Fluctuation-Dissipation Equation and Incompressible Navier-Stokes Equations
نویسنده
چکیده
We review some recent results concerning the hydrodynamical limits of lattice gases, in particular, lattice gases on the cubic lattice Z Z d ; d = 3, with the incompressible Navier-Stokes equations as the hydrodynamical limit. We shall state precisely the law of large numbers theorem stating that for initial distributions corresponding to arbitrary macroscopic L 2 initial data the distributions of the evolving empirical momentum densities are supported entirely on global weak solutions of the incompressible Navier-Stokes equations. Furthermore the rate function for the large deviation will be described. The viscosity will be characterized by the Green-Kubo formula and variational principles. The key analytic inputs are a method solving the uctuation-dissipation equation of the lattice gases and multiscale estimates based on the logarithmic Sobolev inequality. A precise formulation of the uctuation-dissipation equation will be sketched. Some discussions of its relation to the Green-Kubo formula and the variational formulas will also be sketched.
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