Propagation of chaos for the 2D viscous vortex model

We consider a stochastic system of $N$ particles, usually called vortices in that setting, approximating the 2D Navier-Stokes equation written in vorticity. Assuming that the initial distribution of the position and circulation of the vortices has finite (partial) entropy and a finite moment of positive order, we show that the empirical measure of the particle system converges in law to the unique (under suitable a priori estimates) solution of the 2D Navier-Stokes equation. We actually prove a slightly stronger result : the propagation of chaos of the stochastic paths towards the solution of the expected nonlinear stochastic differential equation. Moreover, the convergence holds in a strong sense, usually called entropic (there is no loss of entropy in the limit). The result holds without restriction (but positivity) on the viscosity parameter. The main difficulty is the presence of the singular Biot-Savart kernel in the equation. To overcome this problem, we use the dissipation of entropy which provides some (uniform in $N$) bound on the Fisher information of the particle system, and then use extensively that bound together with classical and new properties of the Fisher information.

[1]  H. Osada.  A stochastic differential equation arising from the vortex problem , 1985 .

[2]  H. Brezis.  Remarks on the preceding paper by M. Ben-Artzi “Global solutions of two-dimensional Navier-Stokes and Euler equations” , 1994 .

[3]  F. Malrieu.  Convergence to equilibrium for granular media equations and their Euler schemes , 2003 .

[4]  Freddy Bouchet,et al.  Statistical mechanics of two-dimensional and geophysical flows , 2011, 1110.6245.

[5]  L. Onsager,et al.  Statistical hydrodynamics , 1949 .

[6]  Camillo De Lellis,et al.  Estimates and regularity results for the DiPerna-Lions flow , 2008 .

[7]  S. Méléard.  Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models , 1996 .

[8]  Uniqueness for the two-dimensional Navier–Stokes equation with a measure as initial vorticity , 2004, math/0406297.

[9]  Vladimir I. Clue.  Harmonic analysis , 2004, 2004 IEEE Electro/Information Technology Conference.

[10]  C. Marchioro.  On the Inviscid Limit for a Fluid with a Concentrated Vorticity , 1998 .

[11]  Shinzo Watanabe,et al.  On the uniqueness of solutions of stochastic difierential equations , 1971 .

[12]  Branching process associated with 2d-Navier Stokes equation , 2001 .

[13]  C. Villani,et al.  Entropy and chaos in the Kac model , 2008, 0808.3192.

[14]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[15]  P. Lions,et al.  On the Fokker-Planck-Boltzmann equation , 1988 .

[16]  Emmanuel Cépa,et al.  Diffusing particles with electrostatic repulsion , 1997 .

[17]  M. Ben-Artzi.  Global solutions of two-dimensional Navier-Stokes and euler equations , 1994 .

[18]  Mario Pulvirenti,et al.  Mathematical Theory of Incompressible Nonviscous Fluids , 1993 .

[19]  H. Osada.  Limit Points of Empirical Distributions of Vorticies with Small Viscosity , 1987 .

[20]  P. Lions,et al.  Existence and Uniqueness of Solutions to Fokker–Planck Type Equations with Irregular Coefficients , 2008 .

[21]  Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation , 2004, math/0402449.

[22]  E. Carlen.  Superadditivity of Fisher's information and logarithmic Sobolev inequalities , 1991 .

[23]  M. Pulvirenti,et al.  Hydrodynamics in two dimensions and vortex theory , 1982 .

[24]  H. Helmholtz.  Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. , 1858 .

[25]  St'ephane Mischler,et al.  On Kac's chaos and related problems , 2014 .

[26]  A. Sznitman.  Topics in propagation of chaos , 1991 .

[27]  S. Mischler,et al.  A new approach to quantitative propagation of chaos for drift, diffusion and jump processes , 2011, 1101.4727.

[28]  H. Osada.  Diffusion processes with generators of generalized divergence form , 1987 .

[29]  P. Lions,et al.  A special class of stationary flows for two-dimensional euler equations: A statistical mechanics description. Part II , 1995 .

[30]  L. J. Savage,et al.  Symmetric measures on Cartesian products , 1955 .

[31]  A. Sznitman.  Équations de type de Boltzmann, spatialement homogènes , 1984 .

[32]  A. Chorin.  Numerical study of slightly viscous flow , 1973, Journal of Fluid Mechanics.

[33]  Derek W. Robinson,et al.  Mean entropy of states in classical statistical mechanics , 1967 .

[34]  Steven Schochet,et al.  THE POINT-VORTEX METHOD FOR PERIODIC WEAK SOLUTIONS OF THE 2-D EULER EQUATIONS , 1996 .

[35]  Paths Clustering and an Existence Result for Stochastic Vortex Systems , 2007 .

[36]  Arnaud Guillin,et al.  Uniform Convergence to Equilibrium for Granular Media , 2012, 1204.4138.

[37]  On the existence and uniqueness of SDE describing an $n$-particle system interacting via a singular potential , 1985 .

[38]  M. Kac.  Foundations of Kinetic Theory , 1956 .

[39]  F. Flandoli,et al.  Full well-posedness of point vortex dynamics corresponding to stochastic 2D Euler equations , 2010, 1004.1407.

[40]  Sylvie Méléard,et al.  Monte-Carlo approximations for 2d Navier-Stokes equations with measure initial data , 2001 .

[41]  Alessio Figalli,et al.  Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients , 2008 .

[42]  T. Gallay.  Interaction of Vortices in Weakly Viscous Planar Flows , 2009, 0908.2518.

[43]  D. Aalto,et al.  Maximal Functions in Sobolev Spaces , 2009 .

[44]  H. Helson.  Harmonic Analysis , 1983 .

[45]  P. Lions,et al.  Ordinary differential equations, transport theory and Sobolev spaces , 1989 .

[46]  Hirofumi Osada,et al.  Propagation of chaos for the two dimensional Navier-Stokes equation , 1986 .

[47]  B. D. Finetti.  La prévision : ses lois logiques, ses sources subjectives , 1937 .