Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces

Carrillo, José A.; Choi, Young-Pil

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 4, pp. 925-954.

We study an asymptotic limit of Vlasov type equation with nonlocal interaction forces where the friction terms are dominant. We provide a quantitative estimate of this large friction limit from the kinetic equation to a continuity type equation with a nonlocal velocity field, the so-called aggregation equation, by employing 2-Wasserstein distance. By introducing an intermediate system, given by the pressureless Euler equations with nonlocal forces, we can quantify the error between the spatial densities of the kinetic equation and the pressureless Euler system by means of relative entropy type arguments combined with the 2-Wasserstein distance. This together with the quantitative error estimate between the pressureless Euler system and the aggregation equation in 2-Wasserstein distance in [Commun. Math. Phys, 365, (2019), 329–361] establishes the quantitative bounds on the error between the kinetic equation and the aggregation equation.

[1] Ambrosio, L.; Gigli, N.; Savaré, G. Gradient Flows: In Metric Spaces and in the Space of Probability Measures, Springer Science & Business Media, 2008 | MR | Zbl

[2] Balagué, D.; Carrillo, J.A.; Laurent, T.; Raoul, G. Nonlocal interactions by repulsive-attractive potentials: radial ins/stability, Physica D, Volume 260 (2013), pp. 5–25 | DOI | MR | Zbl

[3] Bertozzi, A.L.; Carrillo, J.A.; Laurent, T. Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, Volume 22 (2009), pp. 683–710 | DOI | MR | Zbl

[4] Bertozzi, A.L.; Laurent, T.; Rosado, J. Lp theory for the multidimensional aggregation equation, Commun. Pure Appl. Math., Volume 64 (2011), pp. 45–83 | DOI | MR | Zbl

[5] Carrillo, J.A.; McCann, R.J.; Villani, C. Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoam., Volume 19 (2003), pp. 971–1018 | DOI | MR | Zbl

[6] Carrillo, J.A.; Choi, Y.-P.; Hauray, M. The derivation of swarming models: mean-field limit and Wasserstein distances, Collective Dynamics from Bacteria to Crowds, CISM Courses and Lect., vol. 553, Springer, Vienna, 2014, pp. 1–46 | MR

[7] Carrillo, J.A.; Choi, Y.-P.; Tse, O. Convergence to equilibrium in Wasserstein distance for damped Euler equations with interaction forces, Commun. Math. Phys., Volume 365 (2019), pp. 329–361 | DOI | MR | Zbl

[8] Carrillo, J.A.; Choi, Y.-P.; Zatorska, E. On the pressureless damped Euler-Poisson equations with quadratic confinement: critical thresholds and large-time behavior, Math. Models Methods Appl. Sci., Volume 26 (2016), pp. 2311–2340 | DOI | MR | Zbl

[9] Carrillo, J.A.; Rosado, J. Uniqueness of bounded solutions to aggregation equations by optimal transport methods, European Congress of Mathematics, Eur Math. Soc., 2010, pp. 3–16 | MR | Zbl

[10] Cercignani, C.; Illner, R.; Pulvirenti, M. The Mathematical Theory of Dilute Gases, Springer, New York, 1994 | DOI | MR | Zbl

[11] Choi, Y.-P. Global classical solutions and large-time behavior of the two-phase fluid model, SIAM J. Math. Anal., Volume 48 (2016), pp. 3090–3122 | MR | Zbl

[12] Choi, Y.-P. Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces, Nonlinearity, Volume 29 (2016), pp. 1887–1916 | MR | Zbl

[13] Choi, Y.-P.; Kwon, B. The Cauchy problem for the pressureless Euler/isentropic Navier-Stokes equations, J. Differ. Equ., Volume 261 (2016), pp. 654–711 | MR | Zbl

[14] Y.-P. Choi, S.-B. Yun, Existence and hydrodynamic limit for a Paveri-Fontana type kinetic traffic model, preprint. | MR

[15] Coulombel, J.F.; Goudon, T. The strong relaxation limit of the multidimensional isothermal Euler equations, Trans. Am. Math. Soc., Volume 359 (2007), pp. 637–648 | MR | Zbl

[16] Cucker, F.; Smale, S. Emergent behavior in flocks, IEEE Trans. Autom. Control, Volume 52 (2007), pp. 852–862 | DOI | MR | Zbl

[17] Fetecau, R.; Sun, W. First-order aggregation models and zero inertia limits, J. Differ. Equ., Volume 259 (2015), pp. 6774–6802 | DOI | MR | Zbl

[18] Fetecau, R.; Sun, W.; Tan, C. First order aggregation models with alignment, Physica D, Volume 325 (2016), pp. 146–163 | DOI | MR | Zbl

[19] Figalli, A.; Kang, M.-J. A rigorous derivation from the kinetic Cucker-Smale model to the pressureless Euler system with nonlocal alignment, Anal. PDE, Volume 12 (2019), pp. 843–866 | DOI | MR | Zbl

[20] Giesselmann, J.; Lattanzio, C.; Tzavaras, A.E. Relative energy for the Korteweg theory and related Hamiltonian flows in gas dynamics, Arch. Ration. Mech. Anal., Volume 223 (2017), pp. 1427–1484 | DOI | MR | Zbl

[21] Jabin, P.-E. Macroscopic limit of Vlasov type equations with friction, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 17 (2000), pp. 651–672 | Numdam | MR | Zbl

[22] Karper, T.; Mellet, A.; Trivisa, K. Existence of weak solutions to kinetic flocking models, SIAM J. Math. Anal., Volume 45 (2013), pp. 215–243 | DOI | MR | Zbl

[23] Kato, T. Linear evolution equations of “hyperbolic” type II, J. Math. Soc. Jpn., Volume 25 (1973), pp. 648–666 | DOI | MR | Zbl

[24] Karper, T.K.; Mellet, A.; Trivisa, K. Hydrodynamic limit of the kinetic Cucker-Smale flocking model, Math. Models Methods Appl. Sci., Volume 25 (2015), pp. 131–163 | DOI | MR | Zbl

[25] Lattanzio, C.; Tzavaras, A.E. Relative energy for the Korteweg theory and related Hamiltonian flows in gas dynamics, Commun. Partial Differ. Equ., Volume 42 (2017), pp. 261–290

[26] Lin, C.; Coulombel, J.-F. The strong relaxation limit of the multidimensional Euler equations, NoDEA Nonlinear Differ. Equ. Appl., Volume 20 (2013), pp. 447–461 | MR | Zbl

[27] Marcati, P.; Milani, A. The one-dimensional Darcy's law as the limit of a compressible Euler flow, J. Differ. Equ., Volume 84 (1990), pp. 129–147 | DOI | MR | Zbl

[28] Motsch, S.; Tadmor, E. A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., Volume 144 (2011), pp. 923–947 | DOI | MR | Zbl

[29] Nieto, J.; Poupaud, F.; Soler, J. High-field limit for the Vlasov-Poisson-Fokker-Planck system, Arch. Ration. Mech. Anal., Volume 158 (2001), pp. 29–59 | DOI | MR | Zbl

[30] Poupaud, F. Diagonal defect measures, adhesion dynamics and Euler equation, Methods Appl. Anal., Volume 9 (2002), pp. 533–562 | DOI | MR | Zbl

[31] Villani, C. Optimal Transport: Old and New, vol. 338, Springer Science & Business Media, 2008 | MR | Zbl

[32] Villani, C. A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, vol. I, North-Holland, Amsterdam, 2002, pp. 71–305 | MR | Zbl