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

Memory effects in measure transport equations

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.

Citation: Fabio Camilli, Raul De Maio. Memory effects in measure transport equations. Kinetic and Related Models, 2019, 12 (6) : 1229-1245. doi: 10.3934/krm.2019047
References:
[1]

M. AllenL. 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.

[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.

[3]

J. A. CañizoJ. 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.

[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.

[5]

J. H. M. EversS. 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.

[6]

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

[7]

M. HahnK. 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.

[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.

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

[11]

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

[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.

[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.

[14]

A. PiryatinskaA. 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.

[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.

[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.

[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.

show all references

References:
[1]

M. AllenL. 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.

[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.

[3]

J. A. CañizoJ. 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.

[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.

[5]

J. H. M. EversS. 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.

[6]

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

[7]

M. HahnK. 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.

[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.

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

[11]

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

[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.

[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.

[14]

A. PiryatinskaA. 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.

[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.

[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.

[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.

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