By William Snow Burnside, Arthur William Panton
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Extra resources for An Introduction to Determinants, Being a Chapter from The Theory of Equations
3 Asymptotic Fluctuations Multiplying the Boltzmann equation by and letting g ! g as ! 0; ! jvj2 3/ : (12) Notice that, in this case, g is parametrized by its own moments in the v variable, since D hgi ; u D hvgi ; and Â D h. 31 jvj2 1/gi : (13) 54 F. Golse This observation is important in the rigorous derivation of the incompressible Navier-Stokes equations from the Boltzmann equation. Henceforth, we systematically use the following notation. 4 The Incompressibility and Boussinesq Relations The continuity equation (local conservation of mass) reads @t hg i C divx hvg i D 0 ; and passing to the limit in the sense of distributions, we expect that hvg i !
The reader should be aware that the terminology of “incompressible NavierStokes limit” is misleading from the physical viewpoint. It is true that the motion equation satisfied by the velocity field u coincides with the Navier-Stokes equation for an incompressible fluid with constant density. However, the diffusion coefficient in the temperature equation is 3=5 of its value for an incompressible fluid with the same heat capacity and heat conductivity. The difference comes from the work of the pressure: see the detailed discussion of this subtle point in  on pp.
S2 and D is its Legendre dual. e. -L. Lions and N. e. without deriving the heat equation for Â. 6 Incompressible Navier-Stokes Limit Finally, we discuss the case where viscous dissipation and heat diffusion are observed in the fluid dynamic limit, together with the nonlinear convection term. This follows from a scaling assumption where the length and time scale are respectively 1= and 1= 2 (corresponding to the invariance scaling for the heat equation), 24 F. Golse while the size of the fluctuation is precisely of order .
An Introduction to Determinants, Being a Chapter from The Theory of Equations by William Snow Burnside, Arthur William Panton