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December  2019, 12(6): 1229-1245. doi: 10.3934/krm.2019047

Memory effects in measure transport equations

Fabio Camilli and  Raul De Maio

Dip. di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, via Scarpa 16, 00161 Roma, Italy

Received  June 2018 Revised  March 2019 Published  September 2019

Transport equations with a nonlocal velocity field have been introduced as a continuum model for interacting particle systems arising in physics, chemistry and biology. Fractional time derivatives, given by convolution integrals of the time-derivative with power-law kernels, are typical for memory effects in complex systems. In this paper we consider a nonlinear transport equation with a fractional time-derivative. We provide a well-posedness theory for weak measure solutions of the problem and an integral formula which generalizes the classical push-forward representation formula to this setting.

Keywords: Transport equation, measure-valued solution, fractional derivative, anomalous diffusion.
Mathematics Subject Classification: Primary: 35R11, 26A33; Secondary: 35Q35.
Citation: Fabio Camilli, Raul De Maio. Memory effects in measure transport equations. Kinetic & Related Models, 2019, 12 (6) : 1229-1245. doi: 10.3934/krm.2019047
References:
[1]

M. Allen, L. Caffarelli and A. Vasseur, A parabolic problem with a fractional time derivative, Arch. Ration.Mech. Anal., 221 (2016), 603-630.  doi: 10.1007/s00205-016-0969-z.  Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd Edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

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J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.  Google Scholar

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E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, MS&A, Modeling, Simulation and Applications, 12, Springer, Cham, 2014. doi: 10.1007/978-3-319-06620-2.  Google Scholar

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J. H. M. Evers, S. C. Hille and A. Muntean, Measure-valued mass evolution problems with flux boundary conditions and solution-dependent velocities, SIAM J. Math. Anal., 48 (2016), 1929-1953.  doi: 10.1137/15M1031655.  Google Scholar

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M. Fornasier and F. Solombrino, Mean-field optimal control, ESAIM Control Optim. Calc. Var., 20 (2014), 1123-1152.  doi: 10.1051/cocv/2014009.  Google Scholar

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M. Hahn, K. Kobayashi and S. Umarov, SDEs driven by a time-changed Lévy process and their associated time-fractional order pseudo-differential equations, J. Theor. Probab., 25 (2012), 262-279.  doi: 10.1007/s10959-010-0289-4.  Google Scholar

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Y. Luchko and M. Yamamoto, General time-fractional diffusion equation: Some uniqueness and existence results for the initial-boundary-value problems, Fract. Calc. Appl. Anal., 19 (2016), 676-695.  doi: 10.1515/fca-2016-0036.  Google Scholar

[9] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.  doi: 10.1142/9781848163300.  Google Scholar
[10]

M. M. Meerschaert and H.-P. Scheffler, Limit theorems for continuous-time random walks with infinite mean waiting times, J. Appl. Prob., 41 (2004), 623-638.  doi: 10.1239/jap/1091543414.  Google Scholar

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M. M. Meerschaert and P. Straka, Inverse stable subordinators, Math. Model Nat Phenom., 8 (2013), 1-16.  doi: 10.1051/mmnp/20138201.  Google Scholar

[12]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 77 pp. doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[13]

E. Orsingher and L. Beghin, Fractional diffusion equations and processes with randomly varying time, Ann. Probab., 37 (2009), 206-249.  doi: 10.1214/08-AOP401.  Google Scholar

[14]

A. Piryatinska, A. I. Saichev and W. A. Woyczynski, Models of anomalous diffusion: The subdiffusive case, Physica A: Statistical Mechanics and its Applications, 349 (2005), 375-420.  doi: 10.1016/j.physa.2004.11.003.  Google Scholar

[15]

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[16]

F. Poupaud and M. Rascle, Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients, Comm. Partial Differential Equations, 22 (1997), 337-358.  doi: 10.1080/03605309708821265.  Google Scholar

[17]

V. E. Tarasov, Review of some promising fractional physical models, Internat. J. Modern Phys. B, 27 (2013), 1330005, 32 pp. doi: 10.1142/S0217979213300053.  Google Scholar

show all references

References:
[1]

M. Allen, L. Caffarelli and A. Vasseur, A parabolic problem with a fractional time derivative, Arch. Ration.Mech. Anal., 221 (2016), 603-630.  doi: 10.1007/s00205-016-0969-z.  Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd Edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[3]

J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.  Google Scholar

[4]

E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, MS&A, Modeling, Simulation and Applications, 12, Springer, Cham, 2014. doi: 10.1007/978-3-319-06620-2.  Google Scholar

[5]

J. H. M. Evers, S. C. Hille and A. Muntean, Measure-valued mass evolution problems with flux boundary conditions and solution-dependent velocities, SIAM J. Math. Anal., 48 (2016), 1929-1953.  doi: 10.1137/15M1031655.  Google Scholar

[6]

M. Fornasier and F. Solombrino, Mean-field optimal control, ESAIM Control Optim. Calc. Var., 20 (2014), 1123-1152.  doi: 10.1051/cocv/2014009.  Google Scholar

[7]

M. Hahn, K. Kobayashi and S. Umarov, SDEs driven by a time-changed Lévy process and their associated time-fractional order pseudo-differential equations, J. Theor. Probab., 25 (2012), 262-279.  doi: 10.1007/s10959-010-0289-4.  Google Scholar

[8]

Y. Luchko and M. Yamamoto, General time-fractional diffusion equation: Some uniqueness and existence results for the initial-boundary-value problems, Fract. Calc. Appl. Anal., 19 (2016), 676-695.  doi: 10.1515/fca-2016-0036.  Google Scholar

[9] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.  doi: 10.1142/9781848163300.  Google Scholar
[10]

M. M. Meerschaert and H.-P. Scheffler, Limit theorems for continuous-time random walks with infinite mean waiting times, J. Appl. Prob., 41 (2004), 623-638.  doi: 10.1239/jap/1091543414.  Google Scholar

[11]

M. M. Meerschaert and P. Straka, Inverse stable subordinators, Math. Model Nat Phenom., 8 (2013), 1-16.  doi: 10.1051/mmnp/20138201.  Google Scholar

[12]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 77 pp. doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[13]

E. Orsingher and L. Beghin, Fractional diffusion equations and processes with randomly varying time, Ann. Probab., 37 (2009), 206-249.  doi: 10.1214/08-AOP401.  Google Scholar

[14]

A. Piryatinska, A. I. Saichev and W. A. Woyczynski, Models of anomalous diffusion: The subdiffusive case, Physica A: Statistical Mechanics and its Applications, 349 (2005), 375-420.  doi: 10.1016/j.physa.2004.11.003.  Google Scholar

[15]

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[16]

F. Poupaud and M. Rascle, Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients, Comm. Partial Differential Equations, 22 (1997), 337-358.  doi: 10.1080/03605309708821265.  Google Scholar

[17]

V. E. Tarasov, Review of some promising fractional physical models, Internat. J. Modern Phys. B, 27 (2013), 1330005, 32 pp. doi: 10.1142/S0217979213300053.  Google Scholar

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