메뉴 건너뛰기




Volumn 16, Issue 7, 2014, Pages 1423-1466

Propagation of chaos for the 2D viscous vortex model

Author keywords

2D Navier Stokes equation; Entropy dissipation; Fisher information; Propagation of chaos; Stochastic particle systems

Indexed keywords


EID: 84906827741     PISSN: 14359855     EISSN: None     Source Type: Journal    
DOI: 10.4171/JEMS/465     Document Type: Article
Times cited : (127)

References (55)
  • 2
    • 0035731176 scopus 로고    scopus 로고
    • Branching process associated with 2d-Navier Stokes equation
    • Zbl 1013.35063 MR 1891201
    • Benachour, S., Roynette, B., Vallois, P.: Branching process associated with 2d-Navier Stokes equation. Rev. Mat. Iberoamer. 17, 331-373 (2001) Zbl 1013.35063 MR 1891201
    • (2001) Rev. Mat. Iberoamer. , vol.17 , pp. 331-373
    • Benachour, S.1    Roynette, B.2    Vallois, P.3
  • 3
    • 21844500356 scopus 로고
    • Global solutions of two-dimensional Navier-Stokes and Euler equations
    • Zbl 0837.35110 MR 1308857
    • Ben-Artzi, M.: Global solutions of two-dimensional Navier-Stokes and Euler equations. Arch. Ration. Mech. Anal. 128, 329-358 (1994) Zbl 0837.35110 MR 1308857
    • (1994) Arch. Ration. Mech. Anal. , vol.128 , pp. 329-358
    • Ben-Artzi, M.1
  • 4
    • 84875518264 scopus 로고    scopus 로고
    • Uniform convergence to equilibrium for granular media
    • Zbl 1264.35040 MR 3035983
    • Bolley, F., Gentil, I., Guillin, A.: Uniform convergence to equilibrium for granular media. Arch. Ration. Mech. Anal. 208, 429-445 (2013) Zbl 1264.35040 MR 3035983
    • (2013) Arch. Ration. Mech. Anal. , vol.208 , pp. 429-445
    • Bolley, F.1    Gentil, I.2    Guillin, A.3
  • 5
    • 84861341086 scopus 로고    scopus 로고
    • Statistical mechanics of two-dimensional and geophysical flows
    • MR 2929420
    • Bouchet, F., Venaille, A.: Statistical mechanics of two-dimensional and geophysical flows. Phys. Rep. 515, 227-295 (2012) MR 2929420
    • (2012) Phys. Rep. , vol.515 , pp. 227-295
    • Bouchet, F.1    Venaille, A.2
  • 6
    • 0003894892 scopus 로고
    • Masson, Paris, Zbl 0511.46001 MR 0697382
    • Brezis, H.: Analyse fonctionnelle. Masson, Paris (1983) Zbl 0511.46001 MR 0697382
    • (1983) Analyse Fonctionnelle
    • Brezis, H.1
  • 7
    • 21844518574 scopus 로고
    • Remarks on the preceding paper by M. Ben-Artzi: "Global solutions of two-dimensional Navier-Stokes and Euler equations"
    • Zbl 0837.35112 MR 1308858
    • Brezis, H.: Remarks on the preceding paper by M. Ben-Artzi: "Global solutions of two-dimensional Navier-Stokes and Euler equations". Arch. Ration. Mech. Anal. 128, 359-360 (1994) Zbl 0837.35112 MR 1308858
    • (1994) Arch. Ration. Mech. Anal. , vol.128 , pp. 359-360
    • Brezis, H.1
  • 9
    • 0001296888 scopus 로고
    • A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description
    • Zbl 0745.76001 MR 1145596
    • Caglioti, E., Lions, P.-L., Marchioro, C., Pulvirenti, M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Comm. Math. Phys. 143, 501-525 (1992) Zbl 0745.76001 MR 1145596
    • (1992) Comm. Math. Phys. , vol.143 , pp. 501-525
    • Caglioti, E.1    Lions, P.-L.2    Marchioro, C.3    Pulvirenti, M.4
  • 10
    • 0000273265 scopus 로고
    • Superadditivity of Fisher's information and logarithmic Sobolev inequalities
    • Zbl 0732.60020 MR 1132315
    • Carlen, E. A.: Superadditivity of Fisher's information and logarithmic Sobolev inequalities. J. Funct. Anal. 101, 194-211 (1991) Zbl 0732.60020 MR 1132315
    • (1991) J. Funct. Anal. , vol.101 , pp. 194-211
    • Carlen, E.A.1
  • 12
    • 0039758360 scopus 로고    scopus 로고
    • Diffusing particles with electrostatic repulsion
    • Zbl 0883.60089 MR 1440140
    • Cépa, E., Lépingle, D. Diffusing particles with electrostatic repulsion. Probab. Theory Related Fields 107, 429-449 (1997) Zbl 0883.60089 MR 1440140
    • (1997) Probab. Theory Related Fields , vol.107 , pp. 429-449
    • Cépa, E.1    Lépingle, D.2
  • 13
    • 0347786769 scopus 로고
    • Numerical study of slightly viscous flow
    • MR 0395483
    • Chorin, A. J.: Numerical study of slightly viscous flow, J. Fluid Mech. 57, 785-796 (1978) MR 0395483
    • (1978) J Fluid Mech. , vol.57 , pp. 785-796
    • Chorin, A.J.1
  • 14
    • 44949219227 scopus 로고    scopus 로고
    • Estimates and regularity results for the DiPerna-Lions flow
    • Zbl 1160.34004 MR 2369485
    • Crippa, G., De Lellis, C.: Estimates and regularity results for the DiPerna-Lions flow. J. Reine Angew. Math. 616, 15-46 (2008) Zbl 1160.34004 MR 2369485
    • (2008) J. Reine Angew. Math. , vol.616 , pp. 15-46
    • Crippa, G.1    De Lellis, C.2
  • 15
    • 0001918661 scopus 로고
    • La prévision: Ses lois logiques, ses sources subjectives
    • Zbl 0017.07602 MR 1508036
    • de Finetti, B.: La prévision: ses lois logiques, ses sources subjectives. Ann. Inst. H. Poincaré 7, 1-68 (1937) Zbl 0017.07602 MR 1508036
    • (1937) Ann. Inst. H. Poincaré , vol.7 , pp. 1-68
    • De Finetti, B.1
  • 16
    • 0002833142 scopus 로고
    • On the Fokker-Planck-Boltzmann equation
    • Zbl 0671.35068 MR 0972541
    • DiPerna, R. J., Lions, P.-L.: On the Fokker-Planck-Boltzmann equation. Comm. Math. Phys. 120, 1-23 (1988) Zbl 0671.35068 MR 0972541
    • (1988) Comm. Math. Phys. , vol.120 , pp. 1-23
    • DiPerna, R.J.1    Lions, P.-L.2
  • 17
    • 0002306354 scopus 로고
    • Ordinary differential equations, transport theory and Sobolev spaces
    • Zbl 0696.34049 MR 1022305
    • DiPerna, R. J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511-547 (1989) Zbl 0696.34049 MR 1022305
    • (1989) Invent. Math. , vol.98 , pp. 511-547
    • DiPerna, R.J.1    Lions, P.-L.2
  • 18
    • 36649027622 scopus 로고    scopus 로고
    • Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients
    • Zbl 1169.60010 MR 2375067
    • Figalli, A.: Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254, 109-153 (2008) Zbl 1169.60010 MR 2375067
    • (2008) J. Funct. Anal. , vol.254 , pp. 109-153
    • Figalli, A.1
  • 19
    • 79956218437 scopus 로고    scopus 로고
    • Full well-posedness of point vortex dynamics corresponding to stochastic 2D Euler equations
    • Zbl 1221.60082 MR 2802460
    • Flandoli, F., Gubinelli, M., Priola, E.: Full well-posedness of point vortex dynamics corresponding to stochastic 2D Euler equations. Stoch. Process. Appl. 121, 1445-1463 (2011) Zbl 1221.60082 MR 2802460
    • (2011) Stoch. Process. Appl. , vol.121 , pp. 1445-1463
    • Flandoli, F.1    Gubinelli, M.2    Priola, E.3
  • 20
    • 34547419234 scopus 로고    scopus 로고
    • Paths clustering and an existence result for stochastic vortex systems
    • Zbl 1121.82007 MR 2343519
    • Fontbona, J., Martinez, M.: Paths clustering and an existence result for stochastic vortex systems. J. Statist. Phys. 128, 699-719 (2007) Zbl 1121.82007 MR 2343519
    • (2007) J. Statist. Phys. , vol.128 , pp. 699-719
    • Fontbona, J.1    Martinez, M.2
  • 21
    • 18244363856 scopus 로고    scopus 로고
    • Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity
    • Zbl 1096.35102 MR 2178064
    • Gallagher, I., Gallay, T.: Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity. Math. Ann. 332, 287-327 (2005) Zbl 1096.35102 MR 2178064
    • (2005) Math. Ann. , vol.332 , pp. 287-327
    • Gallagher, I.1    Gallay, T.2
  • 22
    • 79954608590 scopus 로고    scopus 로고
    • Interaction of vortices in weakly viscous planar flows
    • Zbl 1229.35177 MR 2787587
    • Gallay, Th.: Interaction of vortices in weakly viscous planar flows. Arch. Ration. Mech. Anal. 200, 445-490 (2011) Zbl 1229.35177 MR 2787587
    • (2011) Arch. Ration. Mech. Anal. , vol.200 , pp. 445-490
    • Gallay, Th.1
  • 23
    • 21244476087 scopus 로고    scopus 로고
    • Global stability of vortex solutions of the two-dimensional Navier-Stokes equation
    • Zbl 1139.35084 MR 2123378
    • Gallay, Th., Wayne, C. E.: Global stability of vortex solutions of the two-dimensional Navier-Stokes equation. Comm. Math. Phys. 255, 97-129 (2005) Zbl 1139.35084 MR 2123378
    • (2005) Comm. Math. Phys. , vol.255 , pp. 97-129
    • Gallay, Th.1    Wayne, C.E.2
  • 24
    • 0000396658 scopus 로고
    • A fast algorithm for particles simulations, J
    • Zbl 0629.65005 MR 0918448
    • Greengard, L., Rokhlin, V.: A fast algorithm for particles simulations, J. Comput. Phys. 73, 325-348 (1987) Zbl 0629.65005 MR 0918448
    • (1987) Comput. Phys. , vol.73 , pp. 325-348
    • Greengard, L.1    Rokhlin, V.2
  • 26
    • 84895932069 scopus 로고    scopus 로고
    • On Kac's chaos and related problems
    • MR 3188710
    • Hauray, M., Mischler, S.: On Kac's chaos and related problems. J. Funct. Anal. 266, 6055-6157 (2014) MR 3188710
    • (2014) J. Funct. Anal. , vol.266 , pp. 6055-6157
    • Hauray, M.1    Mischler, S.2
  • 27
    • 84943471869 scopus 로고
    • Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen
    • ERAM 055.1448cj
    • Helmholtz, H.: Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. J. Reine Angew. Math. 55, 25-55 (1858) ERAM 055.1448cj
    • (1858) J. Reine Angew. Math. , vol.55 , pp. 25-55
    • Helmholtz, H.1
  • 28
    • 84967763082 scopus 로고
    • Symmetric measures on Cartesian products
    • Zbl 0066.29604 MR 0076206
    • Hewitt, E., Savage, L. J.: Symmetric measures on Cartesian products. Trans. Amer. Math. Soc. 80, 470-501 (1955) Zbl 0066.29604 MR 0076206
    • (1955) Trans. Amer. Math. Soc. , vol.80 , pp. 470-501
    • Hewitt, E.1    Savage, L.J.2
  • 31
    • 46349102591 scopus 로고    scopus 로고
    • Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients
    • Zbl 1157.35301 MR 2450159
    • Le Bris, C., Lions, P.-L.: Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients. Comm. Partial Differential Equations 33, 1272-1317 (2008) Zbl 1157.35301 MR 2450159
    • (2008) Comm. Partial Differential Equations , vol.33 , pp. 1272-1317
    • Le Bris, C.1    Lions, P.-L.2
  • 32
    • 0003360649 scopus 로고    scopus 로고
    • Analysis
    • Amer. Math. Soc., Providence, Zbl 0873.26002 MR 1415616
    • Lieb, E., Loss, M.: Analysis. Grad. Stud. Math. 14, Amer. Math. Soc., Providence (1997) Zbl 0873.26002 MR 1415616
    • (1997) Grad. Stud. Math. , vol.14
    • Lieb, E.1    Loss, M.2
  • 33
    • 21144445368 scopus 로고    scopus 로고
    • Convergence to equilibrium for granular media equations and their Euler schemes
    • Zbl 1031.60085 MR 1970276
    • Malrieu, F.: Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13, 540-560 (2003) Zbl 1031.60085 MR 1970276
    • (2003) Ann. Appl. Probab. , vol.13 , pp. 540-560
    • Malrieu, F.1
  • 34
    • 0032474982 scopus 로고    scopus 로고
    • On the inviscid limit for a fluid with a concentrated vorticity
    • Zbl 0911.35086 MR 1643505
    • Marchioro, C.: On the inviscid limit for a fluid with a concentrated vorticity. Comm. Math. Phys. 196, 53-65 (1998) Zbl 0911.35086 MR 1643505
    • (1998) Comm. Math. Phys. , vol.196 , pp. 53-65
    • Marchioro, C.1
  • 35
    • 0000852622 scopus 로고
    • Hydrodynamics in two dimensions and vortex theory
    • Zbl 0527.76021 MR 0667756
    • Marchioro, C., Pulvirenti, M.: Hydrodynamics in two dimensions and vortex theory. Comm. Math. Phys. 84, 483-503 (1982) Zbl 0527.76021 MR 0667756
    • (1982) Comm. Math. Phys. , vol.84 , pp. 483-503
    • Marchioro, C.1    Pulvirenti, M.2
  • 36
    • 0003269966 scopus 로고
    • Mathematical theory of incompressible nonviscous fluids
    • Springer, New York, Zbl 0789.76002 MR 1245492
    • Marchioro, C., Pulvirenti, M.: Mathematical Theory of Incompressible Nonviscous Fluids. Appl. Math. Sci. 96, Springer, New York (1994) Zbl 0789.76002 MR 1245492
    • (1994) Appl. Math. Sci. , vol.96
    • Marchioro, C.1    Pulvirenti, M.2
  • 39
    • 0040431383 scopus 로고    scopus 로고
    • Monte-Carlo approximations for 2d Navier-Stokes equations with measure initial data
    • Zbl 0993.60099 MR 1867427
    • Méléard, S.: Monte-Carlo approximations for 2d Navier-Stokes equations with measure initial data. Probab. Theory Related Fields 121, 367-388 (2001) Zbl 0993.60099 MR 1867427
    • (2001) Probab. Theory Related Fields , vol.121 , pp. 367-388
    • Méléard, S.1
  • 40
    • 84879420862 scopus 로고    scopus 로고
    • Kac's program in kinetic theory
    • Zbl 1274.82048 MR 3069113
    • Mischler, S., Mouhot, C.: Kac's program in kinetic theory. Invent. Math. 193, 1-147 (2013) Zbl 1274.82048 MR 3069113
    • (2013) Invent. Math. , vol.193 , pp. 1-147
    • Mischler, S.1    Mouhot, C.2
  • 42
    • 51249192666 scopus 로고
    • Statistical hydrodynamics
    • MR 0036116
    • Onsager, L.: Statistical hydrodynamics. Il Nuovo Cimento 6, 279-287 (1949) MR 0036116
    • (1949) Il Nuovo Cimento , vol.6 , pp. 279-287
    • Onsager, L.1
  • 43
    • 84906828778 scopus 로고
    • Moment estimates for parabolic equations in the divergence form
    • Zbl 0592.35059 MR 0807493
    • Osada, H.: Moment estimates for parabolic equations in the divergence form. J. Math. Kyoto Univ. 25, 473-488 (1985) Zbl 0592.35059 MR 0807493
    • (1985) J. Math. Kyoto Univ. , vol.25 , pp. 473-488
    • Osada, H.1
  • 44
    • 21844472480 scopus 로고
    • A stochastic differential equation arising from the vortex problem
    • Zbl 0602.60047 MR 0834541
    • Osada, H.: A stochastic differential equation arising from the vortex problem. Proc. Japan Acad. Ser. A Math. Sci. 61, 333-336 (1986) Zbl 0602.60047 MR 0834541
    • (1986) Proc. Japan Acad. Ser. A Math. Sci. , vol.61 , pp. 333-336
    • Osada, H.1
  • 45
    • 0344864782 scopus 로고
    • Propagation of chaos for the two-dimensional Navier-Stokes equation
    • Zbl 0679.76033 MR 0839794
    • Osada, H.: Propagation of chaos for the two-dimensional Navier-Stokes equation. Proc. Japan Acad. Ser. A Math. Sci. 62, 8-11 (1986) Zbl 0679.76033 MR 0839794
    • (1986) Proc. Japan Acad. Ser. A Math. Sci. , vol.62 , pp. 8-11
    • Osada, H.1
  • 46
    • 0002261634 scopus 로고
    • Propagation of chaos for the two-dimensional Navier-Stokes equation
    • Katata/Kyoto, Academic Press, Boston, MA, Zbl 0645.76040 MR 0933829
    • Osada, H.: Propagation of chaos for the two-dimensional Navier-Stokes equation. In: Probabilistic Methods in Mathematical Physics (Katata/Kyoto, 1985), Academic Press, Boston, MA, 303-334 (1987) Zbl 0645.76040 MR 0933829
    • (1985) Probabilistic Methods in Mathematical Physics , pp. 303-334
    • Osada, H.1
  • 48
    • 0040102518 scopus 로고
    • Diffusion processes with generators of generalized divergence form
    • Zbl 0657.35073 MR 0916761
    • Osada, H.: Diffusion processes with generators of generalized divergence form. J. Math. Kyoto Univ. 27, 597-619 (1987) Zbl 0657.35073 MR 0916761
    • (1987) J. Math. Kyoto Univ. , vol.27 , pp. 597-619
    • Osada, H.1
  • 49
    • 0001333283 scopus 로고
    • Mean entropy of states in classical statistical mechanics
    • Zbl 0144.48205 MR 0225553
    • Robinson, D. W., Ruelle, D.: Mean entropy of states in classical statistical mechanics. Comm. Math. Phys. 5, 288-300 (1967) Zbl 0144.48205 MR 0225553
    • (1967) Comm. Math. Phys. , vol.5 , pp. 288-300
    • Robinson, D.W.1    Ruelle, D.2
  • 50
    • 0030520674 scopus 로고    scopus 로고
    • The point vortex method for periodic weak solutions of the 2-D Euler equations
    • Zbl 0862.35092 MR 1399201
    • Schochet, S.: The point vortex method for periodic weak solutions of the 2-D Euler equations. Comm. Pure Appl. Math. 49, 911-965 (1996) Zbl 0862.35092 MR 1399201
    • (1996) Comm. Pure Appl. Math. , vol.49 , pp. 911-965
    • Schochet, S.1
  • 51
    • 0003356317 scopus 로고
    • Harmonic Analysis
    • Princeton Univ. Press, Princeton, NJ, Zbl 0821.42001 MR 1232192
    • Stein, E. M.: Harmonic Analysis. Princeton Math. Ser. 43, Princeton Univ. Press, Princeton, NJ (1993) Zbl 0821.42001 MR 1232192
    • (1993) Princeton Math. Ser. , vol.43
    • Stein, E.M.1
  • 52
    • 34250138528 scopus 로고
    • Équations de type de Boltzmann, spatialement homogènes
    • Zbl 0553.60069 MR 0753814
    • Sznitman, A.-S.: Équations de type de Boltzmann, spatialement homogènes. Z. Wahrsch. Verw. Gebiete 66, 559-592 (1984) Zbl 0553.60069 MR 0753814
    • (1984) Z. Wahrsch. Verw. Gebiete , vol.66 , pp. 559-592
    • Sznitman, A.-S.1
  • 54
    • 34547458684 scopus 로고
    • On the existence and uniqueness of SDE describing an n-particle system interacting via a singular potential
    • Zbl 0579.60049 MR 0831903
    • Takanobu, S.: On the existence and uniqueness of SDE describing an n-particle system interacting via a singular potential. Proc. Japan Acad. Ser. A Math. Sci. 6, 287-290 (1985) Zbl 0579.60049 MR 0831903
    • (1985) Proc. Japan Acad. Ser. A Math. Sci. , vol.6 , pp. 287-290
    • Takanobu, S.1
  • 55
    • 0000827166 scopus 로고
    • On the uniqueness of solutions of stochastic differential equations
    • Zbl 0236.60037 MR 0278420
    • Yamada, T., Watanabe, S.: On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11, 155-167 (1971) Zbl 0236.60037 MR 0278420
    • (1971) J. Math. Kyoto Univ. , vol.11 , pp. 155-167
    • Yamada, T.1    Watanabe, S.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.