Incompressible Navier-Stokes limit for the Enskog equation

K e y w o r d s K i n e t i c equations, Navier-Stokes system, Singularly perturbed problem, Hydrodynamic limit. The Boltzmann equation, in the hydrodynamic limit, as both the Knudsen number and the Mach number are of the same order and tend to 0, is consistent with the incompressible NavierStokes equation (set [1-a]). Our aim is to study the incompressible macroscopic limit for the Enskog kinetic equation--a quite successful model of kinetic theory of moderately dense gases [4-8]. Tile compressible macroscopic limit for the Enskog equation has been studied in [9] at the level of the Enskog-Euler system. To our knowledge, this paper is the first one in which an incompressible macroscopic limit, of the Enskog equation is presented. We consider the kinetic Enskog equation with a constant pair correlation function (the so-called Boltzmann-Enskog equation) in the dimensionless form [101 1 Ot f + l v c 0x f = )-ffE.(f), (1) The work of M.L. was partially supported by the Polish State Committee for Scientific Research. 0893-9659/00/$ see front mat ter @ 2000 Elsevier Science Ltd. All rights reserved. Typeset by AAdN-q~'~X PII: S0893-9659 (00) 00084-7 108 S. JAGODZII~SKI AND M. LACHOWICZ where E~ is the (bilinear) Boltzmann-Enskog collision operator E ~ ( f ) ( t , x , v ) = f S (f( t , x + an, w ' ) / ( t , x, v ') S ( t ,x an, w)S(t, x, v)) ( (n . (w v)) V 0) dn dw, Rs $2, S 2 = {n E R3: Inl = 1}, v' = v + ( ( w v ) . n ) n , w' = w ( ( w v ) . n ) n , the (small) dimensionless parameter 0 < a < c~ is the diameter of particles, Cl V c2 = max {Cl, c2}, f = f ( t , x, v) is the one-particle distribution function, f : R~_ x R3 x ]R 3 --* ]i~, t E R~_ is time, x E ~3--posi t ion and v E R1--velocity variables. Note that for a = 0, equation (1) becomes the Boltzmann equation [1-4,10]. We start with the formal expansion procedure taking a = bG b = const > 0, ~ I 0. (2) This condition corresponds to the case when the particle diameter is of order of the Knudsen number, i.e., N a 3 = b (3) in the limit a ~ 0, N Too, where N is the number of particles. Formally, in limit (2), equation (1) leads to the solution of the form of Maxwellian / Iv u(t , x) i [ y , u, T](t, x, v) -g(t, x) (2~rT(t, x)) -3/2 exp [ 2T(t, x) / / ' \ (4) for some fluid-dynamic parameters p (local density), u (macroscopic velocity), T (temperature). The solution to equation (1) is searched in the form f = M[Qo + eQ1, eu, To + sT1] + e2f (2) + e3f (3) + . . . , (5) where 8o > 0 and To > 0 are constants (cf. [1,2,10]). Let w = w(v) be the global Maxwellian w = M[Qo, 0, To]. Consequently, M[Qo +eQ1 , e u , T o +eT1] = w (1 +c¢1 +~2¢2) + .. , (6) where (~l(V) ---~01 3T 1 v . u T1 v 2, 15T12 3~1T1 lu]2 ( LOl 5 T 1 ) ¢ ~ ( v ) = 8T3 2~oTo 2--G + \ v . + QoT0 2T3 2T3 ( Q1T1 5T?'~ TI[Vl 2 T?]vI 4 + ~ , ~ 4To3 j Ivl 2 + ~ v . u + 8---~0 4 Equation (5) together with (6) and (7) lead to the following set of equations: (V • U) 2 (7b) P (0)V" Ox¢l) ---2b~J(1)(w,w¢l), (8) 2 J ( w , • ± f (2)) = P ± ( w v . 0x¢l) 2 b P ± j ( 1 ) ( w , w ¢ l ) 2 J ( w , w ¢ 2 ) J(oJ¢l ,w¢]) , (9) ~ (~o,¢1 +~v.Ox¢~+V.Oxf ̀ ~') (lO) = 2b2pJ(2)(w, w¢1) + 2bPJ(1)(w, w¢2) + bT)J(')(w¢l, we, ) + 2bT~J O) (w, f(2)) , . . . , Enskog Equation 109 where Y = E.l.=o is the Boltzmann collision operator and j(k) is the symmetrization of the operator 1 .](~:)(f,f)(t,x,v) = ~ / / ( ( ( n . 0x) k f ) ( t , x , w ' ) f (t, x ,v ' ) F~3 $2 ( ( n . Ox)': f ) ( t , x , w ) f (t, x, v)) (n . (w v) V (,) ([i1 (Iw, 7 ) and 7 9± are projections (in L 2 ( w l ( v ) d r ) onto A / = lin {.(~A~, j = 0 . . . . ,4} and R = A/J{f c L2(a;-l(v) dv) : J~3'~zj(v)f(v)dv = 0, Vj = 0 , . . . , 4} , respectively, ~%(v) l/v%~i~o, u'i(v) = vj/Px/-~o-~oTo, (j = 1, 2, 3), and ~%(v) = (Iv[ 2 3 7 } , ) / ( ~ 7 } , ) are the collisi, m invariants, pj.(~,) = ~ (o(k) '~o o (k) 3 + ~ . (11) From equation (8), we obtain the incompressibility relation