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On the Yudovich's type solutions for the 2D Boussinesq system with thermal diffusivity
A gradient flow approach of propagation of chaos
Institut de Mathématiques de Toulouse, UMR 5219, 118 Route de Narbonne, 31400, Toulouse, France |
We provide an estimation of the dissipation of the Wasserstein 2 distance between the law of some interacting $ N $-particle system, and the $ N $ times tensorized product of the solution to the corresponding limit nonlinear conservation law. It then enables to recover classical propagation of chaos results [
References:
[1] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures Math. ETH Zürich, Birkhäuser Verlag, Basel, 2008. |
[2] |
F. Bolley, I. Gentil and A. Guillin,
Convergence to equilibrium in Wasserstein distance for Fokker–Planck equations, J. Funct. Anal., 263 (2012), 2430-2457.
doi: 10.1016/j.jfa.2012.07.007. |
[3] |
F. Bolley, I. Gentil and A. Guillin,
Uniform convergence to equilibrium for granular media, Arch. Rational Mech. Anal., 208 (2013), 429-445.
doi: 10.1007/s00205-012-0599-z. |
[4] |
E. A. Carlen, Superadditivity of Fisher's Information and Logarithmic Sobolev Inequalities, J. Funct. Anal., 101 (1991), 194-211.
doi: 10.1016/0022-1236(91)90155-X. |
[5] |
D. Cordero-Erausquin and A. Figalli,
Regularity of monotone maps between unbounded domains, Discrete & Contin. Dyn. Syst., 39 (2019), 7101-7112.
doi: 10.3934/dcds.2019297. |
[6] |
R. L. Dobrušin,
Vlasov equations, Anal. i Prilozhen., 13 (1979), 48-58.
|
[7] |
A. Durmus, A. Eberle, A. Guillin and R. Zimmer, An elementary approach to uniform in time propagation of chaos., preprint, arXiv: 1805.11387. |
[8] |
A. Eberle,
Reflection couplings and contraction rates for diffusions, Probab. Theory Relat. Fields, 166 (2016), 851-886.
doi: 10.1007/s00440-015-0673-1. |
[9] |
N. Fournier and A. Guillin,
On the rate of convergence in Wasserstein distance of the empirical measure, Probab. Theory Related Fields, 162 (2015), 707-738.
doi: 10.1007/s00440-014-0583-7. |
[10] |
N. Fournier, M. Hauray and S. Mischler,
Propagation of chaos for the 2D viscous vortex model, J. Eur. Math. Soc., 16 (2014), 1423-1466.
doi: 10.4171/JEMS/465. |
[11] |
F. Golse, C. Mouhot and T. Paul,
On the mean field and classical limits of quantum mechanics, Commun. Math. Phys., 343 (2016), 165-205.
doi: 10.1007/s00220-015-2485-7. |
[12] |
M. Hauray and S. Mischler.,
On Kac's chaos and related problems, J. Funct. Anal., 266 (2014), 6055-6157.
doi: 10.1016/j.jfa.2014.02.030. |
[13] |
S. Herrmann and J. Tugaut,
Non-uniqueness of stationary measures for self-stabilizing processes, Stochastic Process. Appl., 120 (2010), 1215-1246.
doi: 10.1016/j.spa.2010.03.009. |
[14] |
T. Holding, Propagation of chaos for Hölder continuous interaction kernels via Glivenko-Cantelli, Preprint, arXiv: 1608.02877, (2016). |
[15] |
P.-E. Jabin and Z. Wang,
Mean field limit and propagation of chaos for Vlasov systems with bounded forces, J. Funct. Anal., 271 (2016), 3588-3627.
doi: 10.1016/j.jfa.2016.09.014. |
[16] |
P.-E. Jabin and Z. Wang,
Quantitative estimate of propagation of chaos for stochastic systems with $W^{-1,\infty}$ kernels, Invent. Math., 214 (2018), 523-591.
doi: 10.1007/s00222-018-0808-y. |
[17] |
F. Malrieu,
Logarithmic Sobolev inequalities for some nonlinear PDE's, Stochastic Process. Appl., 95 (2001), 109-132.
doi: 10.1016/S0304-4149(01)00095-3. |
[18] |
H.-P. McKean Jr.,
A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci. U.S.A., 56 (1966), 1907-1911.
doi: 10.1073/pnas.56.6.1907. |
[19] |
S. Salem, Propagation of chaos for some 2 dimensional fractional Keller Segel equations in diffusion dominated and fair competition cases, preprint, arXiv: 1712.06677. |
[20] |
A.-S. Sznitman, Topics in propagation of chaos, In École d'Été de Probabilités de Saint-Flour XIX–-1989, volume 1464, chapter Lecture Notes in Math., pages 165–251. Springer, Berlin, 1991.
doi: 10.1007/BFb0085169. |
[21] |
C. Villani, Of Triangles, Gases, Prices and Men, Huawei-IHES Workshop on Mathematical Science, Tuesday May 5th 2015. |
[22] |
C. Villani, Optimal Transport, Old and New, Grundlehren Math. Wiss., vol. 338, Springer, Berlin, 2009.
doi: 10.1007/978-3-540-71050-9. |
show all references
References:
[1] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures Math. ETH Zürich, Birkhäuser Verlag, Basel, 2008. |
[2] |
F. Bolley, I. Gentil and A. Guillin,
Convergence to equilibrium in Wasserstein distance for Fokker–Planck equations, J. Funct. Anal., 263 (2012), 2430-2457.
doi: 10.1016/j.jfa.2012.07.007. |
[3] |
F. Bolley, I. Gentil and A. Guillin,
Uniform convergence to equilibrium for granular media, Arch. Rational Mech. Anal., 208 (2013), 429-445.
doi: 10.1007/s00205-012-0599-z. |
[4] |
E. A. Carlen, Superadditivity of Fisher's Information and Logarithmic Sobolev Inequalities, J. Funct. Anal., 101 (1991), 194-211.
doi: 10.1016/0022-1236(91)90155-X. |
[5] |
D. Cordero-Erausquin and A. Figalli,
Regularity of monotone maps between unbounded domains, Discrete & Contin. Dyn. Syst., 39 (2019), 7101-7112.
doi: 10.3934/dcds.2019297. |
[6] |
R. L. Dobrušin,
Vlasov equations, Anal. i Prilozhen., 13 (1979), 48-58.
|
[7] |
A. Durmus, A. Eberle, A. Guillin and R. Zimmer, An elementary approach to uniform in time propagation of chaos., preprint, arXiv: 1805.11387. |
[8] |
A. Eberle,
Reflection couplings and contraction rates for diffusions, Probab. Theory Relat. Fields, 166 (2016), 851-886.
doi: 10.1007/s00440-015-0673-1. |
[9] |
N. Fournier and A. Guillin,
On the rate of convergence in Wasserstein distance of the empirical measure, Probab. Theory Related Fields, 162 (2015), 707-738.
doi: 10.1007/s00440-014-0583-7. |
[10] |
N. Fournier, M. Hauray and S. Mischler,
Propagation of chaos for the 2D viscous vortex model, J. Eur. Math. Soc., 16 (2014), 1423-1466.
doi: 10.4171/JEMS/465. |
[11] |
F. Golse, C. Mouhot and T. Paul,
On the mean field and classical limits of quantum mechanics, Commun. Math. Phys., 343 (2016), 165-205.
doi: 10.1007/s00220-015-2485-7. |
[12] |
M. Hauray and S. Mischler.,
On Kac's chaos and related problems, J. Funct. Anal., 266 (2014), 6055-6157.
doi: 10.1016/j.jfa.2014.02.030. |
[13] |
S. Herrmann and J. Tugaut,
Non-uniqueness of stationary measures for self-stabilizing processes, Stochastic Process. Appl., 120 (2010), 1215-1246.
doi: 10.1016/j.spa.2010.03.009. |
[14] |
T. Holding, Propagation of chaos for Hölder continuous interaction kernels via Glivenko-Cantelli, Preprint, arXiv: 1608.02877, (2016). |
[15] |
P.-E. Jabin and Z. Wang,
Mean field limit and propagation of chaos for Vlasov systems with bounded forces, J. Funct. Anal., 271 (2016), 3588-3627.
doi: 10.1016/j.jfa.2016.09.014. |
[16] |
P.-E. Jabin and Z. Wang,
Quantitative estimate of propagation of chaos for stochastic systems with $W^{-1,\infty}$ kernels, Invent. Math., 214 (2018), 523-591.
doi: 10.1007/s00222-018-0808-y. |
[17] |
F. Malrieu,
Logarithmic Sobolev inequalities for some nonlinear PDE's, Stochastic Process. Appl., 95 (2001), 109-132.
doi: 10.1016/S0304-4149(01)00095-3. |
[18] |
H.-P. McKean Jr.,
A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci. U.S.A., 56 (1966), 1907-1911.
doi: 10.1073/pnas.56.6.1907. |
[19] |
S. Salem, Propagation of chaos for some 2 dimensional fractional Keller Segel equations in diffusion dominated and fair competition cases, preprint, arXiv: 1712.06677. |
[20] |
A.-S. Sznitman, Topics in propagation of chaos, In École d'Été de Probabilités de Saint-Flour XIX–-1989, volume 1464, chapter Lecture Notes in Math., pages 165–251. Springer, Berlin, 1991.
doi: 10.1007/BFb0085169. |
[21] |
C. Villani, Of Triangles, Gases, Prices and Men, Huawei-IHES Workshop on Mathematical Science, Tuesday May 5th 2015. |
[22] |
C. Villani, Optimal Transport, Old and New, Grundlehren Math. Wiss., vol. 338, Springer, Berlin, 2009.
doi: 10.1007/978-3-540-71050-9. |
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