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Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations
Equivalence between duality and gradient flow solutions for one-dimensional aggregation equations
1. | Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Université d'Orléans & CNRS UMR 7349, Fédération Denis Poisson, Université d'Orléans & CNRS FR 2964, 45067 Orléans Cedex 2, France |
2. | UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, INRIA Paris-Rocquencourt, EPI MAMBA, F-75005, Paris |
References:
[1] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, 2000. |
[2] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric space of probability measures, Lectures in Mathematics, Birkäuser, 2008. |
[3] |
D. Balagué and J. A. Carrillo, Aggregation equation with growing at infinity attractive-repulsive potentials, in Hyperbolic Problems-Theory, Numerics and Applications, Vol. 1, Ser. Contemp. Appl. Math. CAM, 17, World Sci. Publishing, Singapore, 2012, 136-147. |
[4] |
D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media, RAIRO Model. Math. Anal. Numer., 31 (1997), 615-641. |
[5] |
A. L. Bertozzi and J. Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation, Comm. Math. Sci., 8 (2010), 45-65.
doi: 10.4310/CMS.2010.v8.n1.a4. |
[6] |
A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equation with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710.
doi: 10.1088/0951-7715/22/3/009. |
[7] |
A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in $\mathbb{R}^{N}$, Comm. Math. Phys., 274 (2007), 717-735.
doi: 10.1007/s00220-007-0288-1. |
[8] |
A. L. Bertozzi, T. Laurent and F. Léger, Aggregation and spreading via the Newtonian potential: The dynamics of patch solutions, Math. Models Methods Appl. Sci., 22 (2012), 1140005, 39pp.
doi: 10.1142/S0218202511400057. |
[9] |
A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation, Comm. Pure Appl. Math., 64 (2011), 45-83.
doi: 10.1002/cpa.20334. |
[10] |
S. Bianchini and M. Gloyer, An estimate on the flow generated by monotone operators, Comm. Partial Diff. Eq., 36 (2011), 777-796.
doi: 10.1080/03605302.2010.534224. |
[11] |
A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the sub-critical Patlak-Keller-Segel model, SIAM J. Numer. Anal., 46 (2008), 691-721.
doi: 10.1137/070683337. |
[12] |
G. A. Bonaschi, J. A. Carrillo, M. Di Francesco and M. A. Peletier, Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D, ESAIM Control Optim. Calc. Var., 21 (2015), 414-441.
doi: 10.1051/cocv/2014032. |
[13] |
F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Analysis TMA, 32 (1998), 891-933.
doi: 10.1016/S0362-546X(97)00536-1. |
[14] |
F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Eq., 24 (1999), 2173-2189.
doi: 10.1080/03605309908821498. |
[15] |
F. Bouchut, F. James and S. Mancini, Uniqueness and weak stability for multidimensional transport equations with one-sided Lipschitz coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci.(5), 4 (2005), 1-25. |
[16] |
Y. Brenier, Polar factorization and monotone rearrangement of vectorvalued functions, Comm. Pure and Appl. Math., 44 (1991), 375-417.
doi: 10.1002/cpa.3160440402. |
[17] |
M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions, Nonlinear Analysis RWA, 8 (2007), 939-958.
doi: 10.1016/j.nonrwa.2006.04.002. |
[18] |
J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271.
doi: 10.1215/00127094-2010-211. |
[19] |
J. A. Carrillo, R. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Rev. Matematica Iberoamericana, 19 (2003), 971-1018.
doi: 10.4171/RMI/376. |
[20] |
R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34pp.
doi: 10.1142/S0218202511500230. |
[21] |
G. Crippa and M. Lécureux-Mercier, Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 523-537.
doi: 10.1007/s00030-012-0164-3. |
[22] |
Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodyna.mic limits and spatio-temporal mechanisms, J. Math. Biol., 51 (2005), 595-615.
doi: 10.1007/s00285-005-0334-6. |
[23] |
F. Filbet, Ph. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189-207.
doi: 10.1007/s00285-004-0286-2. |
[24] |
A. F. Filippov, Differential equations with discontinuous right-hand side, A.M.S. Transl.(2), 42 (1964), 199-231. |
[25] |
F. James and N. Vauchelet, A remark on duality solutions for some weakly nonlinear scalar conservation laws, C. R. Acad. Sci. Paris, Sér. I, 349 (2011), 657-661.
doi: 10.1016/j.crma.2011.05.004. |
[26] |
F. James and N. Vauchelet, Chemotaxis: from kinetic equations to aggregation dynamics, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101-127.
doi: 10.1007/s00030-012-0155-4. |
[27] |
F. James and N. Vauchelet, Numerical methods for one-dimensional aggregation equation, SIAM J. Numer. Anal., 53 (2015), 895-916.
doi: 10.1137/140959997. |
[28] |
R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
[29] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[30] |
H. Li and G. Toscani, Long time asymptotics of kinetic models of granular flows, Arch. Rat. Mech. Anal., 172 (2004), 407-428.
doi: 10.1007/s00205-004-0307-8. |
[31] |
D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50 (2005), 49-66.
doi: 10.1007/s00285-004-0279-1. |
[32] |
J. Nieto, F. Poupaud and J. Soler, High field limit for Vlasov-Poisson-Fokker-Planck equations, Arch. Rational Mech. Anal., 158 (2001), 29-59.
doi: 10.1007/s002050100139. |
[33] |
F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Diff. Eq., 26 (2001), 101-174.
doi: 10.1081/PDE-100002243. |
[34] |
A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives, Springer, Berlin, 2001.
doi: 10.1007/978-1-4757-4978-6. |
[35] |
C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[36] |
F. Poupaud and M. Rascle, Measure solutions to the linear multidimensional transport equation with discontinuous coefficients, Comm. Partial Diff. Equ., 22 (1997), 337-358.
doi: 10.1080/03605309708821265. |
[37] |
F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Methods Appl. Anal., 9 (2002), 533-561.
doi: 10.4310/MAA.2002.v9.n4.a4. |
[38] |
S. T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. I. Theory, Probab. Appl. (N. Y.), Springer-Verlag, New York, 1998. |
[39] |
C. Villani, Optimal Transport, Old and New, Grundlehren der Mathematischen Wissenschaften, 338, Springer, 2009.
doi: 10.1007/978-3-540-71050-9. |
[40] |
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, Amer. Math. Soc, Providence, 2003. |
[41] |
A. I. Vol'pert, The spaces BV and quasilinear equations, Math. USSR Sb., 73 (1967), 255-302. |
show all references
References:
[1] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, 2000. |
[2] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric space of probability measures, Lectures in Mathematics, Birkäuser, 2008. |
[3] |
D. Balagué and J. A. Carrillo, Aggregation equation with growing at infinity attractive-repulsive potentials, in Hyperbolic Problems-Theory, Numerics and Applications, Vol. 1, Ser. Contemp. Appl. Math. CAM, 17, World Sci. Publishing, Singapore, 2012, 136-147. |
[4] |
D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media, RAIRO Model. Math. Anal. Numer., 31 (1997), 615-641. |
[5] |
A. L. Bertozzi and J. Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation, Comm. Math. Sci., 8 (2010), 45-65.
doi: 10.4310/CMS.2010.v8.n1.a4. |
[6] |
A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equation with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710.
doi: 10.1088/0951-7715/22/3/009. |
[7] |
A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in $\mathbb{R}^{N}$, Comm. Math. Phys., 274 (2007), 717-735.
doi: 10.1007/s00220-007-0288-1. |
[8] |
A. L. Bertozzi, T. Laurent and F. Léger, Aggregation and spreading via the Newtonian potential: The dynamics of patch solutions, Math. Models Methods Appl. Sci., 22 (2012), 1140005, 39pp.
doi: 10.1142/S0218202511400057. |
[9] |
A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation, Comm. Pure Appl. Math., 64 (2011), 45-83.
doi: 10.1002/cpa.20334. |
[10] |
S. Bianchini and M. Gloyer, An estimate on the flow generated by monotone operators, Comm. Partial Diff. Eq., 36 (2011), 777-796.
doi: 10.1080/03605302.2010.534224. |
[11] |
A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the sub-critical Patlak-Keller-Segel model, SIAM J. Numer. Anal., 46 (2008), 691-721.
doi: 10.1137/070683337. |
[12] |
G. A. Bonaschi, J. A. Carrillo, M. Di Francesco and M. A. Peletier, Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D, ESAIM Control Optim. Calc. Var., 21 (2015), 414-441.
doi: 10.1051/cocv/2014032. |
[13] |
F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Analysis TMA, 32 (1998), 891-933.
doi: 10.1016/S0362-546X(97)00536-1. |
[14] |
F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Eq., 24 (1999), 2173-2189.
doi: 10.1080/03605309908821498. |
[15] |
F. Bouchut, F. James and S. Mancini, Uniqueness and weak stability for multidimensional transport equations with one-sided Lipschitz coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci.(5), 4 (2005), 1-25. |
[16] |
Y. Brenier, Polar factorization and monotone rearrangement of vectorvalued functions, Comm. Pure and Appl. Math., 44 (1991), 375-417.
doi: 10.1002/cpa.3160440402. |
[17] |
M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions, Nonlinear Analysis RWA, 8 (2007), 939-958.
doi: 10.1016/j.nonrwa.2006.04.002. |
[18] |
J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271.
doi: 10.1215/00127094-2010-211. |
[19] |
J. A. Carrillo, R. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Rev. Matematica Iberoamericana, 19 (2003), 971-1018.
doi: 10.4171/RMI/376. |
[20] |
R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34pp.
doi: 10.1142/S0218202511500230. |
[21] |
G. Crippa and M. Lécureux-Mercier, Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 523-537.
doi: 10.1007/s00030-012-0164-3. |
[22] |
Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodyna.mic limits and spatio-temporal mechanisms, J. Math. Biol., 51 (2005), 595-615.
doi: 10.1007/s00285-005-0334-6. |
[23] |
F. Filbet, Ph. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189-207.
doi: 10.1007/s00285-004-0286-2. |
[24] |
A. F. Filippov, Differential equations with discontinuous right-hand side, A.M.S. Transl.(2), 42 (1964), 199-231. |
[25] |
F. James and N. Vauchelet, A remark on duality solutions for some weakly nonlinear scalar conservation laws, C. R. Acad. Sci. Paris, Sér. I, 349 (2011), 657-661.
doi: 10.1016/j.crma.2011.05.004. |
[26] |
F. James and N. Vauchelet, Chemotaxis: from kinetic equations to aggregation dynamics, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101-127.
doi: 10.1007/s00030-012-0155-4. |
[27] |
F. James and N. Vauchelet, Numerical methods for one-dimensional aggregation equation, SIAM J. Numer. Anal., 53 (2015), 895-916.
doi: 10.1137/140959997. |
[28] |
R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
[29] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[30] |
H. Li and G. Toscani, Long time asymptotics of kinetic models of granular flows, Arch. Rat. Mech. Anal., 172 (2004), 407-428.
doi: 10.1007/s00205-004-0307-8. |
[31] |
D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50 (2005), 49-66.
doi: 10.1007/s00285-004-0279-1. |
[32] |
J. Nieto, F. Poupaud and J. Soler, High field limit for Vlasov-Poisson-Fokker-Planck equations, Arch. Rational Mech. Anal., 158 (2001), 29-59.
doi: 10.1007/s002050100139. |
[33] |
F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Diff. Eq., 26 (2001), 101-174.
doi: 10.1081/PDE-100002243. |
[34] |
A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives, Springer, Berlin, 2001.
doi: 10.1007/978-1-4757-4978-6. |
[35] |
C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[36] |
F. Poupaud and M. Rascle, Measure solutions to the linear multidimensional transport equation with discontinuous coefficients, Comm. Partial Diff. Equ., 22 (1997), 337-358.
doi: 10.1080/03605309708821265. |
[37] |
F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Methods Appl. Anal., 9 (2002), 533-561.
doi: 10.4310/MAA.2002.v9.n4.a4. |
[38] |
S. T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. I. Theory, Probab. Appl. (N. Y.), Springer-Verlag, New York, 1998. |
[39] |
C. Villani, Optimal Transport, Old and New, Grundlehren der Mathematischen Wissenschaften, 338, Springer, 2009.
doi: 10.1007/978-3-540-71050-9. |
[40] |
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, Amer. Math. Soc, Providence, 2003. |
[41] |
A. I. Vol'pert, The spaces BV and quasilinear equations, Math. USSR Sb., 73 (1967), 255-302. |
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