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Sonic-supersonic solutions for the two-dimensional pseudo-steady full Euler equations
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 |
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.
References:
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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. |
| [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. |
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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. |
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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. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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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