
  • Cited by (0)

[Back to Top]