Mean-Field Limits for Some Riesz Interaction Gradient Flows

This paper is concerned with the mean-field limit for the gradient flow evolution of particle systems with pairwise Riesz interactions, as the number of particles tends to infinity. Based on a modulated energy method, using regularity and stability properties of the limiting equation, as inspired by the work of Serfaty in the context of the Ginzburg-Landau vortices, we prove a mean-field limit result in dimensions 1 and 2 in cases for which this problem was still open.

[1]  J. Vázquez,et al.  Nonlinear Porous Medium Flow with Fractional Potential Pressure , 2010, 1001.0410.

[2]  S. Serfaty.  Mean Field Limits of the Gross-Pitaevskii and Parabolic Ginzburg-Landau Equations , 2015, 1507.03821.

[3]  Chunting Lu,et al.  Hydrodynamic Limits of the Boltzmann Equation , 2018 .

[4]  Etienne Sandier,et al.  Lower Bounds for the Energy of Unit Vector Fields and Applications , 1998 .

[5]  Well-posedness of a porous medium flow with fractional pressure in Sobolev spaces , 2016, 1603.01818.

[6]  Ping Zhang,et al.  On the hydrodynamic limit of Ginzburg-Landau vortices , 1999 .

[7]  R. Monneau,et al.  Nonlinear Diffusion of Dislocation Density and Self-Similar Solutions , 2008, 0812.4979.

[8]  J. Vázquez,et al.  Regularity of solutions of the fractional porous medium flow , 2012, 1409.8190.

[9]  Y. Brenier,et al.  convergence of the vlasov-poisson system to the incompressible euler equations , 2000 .

[10]  L. Caffarelli,et al.  An Extension Problem Related to the Fractional Laplacian , 2006, math/0608640.

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

[12]  Large Deviation Techniques Applied to Systems with Long-Range Interactions , 2004, cond-mat/0406358.

[13]  Sylvia Serfaty,et al.  Large deviation principle for empirical fields of Log and Riesz gases , 2015, Inventiones mathematicae.

[14]  S. Serfaty,et al.  NEXT ORDER ASYMPTOTICS AND RENORMALIZED ENERGY FOR RIESZ INTERACTIONS , 2014, Journal of the Institute of Mathematics of Jussieu.

[15]  Steven Schochet,et al.  The weak vorticity formulation of the 2-D Euler equations and concentration-cancellation , 1995 .

[16]  J. Delort.  Existence de nappes de tourbillon en dimension deux , 1991 .

[17]  R. Berman,et al.  Propagation of chaos, Wasserstein gradient flows and toric Kahler-Einstein metrics , 2015, 1501.07820.

[18]  Horng-Tzer Yau,et al.  Relative entropy and hydrodynamics of Ginzburg-Landau models , 1991 .

[19]  R. Jerrard.  Lower bounds for generalized Ginzburg-Landau functionals , 1999 .

[20]  E. Saff,et al.  Discretizing Manifolds via Minimum Energy Points , 2004 .

[21]  M. SIAMJ.,et al.  EXISTENCE OF WEAK SOLUTIONS TO SOME VORTEX DENSITY MODELS , 2003 .

[22]  Martial Mazars,et al.  Long ranged interactions in computer simulations and for quasi-2D systems , 2011 .

[23]  J. A. Carrillo,et al.  The derivation of swarming models: Mean-field limit and Wasserstein distances , 2013, 1304.5776.

[24]  Sylvia Serfaty,et al.  Vortices in the Magnetic Ginzburg-Landau Model , 2006 .

[25]  L. Ambrosio,et al.  A gradient flow approach to an evolution problem arising in superconductivity , 2008 .

[26]  M. Hauray.  WASSERSTEIN DISTANCES FOR VORTICES APPROXIMATION OF EULER-TYPE EQUATIONS , 2009 .

[27]  Andrea L. Bertozzi,et al.  AGGREGATION AND SPREADING VIA THE NEWTONIAN POTENTIAL: THE DYNAMICS OF PATCH SOLUTIONS , 2012 .

[28]  J. Vázquez,et al.  A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators , 2014 .