Content-Type: multipart/mixed; boundary="-------------0104031250698" This is a multi-part message in MIME format. ---------------0104031250698 Content-Type: text/plain; name="01-133.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-133.comments" Journal of Statistical Physics 101, 1087 (2000) ---------------0104031250698 Content-Type: text/plain; name="01-133.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-133.keywords" phase segregation kinetics, binary fluids, Vlasov-Boltzmann, kinetic equations, Vlasov-Euler, Vlasov-Navier-Stokes, hydrodynamic equations, Chapman-Enskog ---------------0104031250698 Content-Type: application/x-tex; name="jsp2000.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="jsp2000.tex" %&Plain \newcount\driver \newcount\mgnf \newcount\tipi \newskip\ttglue %%cm completo \def\TIPITOT{ \font\quattordicirm=cmr12 scaled\magstep2 \font\quattordicibf=cmbx12 scaled\magstep2 \font\quattordicisy=cmsy10 scaled\magstep2 \font\quattordiciex=cmex10 scaled\magstep2 \font\dodicirm=cmr12 \font\dodicii=cmmi12 \font\dodicisy=cmsy10 scaled\magstep1 \font\dodiciex=cmex10 scaled\magstep1 \font\dodiciit=cmti12 \font\dodicitt=cmtt12 \font\dodicibf=cmbx12 scaled\magstep1 \font\dodicisl=cmsl12 \font\ninerm=cmr9 \font\ninesy=cmsy9 \font\eightrm=cmr8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightbf=cmbx8 \font\eighttt=cmtt8 \font\eightsl=cmsl8 \font\eightit=cmti8 \font\seirm=cmr6 \font\seibf=cmbx6 \font\seii=cmmi6 \font\seisy=cmsy6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \font\dodicitruecmr=cmr10 scaled\magstep1 \font\dodicitruecmsy=cmsy10 scaled\magstep1 \font\tentruecmr=cmr10 \font\tentruecmsy=cmsy10 \font\eighttruecmr=cmr8 \font\eighttruecmsy=cmsy8 \font\seventruecmr=cmr7 \font\seventruecmsy=cmsy7 \font\seitruecmr=cmr6 \font\seitruecmsy=cmsy6 \font\fivetruecmr=cmr5 \font\fivetruecmsy=cmsy5 %%%% definizioni per 10pt %%%%%%%% 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\or {\rm ottobre} \or {\rm novembre} \or {\rm dicembre} \fi\,\number\year} \newcount\pgn \pgn=1 \def\foglio{\number\numsec:\number\pgn \global\advance\pgn by 1} \def\foglioa{A\number\numsec:\number\pgn \global\advance\pgn by 1} \def\footnormal{ \footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm \foglio\hss} } \def\footappendix{ \footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm \foglioa\hss} } \global\newcount\numsec\global\newcount\numfor %\global\newcount\numfig \gdef\profonditastruttura{\dp\strutbox} \def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax} \def\SIA #1,#2,#3 {\senondefinito{#1#2}% \expandafter\xdef\csname #1#2\endcsname{#3}\else \write16{???? ma #1,#2 e' gia' stato definito !!!!} \fi} \def\etichetta(#1){(\veroparagrafo.\veraformula)% \SIA e,#1,(\veroparagrafo.\veraformula) % \global\advance\numfor by 1% \write15{\string\FU (#1){\equ(#1)}}% %\write16{ EQ \equ(#1) <==#1 } } \def\FU(#1)#2{\SIA fu,#1,#2 } 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%%%%%%%%%%%%%%% DEFINIZIONI LOCALI \let\ciao=\bye \def\fiat{{}} \def\pagina{{\vfill\eject}} \def\\{\noindent} \def\bra#1{{\langle#1|}} \def\ket#1{{|#1\rangle}} \def\media#1{{\langle#1\rangle}} \def\ie{\hbox{\it i.e.\ }} \let\ii=\int \let\ig=\int \let\io=\infty %\let\i=\infty \let\dpr=\partial \let \pt=\partial \let \imp=\Longrightarrow \let \n\nabla \def\V#1{\vec#1} \def\Dp{\V\dpr} \def\oo{{\V\o}} \def\OO{{\V\O}} \def\uu{{\V\y}} \def\xxi{{\V \xi}} \def\xx{{\V x}} \def\yy{{\V y}} \def\kk{{\V k}} \def\zz{{\V z}} \def\rr{{\V r}} \def\pp{{\V p}} \def\tende#1{\vtop{\ialign{##\crcr\rightarrowfill\crcr \noalign{\kern-1pt\nointerlineskip} \hskip3.pt${\scriptstyle #1}$\hskip3.pt\crcr}}} \def\otto{{\kern-1.truept\leftarrow\kern-5.truept\to\kern-1.truept}} \def\guida{\ ...\ } \def\Z{{\bf Z}}\def\R{{\bf R}}\def\tab{}\def\nonumber{} \def\mbox{\hbox}\def\lis#1{{\overline#1}}\def\nn{{\V n}} \def\Tr{{\rm Tr}\,}\def\EE{{\cal E}} \def\Veff{{V_{\rm eff}}}\def\Pdy{{P(d\psi)}}\def\const{{\rm const}} %\def\RR{{\cal R}} \def\NN{{\cal N}}\def\ZZ#1{{1\over Z_{#1}}} \def\OO{{\cal O}} \def\GG{{\cal G}} \def\LL{{\cal L}} \def\DD{{\cal D}} \def\fra#1#2{{#1\over#2}} \def\ap{{\it a priori\ }} \def\rad#1{{\sqrt{#1}\,}} \def\eg{{\it e.g.\ }} \def\={{\equiv}} \def\ch{{\chi}} \def\NoBlackBoxes{\global\overfullrule0pt} \def\vec{\overrightarrow} \def\n{\nabla} \def\supt{\sup_{t\in [0, \bar t]}} \def\initfiat#1#2#3{ \mgnf=#1 \driver=#2 \tipi=#3 \TIPITOT %\ifnum\driver=0 \special{ps: plotfile ini.pst global} \fi %\ifnum\driver=1 \special{header=ini.pst} \fi %%%%%%%%%%%%%%% FORMATO \ifnum\mgnf=0 \magnification=\magstep0\hoffset=0.cm \voffset=-1truecm\hoffset=-.5truecm\hsize=16.5truecm \vsize=25.truecm \baselineskip=15pt % plus0.1pt minus0.1pt \parindent=12pt \lineskip=4pt\lineskiplimit=0.1pt \parskip=0.1pt plus1pt \def\ds{\displaystyle}\def\st{\scriptstyle}\def\sst{\scriptscriptstyle} \font\seven=cmr7 \fi \ifnum\mgnf=1 \magnification=\magstep1\hoffset=0.cm \voffset=-1truecm\hoffset=-.5truecm\hsize=16.5truecm \vsize=24.truecm \baselineskip=15pt % plus0.1pt minus0.1pt \parindent=12pt \lineskip=4pt\lineskiplimit=0.1pt \parskip=0.1pt plus1pt \def\ds{\displaystyle}\def\st{\scriptstyle}\def\sst{\scriptscriptstyle} \font\seven=cmr7 \fi \setbox200\hbox{$\scriptscriptstyle \data $} } %\def\reset{\pgn=1\footline={\hss{\tenrm \foglio}\hss}} \def\reset{} \def \II{\ \hbox{I}\!\!\!\hbox{I}\,} \def\11{\hbox{l}\!\!\!\hbox{1}\,} \def\ra{\rightarrow} \def\pa{\parallel} \def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt \hbox{\vrule width.#2pt height #1pt \kern#1pt \vrule width.#2pt} \hrule height.#2pt}}}} \def\square{\mathchoice\sqr56\sqr56\sqr{2.1}3\sqr{1.5}3} \def\qed{$\square$} \def\picture #1 by #2 (#3){ \vbox to #2{ \hrule width #1 height 0pt depth 0pt \vfill \special{picture #3} } } \def\scaledpicture #1 by #2 (#3 scaled #4){{ \dimen0=#1 \dimen1=#2 \divide\dimen0 by 1000 \multiply\dimen0 by #4 \divide\dimen1 by 1000 \multiply\dimen1 by #4 \picture \dimen0 by \dimen1 (#3 scaled #4)} } \def\union{\mathop{\bigcup}} \def\und(#1){$\underline{\hbox{#1}}$} \def \div{\hbox{\rm div\/\/}} \def \rot{\hbox{\rm rot\/\/}} \def\xp(#1){\hbox{\rm e}^{#1}} %\input amssym.tex \input amssym.def %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% GRECO \let\a=\alpha \let\b=\beta \let\c=\chi \let\d=\delta \let\e=\varepsilon \let\f=\varphi \let\g=\gamma \let\h=\eta \let\k=\kappa \let\l=\lambda \let\m=\mu %\let\n=\nu \let\o=\omega \let\p=\pi \let\ps=\psi \let\r=\rho \let\s=\sigma \let\t=\tau \let\th=\vartheta \let\y=\upsilon \let\x=\xi \let\z=\zeta \let\D=\Delta \let\F=\Phi \let\G=\Gamma \let\L=\Lambda \let\Th=\Theta \let\O=\Omega \let\P=\Pi \let\Ps=\Psi \let\Si=\Sigma \let\X=\Xi \let\Y=\Upsilon %%%%%%%%%%%%%%% DEFINIZIONI LOCALI \def\media#1{{\Big\langle#1\Big\rangle}} \def\ie{\hbox{\it i.e.\ }} \def\mbox{\hbox}\def\lis#1{{\overline#1}}\def\nn{{\V n}} \def\Tr{{\rm Tr}\,} \def\const{{\rm const}} \def\eg{{\it e.g.\ }} \def\={{\equiv}}\def\ch{{\chi}} \def\W#1{#1_{{\kern-3pt\lower6.5pt\hbox{$\widetilde{}$}}% \kern3pt_{\raise0pt\hbox{\kern0.7pt\ottopunti${}$}}}} \def\nequ {=\hskip-3 mm/} \def\neque {=\hskip-2 mm/} \def\neq{=\hskip-.32cm/\hskip.18cm} \def\sneq{=\hskip-.18cm/\hskip.07cm} %\head \def\head (#1,#2){\vskip 1cm \numsec= #1 \numfor= 1 {\noindent\bf #1. #2.}\vskip .7cm} %\theorem \def\theorem(#1,#2){\vskip.3cm\noindent{\bf Theorem #1.\ }{\it #2.}} %\lemma \def\lemma(#1,#2){\vskip.3cm\noindent{\bf Lemma #1.\ }{\it #2.}} %\propos \def\propos(#1,#2){\vskip.3cm\noindent{\bf Proposition #1.\ }{\it #2.}} %\un \def\un#1{\underline{#1}} %\proof \def\proof{\noindent$\un{\hbox{\rm Proof}}$:\hskip.3cm} %\div \def\div{\hbox{\rm div }} \let\pt=\partial \def\ex#1{\hbox{\rm e}^{#1}} \def\defi{\,{\buildrel def \over =}\,} \def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt \hbox{\vrule width.#2pt height #1pt \kern#1pt \vrule width.#2pt} \hrule height.#2pt}}}} \def\square{\mathchoice\sqr56\sqr56\sqr{2.1}3\sqr{1.5}3} \def\qed{$\square$} \def\v{\varphi} \def\bstar{\bigcirc\hskip-.34cm *\hskip.2cm} %\def \IR{\Bbb R} %\def\IZ{\Bbb Z} \def\IR{{\rm I\kern -1.8pt{\rm R}}} \def\IP{{\rm I\kern -1.6pt{\rm P}}} \def\IZ{{\rm Z\kern -4.0pt{\rm Z}}} \def\IC{\ {\rm I\kern -6.0pt{\rm C}}} \def\tf{\tilde\f} \NoBlackBoxes \initfiat {1}{1}{2} %\BOZZA \magnification=1200 \baselineskip15pt \centerline{\bf Binary Fluids with Long Range Segregating Interaction I:} \centerline{\bf Derivation of Kinetic and Hydrodynamic Equations.} \vskip.3cm \centerline{by} \vskip1cm \centerline{S. Bastea\footnote{$^\dagger$} {\eightrm Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, CA 94550, USA},\hskip.2cm R. Esposito\footnote {$^*$} {\eightrm Dipartimento di Matematica, Universit\'a degli Studi di L'Aquila, Coppito, 67100 L'Aquila, Italy}, \hskip .2cm J. L. Lebowitz\footnote {$^+$} {\eightrm Departments of Mathematics and Physics, Rutgers University, New Brunswick, NJ 08903, USA} \hskip.1cm and \hskip.1cm R. Marra\footnote {$^\#$} {\eightrm Dipartimento di Fisica and Unita INFM, Universit\`a di Roma Tor Vergata, 00133 Roma, Italy.} } \vskip .8cm \bigskip \bigskip {\baselineskip = 10pt\rightskip1.4cm\leftskip 1.4cm \ottopunti {\noindent {\bf Abstract:}\/} We study the evolution of a two component fluid consisting of ``blue'' and ``red'' particles which interact via strong short range (hard core) and weak long range pair potentials. At low temperatures the equilibrium state of the system is one in which there are two coexisting phases. Under suitable choices of space-time scalings and system parameters we first obtain (formally) a mesoscopic kinetic Vlasov-Boltzmann equation for the one particle position and velocity distribution functions, appropriate for a description of the phase segregation kinetics in this system. Further scalings then yield Vlasov-Euler and incompressible Vlasov-Navier-Stokes equations. We also obtain, via the usual truncation of the Chapman-Enskog expansion, compressible Vlasov-Navier-Stokes equations. \bigskip \bigskip \bigskip} \baselineskip=18pt \head(1,Introduction) The process of phase segregation in which a system evolves from an initial unstable homogeneous state into a final equilibrium state consisting of two coexisting phases is of continuing theoretical and practical interest [GSS], [FLP], [L]. Such a process occurs whenever the system, which is initially at values of the thermodynamic parameters, say temperature $T_0$ and pressure $p_0$, corresponding to a single homogeneous phase has its parameters ``suddenly'' changed to new values, say $T$ and $p$, at which there is a coexistence of phases. This happens, for example, when an alloy is `quenched' from a high temperature melt or solid to a low temperature solid state by sudden cooling [GSS]. After such a quench the system finds itself in an unstable (or metastable) situation, as far as the spatial concentrations, which have not been able to adjust rapidly enough to the "sudden" quench, are concerned and domains of the two equilibrium phases form and start growing in time. This proceeds until there are ``in the final state'' only two regions of pure equilibrium phases separated by an interface. Since the kinetics of the domain growth have a profound influence on the properties of the alloy, this problem has been and continues to be extensively studied both theoretically and experimentally [GSS]. For such alloy systems the segregation process takes place mainly through the (anti)diffusion of the two components---from a uniformly mixed state to a demixed one. There are no macroscopic matter or energy flows since the system is a solid and has a high heat conductivity which keeps the temperature equal to some constant ambient value. The only relevant conserved quantities are therefore the particle numbers of the two components and the macroscopic equations describing the process are fairly well established: these are the well known Cahn-Hilliard equations [CH] and variations on them. We refer the reader to reviews on this subject [GSS]. The situation is much less clear for phase segregation in fluids where macroscopic flows of matter and heat are important. There are now additional conservation laws for momentum and energy and there is no general consensus even on what hydrodynamical equations are most appropriate for describing the macroscopic evolution of the system [S], [OP], [AGA]. In particular, it is not clear which is the correct coupling between the Cahn-Hilliard equation for the order parameter and the Navier-Stokes equation for the fluid velocity. To make a start on the mathematical analysis of such processes we investigate a model binary fluid introduced in [BL] where the process of phase segregation was studied numerically. In the present work we derive general equations appropriate both in the one phase and in the coexistence region. In part II we consider applications to the segregation process including an analysis of new numerical results. Many of our discussions here will be semi-heuristic. In particular, we will not go into detail about the domain of validity of the technical conditions necessary for the rigorous mathematical establishment of the results. The model we study is composed of two types of particles, call them red and blue. There are $N_r$ red and $N_b$ blue particles in a cubic box of volume $\Lambda=L^d$; we will generally consider $d=3$ and use periodic boundary conditions. The particles all have unit mass and hard core diameter $a$. Particles of different kind also interact with each other through a long range pair potential of the Kac type, having a range $\ell$ and a strength $A_\ell$. By properly choosing $A_\ell$, we obtain, in the limit $\ell\to\infty$, a system whose equilibrium properties are described by a mean-field type phase diagram exhibiting a demixing phase transition for temperature $T\d ,\quad i\neq j\},$$ where $K(|x-y|)=-(\nabla U)(|x-y|)$, $N=N_r+N_b$, and we use the notation $q_i, v_i$, $i=1,\dots,N$ when the color is irrelevant. When two particles are in contact (namely at distance $\d$) they undergo an elastic collision regardless of their color. We neglect the event that more that two particles are in contact because it has vanishing Lebesgue measure. Hence the evolution \equ(?) is defined only almost everywhere. Note that when $N\sim \ell^2$ then the mean force on each particle on the kinetic scale \equ(?) is of order unity. This is the reason why our original choice of the strength $A_{\ell}$ of the potential in \equ(o.2) was like $\ell^{-2}$ rather than $\ell^{-3}$ as in the usual case [LP]. With this scaling we can get, at least formally, in the limit $\d\to 0$ the Vlasov-Boltzmann equation for a binary mixture of hard core particles interacting via a weak long range potential. A formal proof of this is given in Appendix C. The rigorous proof would require the extension of the Lanford argument to this case, an extension that is not obvious because the Vlasov part is not well controlled in the Lanford norms. Even if we have discussed the derivation of the Vlasov-Boltzmann equation only for hard spheres, from now on we consider the Vlasov-Boltzmann equations in full generality. The function $f^{r}(q,v,\t)$ (resp. $f^{b}(q,v,\t)$) is proportional to the probability density of finding a red (resp. blue) particle at $q\in \O\subset \IR^3$, with velocity $v\in \IR^3$ at time $\t\ge 0$. We notice that the relation between the $f^\a$'s and the microscopic one particle densities $\rho_{1}^\a (\xi, v, \tau_m)$ (normalized to $N_\a$) is given by $$f^\a (q, v, \tau) =\lim_{\d\to 0}\delta^{-1}\rho_{1}^\a (\d^{-1} q, v, \d^{-1}\tau).$$ The functions $f^{r}$ and $f^{b}$ are positive and normalized to $\gamma^{-3}$ for any value of $\t$. They are solutions to the equations $$\eqalign{&\pt_\t f^{r} +v\cdot \n_qf^{r} +F^{r}\cdot \n_v f^{r} = J(f^{r},f^{r})+J(f^{r},f^{b}),\cr& \pt_\t f^{b} +v\cdot \n_qf^{b} +F^{b}\cdot \n_v f^{b} = J(f^{b},f^{b})+J(f^{b},f^{r}).} \Eq(2.1) $$ The Vlasov force acting on each particle is of the Kac type, meaning that for any $\g>0$, the forces are conservative non local forces with range $\g^{-1}$ defined by the position % $$F^{\a}(q,\t)= -\n_q\int_\O d\/q' \g^3 U(\g|q-q'|)n^{\b}(q',\t),\quad \a=r,b, \quad\a\ne \b\Eq(2.2)$$ with $U(|q|)$ a smooth, non negative function of compact support and $n^r$, $n^b$ are the rescaled spatial densities of the red and blue particles: $$ n^{\a}(q,\t)=\int_{\IR^3} d\/v f^{(\a)}(q,v,\t),\qquad \int_{\Omega} d\/q\ n^{(\a)}(q,\t)=\gamma^{-3}. \Eq(2.4) $$ % For any positive functions $f$ and $g$, $J(f,g)$ denotes the effect of the collisions of particles distributed according to $g$ on the distribution $f$. Its expression is given by $$J(f,g)=\int_{\IR^3} d\/v_*\int_{S^2}d\/\omega b(|v-v_*|,\omega) [f(v')g(v'_*)-f(v)g(v_*)]. \Eq(2.3)$$ Here $b(|v|,\o)$ is the differential cross section of the short range interaction, $\o\in S_2$ is the impact parameter and $v',v'_*$ are the incoming velocities corresponding to an elastic collision with outgoing velocities $v,v_*$ and impact parameter $\o$. We assume the Grad's (see [Gra]) angular cutoff condition that $b(|v|,\o)$ is a smooth function growing at most linearly for large $|v|$, i.e. $b(|v|,\o)=|v|^\s h(\o)$ with $0\le \s\le 1$ and $h$ a smooth bounded function on $S_2$. \vskip.2cm An important property of the collisions is the entropy production inequality: let $${\cal N}_\a=\int_{\IR^3} dv J(f^\a,f^\a)\log f^\a, \quad \a=1,2,$$ $${\cal N}_{\a,\b}=\int_{\IR^3} dv J(f^\a,f^\b)\log f^\a, \quad \a,\b=1,2.$$ Then ${\cal N}_\a$'s as well as ${\cal N}_{1,2}+{\cal N}_{2,1}$ are non negative. Moreover $N_\a$ vanishes as usual if and only if the $f^\a$'s are Maxwellians: $$f^\a= M(n^\a,u^\a,T^\a;v),\quad \a=1,2$$ with $$M(n,u,T;v):={n\over(2\pi T)^{3/2}}{\displaystyle\hbox{\rm e}^{-(v-u)^2/2T}}.\Eq(maxw)$$ Furthermore ${\cal N}_{1,2}+{\cal N}_{2,1}$ vanishes if and only if the two Maxwellians have the same local temperature and mean velocities: $$u^\a=u, \quad T^\a=T, \quad \a=1,2.$$ This implies that the only solutions of the equations $$\eqalign{&J(f_1,f_1)+J(f_1,f_2)=0, \cr&J(f_2,f_2)+J(f_2,f_1)=0}$$ are Maxwellians with the same mean velocity and temperature. General arguments suggest that all the stationary solutions of equations \equ(2.1) will be Maxwellians with $u=0$, T(q)=T, and densities satisfying the equations $$ \eqalign{ T\log n^1(q)&+\int dq' \gamma^3U(\gamma |q-q'|) n^2(q')=C_1\cr T\log n^2(q)&+\int dq' \gamma^3U(\gamma |q-q'|) n^1(q')=C_2. } \Eq(1.1)$$ Beyond the spatially constant equilibria, there may be other spatially non homogeneous solutions. For example, by prescribing the boundary conditions in one dimension $$\lim_{z\to \pm \infty}n^\a(z)= \bar n^\a_{\pm},$$ one gets at small values of $T$ a solitonic solution describing the interface profile [BL]. We shall leave a discussion of this part for [BELMII] and focus here on deriving macroscopic equations for the evolution of the conserved quantities. Before closing this section, let us define $$f(q,v,\t)= {1\over 2}[f^r(q,v,\t)+f^b(q,v,\t)]$$ as the density of finding a particle at $q$ with velocity $v$ at time $\t$, independently of its color. Moreover, we set $$\phi(q,v,\t)= {1\over 2}[f^r(q,v,\t)-f^b(q,v,\t)].$$ The system \equ(2.1) can be written in the following equivalent form: $$\eqalign{&\pt_\t f +v\cdot \n_q f +2F\cdot \n_v f+ 2W\cdot \n_v\phi = 4J(f,f), \cr&\pt_\t \phi +v\cdot \n_q \phi +2F\cdot \n_v \phi+ 2W\cdot \n_vf = 4J(\phi,f),} \Eq(2.5)$$ where $F=F^{r}+F^{b}$, $W=F^{r}-F^{b}$. We can absorb the numerical factors by redefining $U$ as $U/2$ and $b$ as $b/4$ so obtaining $$\eqalign{&\pt_\t f +v\cdot \n_q f +F\cdot \n_v f+ W\cdot \n_v\phi = J(f,f), \cr&\pt_\t \phi +v\cdot \n_q \phi +F\cdot \n_v \phi+ W\cdot \n_vf = J(\phi,f).} \Eq(2.5')$$ \vskip.3cm %\vfill\eject \numsec=3 \numfor=1 \head(3, Compressible Hydrodynamics) We are interested in the behavior of the system on the macroscopic scale. To this end we introduce a scaling parameter $\e$ representing the ratio between the kinetic and macroscopic space units and, for any $t\ge 0$ and $x\in \e\O$ we set $$\t=\e^{-1}t,\quad q= \e^{-1}x.$$ We assume that at time zero the densities vary slowly on the microscopic scale $ f^i(q,v,0)=\tilde f(\e q, v, 0)$ and look for solutions of \equ(2.5') such that $$f^i(q,v,\t)=\tilde f^i(\e q, v, \e \t),\quad i=r,b$$ with $\tilde f^i$ smooth functions on $\e\O\times \IR^3\times \IR_+$. For the force we have $$F^r(\e^{-1}x,\e^{-1}t)=-\e\n_x\int_{\e \O}dx'\left({\g\over \e}\right)^d U\left[\left({\g\over \e}\right)|x-x'|\right]\tilde n^b(x',t),$$ and a similar relation for $F^b$. Therefore, if we assume $\g=\e$, also the forces are slowly varying functions and the $\tilde f$ satisfy the following system, where we remove the ``tilde's'' because in the sequel we shall always consider only the macroscopic variables: $$\eqalign{&\pt_t f +v\cdot \n_x f +F\cdot \n_v f+ W\cdot \n_v\phi = \e^{-1}J(f,f),\cr& \pt_t \phi +v\cdot \n_x \phi +F\cdot \n_v \phi+ W\cdot \n_vf = \e^{-1}J(\phi,f).} \Eq(2.7)$$ We shall use the notation $$F={\bf K}\bstar f, \quad W=-{\bf K}\bstar \phi,\Eq(2.8)$$ where, $${\bf K}(x)=-\n_x U(|x|)\Eq(2.8.0)$$ and, for any function $g$ we set $$({\bf K}\bstar g)(x,t)\defi \int_\O dx'{\bf K}(|x-x'|)\int_{\IR^3} dv g(x',v,t).\Eq(2.8.1)$$ We will show that the solution of the system \equ(2.7) is close for $\e$ small to the local equilibrium with parameters $\r^{(1)}$, $\r^{(2)}$, $u$ and $T$ satisfying the following set of hydrodynamic equations: $$\eqalign{&\pt_t \r +\n\cdot [\r u]=0,\cr& \partial_t \varphi + \nabla\cdot(\varphi u) =\e \nabla\cdot(D Q), \cr&\rho D_t u +\nabla P -{\rho\bf K}*\rho + \varphi{\bf K}*\varphi= -\e\nabla \sigma, \cr&{3\over 2} \rho D_t T +P \nabla\cdot u=\e\nabla(\kappa\nabla T) -\e\sigma: \nabla u -\e{\bf K}*\f\cdot D Q.} \Eq(4.11)$$ Here $\r=\r^{(1)}+\r^{(2)}$ is the total density, $\f=\r^{(1)}-\r^{(2)}$, $P=\rho T$ $$D_t:=\pt_t + u\cdot \n,$$ $$\sigma:=-\nu(\n u+\n u^\dagger-{2\over 3}\Bbb I\,\n\cdot u)$$ $$Q:=\nabla{\v\over\rho} + {1\over \rho^2T} (\rho^2-\f ^2){\bf K}*\f.\Eq(4.11.1)$$ $\n u^\dagger$ is the adjoint of the matrix $\n u$, $\s:\n u= \Tr (\s\n u)$, $\Bbb I$ is the unit matrix, $\nu$ and $D/\rho$ are the viscosity and the diffusion coefficients and $\kappa$ is the heat conductivity. These are computed from the VBE. The above equations, with $\e=0$ will be referred to as the Vlasov-Euler equations (VE). We assume that the initial value problem for such equations, with suitable initial data, has a sufficiently smooth solution at least on a time interval $[0,\bar t]$. Under such conditions we will prove in the next section and in Appendix A that the solution to the VBE for the binary fluid, under the Euler scaling, converges to the Maxwellian local equilibrium with parameters satisfying the VE equations in the interval $[0,\bar t]$, with an error of order $\e$ (Proposition 4.1 and Corollary 4.2). When $\e>0$, the above equations will be referred to as the Vlasov-Navier-Stokes equations (VNS). In Section 5, using also the arguments of Appendix A we will show that their solutions provide an approximation up to the order $\e^2$ to the solutions of the VBE in the Euler scaling, provided that the initial value problem for such equations has suitably smooth solutions as before. The precise statement is given in Proposition 5.1 and Corollary 5.2. Like for the usual Navier-Stokes equations, which are frequently and successfully used with $\e=1$ in physical and engineering applications, although their derivation is restricted to small values of $\e$, we will consider the VNS equations with $\e=1$ and analyze some of their properties in [BELMII]. In order to get diffusive effects as sharp limits of the VBE, it is necessary to go to the parabolic scaling where $\tau=\e^{-2}t$ and consider simultaneously a low Mach number situation. This will be discussed in Section 6. \vskip.8cm \numsec=4 \numfor=1 \head(4, Euler limit) We outline the proof of the convergence of the Vlasov-Boltzmann system to the VE equations. The proof will be completed in Appendix A. We fix the Maxwellian $M(\r,u,t)$ with $\r,u,T$ possibly depending on space and time and denote $${\cal L} f= J(M,f)+J(f,M) \Eq(2.9)$$ and $$\Gamma f=J(f,M).\Eq(2.9.5)$$ Moreover, we set $$Q(f,g)={1\over 2}[J(g,f)+J(f,g)] \Eq(2.10)$$ It is easy to check that, as for the one-component Boltzmann equation, $${\cal L} f=0\quad \hbox{ iff } \quad f=M\chi_\a,\quad \a=0,\dots,4, \Eq(2.9.7)$$ where $$\chi_0=1,\quad \chi_i=v_i,\quad i=1,\dots,3,\quad \chi_4=v^2/2 \Eq(2.11).$$ Moreover, along the same lines one gets $$\Gamma f=0 \hbox{ iff } \quad f=aM, \quad a\in \IR.\Eq(2.11.2)$$ We shall try to solve \equ(2.7) following [Ca80], in terms of a truncated Hilbert expansion of the form $$\eqalign{&f=\sum_{n=0}^K\e^n f_n+\e^m R_f,\cr&\phi=\sum_{n=0}^{K}\e^n \phi_n+\e^m R_\phi,}\Eq(2.12.5)$$ with suitably chosen positive integers $K$ and $m$. The functions $f_n$ and $\phi_n$ are computed using a Hilbert expansion and the remainders $R_f$ and $R_\phi$ are defined as the difference between the solution and the truncated expansion. \vskip.5cm We substitute in \equ(2.7) the formal power series $$f=\sum_{n=0}^\infty \e^n f_n, \quad \phi=\sum_{n=0}^\infty \e^n \phi_n,\Eq(3.1)$$ $$F=\sum_{n=0}^\infty \e^n F_n=\sum_{n=0}^\infty \e^n {\bf K}\bstar f_n,\quad W=\sum_{n=0}^\infty \e^n W_n=\sum_{n=0}^\infty \e^n {\bf K}\bstar \phi_n,\Eq(3.2)$$ and denote by $D_t$ the time derivative along the trajectories: $$D_t=\pt_t+v\cdot\n_x .$$ We have: $$\e^{-1} Q(f_0,f_0)+\sum_{n=0}^\infty\e^n\Big[ 2Q(f_0,f_{n+1})+{\cal S}_n\Big]=0,\Eq(3.4)$$ $$\e^{-1} J(\phi_0,f_0)+\sum_{n=0}^\infty\e^n\Big[ J(\phi_{n+1},f_0)+ J(\phi_0,f_{n+1})+{\cal T}_n\Big]=0,\Eq(3.5)$$ where $${\cal S}_n=\sum_{\st (h,h'):\ h, h'\ge 1 \atop \st h+h'=n+1}Q(f_h,f_{h'})-\sum_{\st (h,h'):\ h,h'\ge 0\atop h+h'=n} \big[F_h\cdot \n_v f_{h'} +W_h \cdot \n_v\phi_{h'}\big]-D_t f_{n}.\Eq(3.6)$$ $${\cal T}_n=\sum_{\st (h,h'):\ h, h'\ge 1 \atop \st h+h'=n+1}J(\phi_h,f_{h'})-\sum_{\st (h,h'):\ h,h'\ge 0\atop h+h'=n} \big[F_h\cdot \n_v \phi_{h'} +W_h \cdot \n_vf_{h'}\big]-D_t \phi_{n}.\Eq(3.7)$$ In order that the formal series solve \equ(2.7) the coefficients have to satisfy the conditions: $$\eqalign{&Q(f_0,f_0)=0,\cr& J(\phi_0,f_0)=0}\Eq(3.8)$$ and, for any $n\ge 0$, $$\eqalign{&2Q(f_0,f_{n+1})+{\cal S}_n=0,\cr& J(\phi_{n+1},f_0)+ J(\phi_0,f_{n+1})+{\cal T}_n=0.}\Eq(3.9)$$ As remarked in the previous section, the first of the conditions \equ(3.8), implies that $f_0$ is a Maxwellian with parameters depending on $x,t$: $$f_0(x,v,t)=M(\r(x,t),u(x,t),T(x,t);v):=M(v).\Eq(3.10)$$ Moreover from the second eqn. of \equ(3.8) we get $$\phi_0(x,v,t)={\f(x,t)\over \r(x,t)}M(\r(x,t),u(x,t),T(x,t);v),$$ for some suitable function $\f(x,t)$. Using \equ(2.9) and \equ(2.9.5) we can write \equ(3.9) as $$\eqalign{&{\cal L}f_{n+1}=-{\cal S}_n,\cr& \G\phi_{n+1}=- J(\phi_0,f_{n+1})-{\cal T}_n.}\Eq(3.11)$$ Since ${\cal S}_n$ and ${\cal T}_n$ only depend on the $f_k$ and $\phi_k$ for $k\le n$, we have first to solve the first eqn. and then, once $f_{n+1}$ is determined, we solve the second one for $\phi_{n+1}$. In order to check the solvability of these equations we introduce the Hilbert space ${\cal H}$ of measurable functions on $\IR^3$ such that the scalar product $$(f,g)=\int_{\IR^3}dv f(v) g(v) M^{-1}(v),\Eq(3.12)$$ is finite. In this Hilbert space the operators ${\cal L}$ and $\G$ are densely defined and symmetric. Moreover, the null spaces of ${\cal L}$ and $\G$ are the five-dimensional subspace spanned by $\{M\chi_\a, \a=0,\dots,4\}$ introduced in \equ(2.11) and the one-dimensional space spanned by $M\chi_0$ respectively. We denote by ${\cal P}$ and ${\cal K}$ the projectors on such subspaces and by ${\cal P}^\perp=1-{\cal P}$ and ${\cal K}^\perp=1-{\cal K}$ the projectors on their orthogonal complements. From the properties of ${\cal L}$ and $\G$ it is immediate to check that $${\cal P}{\cal L}=0, \quad {\cal K}\G=0.\Eq(3.13)$$ Both ${\cal L}$ and $\G$ are non positive and there are positive constants $\d$ and $\d'$ $$(f,{\cal L}f)\le -\d\|{\cal P}^\perp f\|^2,\quad (f,{\G}f)\le -\d'\|{\cal K}^\perp f\|^2.\Eq(3.14)$$ Moreover $$\eqalign{&{\cal L}=-\nu + K,\cr& \G= -\nu+\Theta,}\Eq(3.15)$$ where $$\nu(x,v,t)=\int_{Re^3}dv_* b(|v-v_*|)M(\r(x,t),u(x,t),T(x,t);v_*),\Eq(3.16)$$ is a strictly positive function such that $$\nu_0(1+|v|)^\s\le \nu(x,v,t)\le \nu_1(1+|v|)^\s\Eq(3.17)$$ for some positive constants $\nu_0$ and $\nu_1$ provided that $\r$ and $T$ are bigger than some fixed positive constants. Furthermore, $K$ and $\Theta$ are compact operators on ${\cal H}$. Therefore, using the Fredholm alternative theorem we can conclude the existence of solutions to \equ(3.11) provided that ${\cal S}_n\in{\cal P}^\perp{\cal H} $ and ${\cal T}_n+J(\phi_0,f_{n+1})\in{\cal K}^\perp{\cal H}$. These conditions can be verified inductively, as in the usual one-component Boltzmann equation. We now write the conditions for $n=0$ which determine the macroscopic equations for $\r$, $u$, $T$ and $\f$: Since ${\cal P}[\n_v f_0 ]$ and ${\cal P}[\n_v\phi_0]$ have no component along $\chi_0$, it is easy to check that the condition ${\cal P}{\cal {\cal S}}_0=0$ can be written explicitly as $$\eqalign{&\pt_t\r +\n_x\cdot[\r u]=0,\cr& \r[\pt_t u+(u\cdot \n_x)u]=-\n_x P +\r {\bf K}*\r - \f{\bf K}*\f,\cr& \r[\pt_t e+(u\cdot \n_x)e] + P\n_x \cdot u=0},\Eq(3.18)$$ where $*$ denotes the usual convolution, $P=\r T$ is the equation of state for the pressure in the perfect gas and $e=3T/2$ is its internal kinetic energy. On the other hand, since ${\cal K}J=0$ and ${\cal K}\n_v=0$, the condition ${\cal K} {\cal T}_0=0$ becomes ${\cal K}D_t\phi_0=0$ which is explicitly written as $$\pt_t \f+\n_x\cdot[\f u]=0,\Eq(3.19)$$ Equations \equ(3.18) and \equ(3.19) represent the Euler equations for the binary mixture. They differ from the usual Euler equations by the presence of the equation \equ(3.19) for $\f$ and for the nonlinear self consistent force terms due to the long range Kac interaction. We will refer to them as the {\it Vlasov-Euler equations (VE)}. Existence of solutions to the initial value problem for the system \equ(3.18)-\equ(3.19) requires some analysis but we do not discuss this. We simply assume that, for sufficiently smooth initial data a unique solution of the system exists and stays smooth up to some time $\bar t$. Given such a solution, the functions $f_1$ and $\phi_1$ can be found by solving \equ(3.11) with $n=0$. In consequence, $f_1$ is determined up to $p_1\in {\cal P}{\cal H}$ and $\phi_1$ up to $q_1\in{\cal K}{\cal H}$. The procedure can then continue by taking advantage of the arbitrariness of $p_1$ and $q_1$ to satisfy the conditions ${\cal P}{\cal {\cal S}}_1=0$, ${\cal K}{\cal T}_1=0$. In this way the functions $f_n$ and $\phi_n$ can be found for any $n$. Classical results by Grad [Gra] provide the smoothness and decay properties we use below. Now we go back to the truncated expansions \equ(2.12.5). Once $\f_n$ and $\phi_n$ are computed for $n=0,\dots, K$, we can look for the equations for the remainders $R_f$ and $R_\phi$. A straightforward calculation shows that, in order that $f$ and $\phi$ satisfy \equ(2.7), $R_f$ and $R_\phi$ have to solve the equations $$\eqalign{ D_t R_f + F\cdot\n_v R_f+W\cdot \n_v R_\phi=&\e^{-1} {\cal L}R_f +{\cal L}^{(1)}R_f+\e^{m-1}[J(R_f,R_f)+A_f],\cr D_t R_\phi + F\cdot\n_v R_\phi+W\cdot \n_v R_f=&\e^{-1} \G R_\phi+\e^{-1}\tilde\Theta R_f +\G^{(1)}R_\phi+ \cr&\e^{m-1}[J(R_\phi,R_f)+A_\phi],}\Eq(3.20)$$ where $$\eqalign{&{\cal L}^{(1)} g = \sum_{h=1}^K \e^{h-1}[J(f_h,g)+J(g,f_h)],\cr& \tilde\Theta g= J(\sum_{n=0}^K\e^n \phi_n, g),\qquad \G^{(1)} g = J(g,\sum_{h=1}^K \e^{h-1}f_h),}\Eq(3.21)$$ $$\eqalign{&A_f=\e^{K-2m+1}\Big( \sum_{\st (h,h'):\ h, h'\ge 1 \atop \st h+h'> K+1}\e^{h+h'-K-1} Q(f_h,f_{h'})\cr&-\sum_{\st (h,h'):\ h,h'\ge 0\atop h+h'> K}\e^{h+h'-K} \big[F_h\cdot \n_v f_{h'} +W_h \cdot \n_v\phi_{h'}\big]-D_t f_{K}\Big)\cr& A_\phi=\e^{K-2m+1}\Big( \sum_{\st (h,h'):\ h, h'\ge 1 \atop \st h+h'> K+1}\e^{h+h'-K-1} J(\phi_h,f_{h'})\cr&-\sum_{\st (h,h'):\ h,h'\ge 0\atop h+h'> K}\e^{h+h'-K} \big[W_h\cdot \n_v f_{h'} +F_h \cdot \n_v\phi_{h'}\big]-D_t \phi_{K}\Big),}\Eq(3.21.1) $$ and $$\eqalign{&F=\sum_{n=0}^K\e^n F_n+ \e^m{\bf K}\bstar R_f, \cr& W= \sum_{n=0}^K \e^n W_n +\e^m{\bf K}\bstar R_\phi.}\Eq(3.22)$$ The expressions of $A_f$ and $A_\phi$ show that it is convenient to choose $K\ge 2m-1$ in order to get them bounded as $\e\to 0$. The construction of the solution $R_f$, $R_\phi$ of \equ(3.20) is obtained using a fixed point argument to handle the nonlinear terms in the equations. In Appendix A we shall sketch the proof of the Proposition below, which extends the similar result proved for the one-component Boltzmann gas without long-range interactions in [Ca80] and [Lac]. We assume for simplicity periodic boundary conditions, namely $\O$ is the $3$-dimensional torus of unit side. The potential of the long range force is assumed $C^\infty$, non negative and of compact support. Of course such assumptions could be relaxed, but we will not try to examine the most general setup. Moreover we use the norm $$\|f\|_{\a,\ell,s}=\sup_{v\in \IR^{3}}\Big[\hbox{\rm e}^{\a v^{2}}(1+|v|^2)^{\ell/2}|f(\,\cdot\, , v)|_s \Big]\Eq(3.25)$$ and $|f|_s$ is the Sobolev norm of order $s$. We will refer below to sufficiently smooth solutions to the VE equations meaning solutions which are in $H_s$ for some sufficiently large $s$ and such that the inequalities $$T_0\le T\le T_1, \quad \rho_0\le \rho\pm \f\le \rho_1$$ are verified for suitable positive constants $T_0$, $T_1$, $\rho_0$ and $\rho_1$. \vskip.3cm \noindent{\bf Proposition 4.1}. {\it Suppose that $(\r,u,T,\f)$ is a solution to the Vlasov-Euler equations \equ(3.18), \equ(3.19) sufficiently smooth in the time interval $[0,\bar t]$. Then there are positive constants $\e_0$ and $C$ such that, for $\e<\e_0$ a unique classical solution to the system \equ(3.20) with $m\ge 4$ exists and satisfies the bounds $$\eqalign{&\supt\|R_f(\,\cdot\, ,t)\|_{\a,\ell,s}\le C\e \supt\big[\|A_f(\,\cdot\, ,t)\|_{\a,\ell,s}+\|A_\phi(\,\cdot\, ,t)\|_{\a,\ell,s}\big],\cr& \supt\|R_\phi(\,\cdot\, ,t)\|_{\a,\ell,s}\le C\e \supt\big[\|A_f(\,\cdot\, ,t)\|_{\a,\ell,s}+\|A_\phi(\,\cdot\, ,t)\|_{\a,\ell,s}\big],}\Eq(3.26)$$ for any positive $\a<\bar T/2$, $\bar T\defi\sup_{x\in \O, t\in [0,\bar t]} T(x,t)$, $\ell>3$, $s\ge 2$.} \vskip.3cm \noindent{\bf Corollary 4.2}. {\it Under the assumptions of Proposition 3.1, for $\e<\e_0$ there is a smooth solution $(f^\e_t,\phi^\e_t)$ to the rescaled Vlasov-Boltzmann equations \equ(2.7) and moreover, denoting by $M_t$ the Maxwellian with parameters evolving according to the Euler equations, it satisfies: $$\sup_{0\le t\le \bar t}[\|f^\e_t-M_t\|_{\a,\ell,s}+\|\phi^\e_t-{\f_t\over \r_t}M_t\|_{\a,\ell,s}]\le C\e. $$} % % \vskip.8cm\goodbreak \numsec= 5 \numfor= 1 \head(5, Navier Stokes correction) \vskip.2cm The Navier-Stokes corrections to the hydrodynamical equations on the Euler scale are usually obtained by means of a suitable resummation of the Hilbert series expansion called the {\it Chapman-Enskog expansion}. For our purposes it is convenient to look at a modified version of the expansion proposed by Caflisch [Ca87]. \vskip.1cm We use the notation: for $n\ge 0$, $\hat f_n={\cal P} f_n$, $\bar f_n= {\cal P}^\perp f_n$, $\hat\phi_n={\cal K}\phi_n$, $\bar\phi_n={\cal K}^\perp\phi_n$, $F_n={\bf K}\bstar f_n$, $ W_n={\bf K}\bstar\phi_n$. The terms in the expansions are given as follows: we set $M_s:=M(1,u,T)$; $$ f_0=\r M_s, \quad \phi_0={\f} M_s,\quad \hat f_1=0,\quad \hat\phi_1=0 \Eq(2.12)$$ $${\cal L}\bar f_1={\cal P}^\bot\Big[D_t f_0 +F_0\cdot\nabla_v f_0 +W_0 \cdot\nabla_v \phi_0 \Big ] \Eq(2.14.1)$$ $$\Gamma\bar\phi_1= -J(\phi_0,\bar f_1)+{\cal K}^\bot\Big[ D_t\phi_0-\e{\f\over \r^2}{\cal P}[D_t \bar f_1] +W_0\cdot\nabla_v f_0 + F_0\cdot\nabla_v \phi_0\Big].\Eq(2.16)$$ $${\cal P}\Big[ D_t(f_0+\e \bar f_1) + F_0\cdot\nabla_v (f_0 +\e\bar f_1)+W_0\cdot\nabla_v (\phi_0 +\e\bar\phi_1) \Big ]=0 \Eq(2.14)$$ $${\cal K}\Big[ D_t(\phi_0+\e \bar \phi_1)+W_0\cdot\nabla_v (f_0 +\e\bar f_1)+ F_0\cdot\nabla_v (\phi_0 +\e\bar\phi_1)\Big]=0 \Eq(2.15)$$ $$\eqalign{&{\cal L}\bar f_{2}=-2Q( f_{1},f_1) \cr&+{\cal P}^\bot\Big[ D_t ( \bar f_1 +\e \hat f_{2})+ \big[F_0\cdot \n_v f_{1}+F_1\cdot \n_v f_{0} +W_0 \cdot \n_v\phi_{1}+W_1 \cdot \n_v\phi_{0}\big]\Big ]} \Eq(2.18)$$ $$\eqalign{&\Gamma\bar\phi_{2}=-\varphi J(M_s, f_{2}) -J(\phi_1,f_1) +{\cal K}^\bot\Big[ D_t\bar \phi_n+{\f\over \r^2}{\cal P}[D_t \bar f_1]+\e D_t\hat\phi_{2}\cr&+ \big[W_0\cdot \n_v f_{1}+W_1\cdot \n_v f_{0} +F_0 \cdot \n_v\phi_{1}+F_1 \cdot \n_v\phi_{0}\big]\Big ] } \Eq(2.20)$$ $${\cal P}\Big[ D_t f_{2} + \sum_{\st (h,h'):\ h\ge 0,h'>0\atop h+h'=2} \big[F_h\cdot \n_v f_{h'} +W_h \cdot \n_v\phi_{h'}\big]\Big ]=0 \Eq(2.17)$$ $${\cal K}\Big[ D_t \phi_{2}+\sum_{\st (h,h'):\ h\ge 0,h'>0\atop h+h'=2} \big[F_h\cdot \n_v \phi_{h'} +W_h \cdot \n_vf_{h'}\big]\Big]=0 \Eq(2.19)$$ Before giving conditions defining the higher order terms of the expansion let us comment about previous conditions: Equations \equ(2.12)--\equ(2.19) have to be solved in the order they are written: \equ(2.14.1) and \equ(2.16) are used to determine $\bar f_1$ and then $\bar \phi_1$ in terms of the hydrodynamical parameters $(\r,u,T,\f)$ and their derivatives; as a consequence \equ(2.14) and \equ(2.15) only involve $(\r,u,T,\f)$ and represent the hydrodynamical equations we are looking for. Because of \equ(2.12) $f_1$ and $\phi_1$ are completely determined. Then \equ(2.18) and \equ(2.20) can be solved to find $\bar f_2$ and $\bar\phi_2$, depending only on $f_0,f_1,\phi_0, \phi_1$ and on the hydrodynamical parts of $f_2$ and $\phi_2$. Finally, \equ(2.17) and \equ(2.19) are linear equations in the hydrodynamical part of $f_2$ and $\phi_2$ which can be used to determine them. We notice that the term proportional to $\e$ in \equ(2.16) has been included to avoid third order derivatives in the hydrodynamical equations. This is usually done in the standard Chapman-Enskog expansion by {\it expanding the time derivatives in powers of $\e$}. For $n\ge 2$ we set: $$\eqalign{&{\cal L}\bar f_{n+1}=-\sum_{\st k,j\ge 1 \atop\st k+j= n+1}2Q( f_{j},f_k) \cr&+{\cal P}^\bot\Big[ D_t ( \bar f_n +\e \hat f_{n+1})+ \sum_{\st (h,h'):\ h,h'\ge 0\atop h+h'=n} \big[F_h\cdot \n_v f_{h'} +W_h \cdot \n_v\phi_{h'}\big]\Big ]} \Eq(2.18')$$ $$\eqalign{&\Gamma\bar\phi_{n+1}=-\varphi J(M_s, f_{n+1}) -\sum_{\st k,j\ge 1 \atop\st k+j= n+1}J(\phi_k,f_j)\cr& +{\cal K}^\bot\Big[ D_t\bar \phi_n+\e D_t\hat\phi_{n+1}+ \sum_{\st (h,h'):\ h,h'\ge 0\atop h+h'=n} \big[F_h\cdot \n_v \phi_{h'} +W_h\cdot \n_vf_{h'}\big]\Big] } \Eq(2.20')$$ $${\cal P}\Big[ D_t f_{n+1} + \sum_{\st (h,h'):\ h\ge 0,h'>0\atop h+h'=n+1} \big[F_h\cdot \n_v f_{h'} +W_h \cdot \n_v\phi_{h'}\big]\Big ]=0 \Eq(2.17')$$ $${\cal K}\Big[ D_t \phi_{n+1}+\sum_{\st (h,h'):\ h\ge 0,h'>0\atop h+h'=n+1} \big[F_h\cdot \n_v \phi_{h'} +W_h \cdot \n_vf_{h'}\big]\Big]=0 \Eq(2.19')$$ The procedure used for $f_2$ and $\phi_2$ can be repeated to get $f_n$ and $\phi_n$ for any $n>2$. \vskip.2cm As for the Euler limit discussed in the previous section, instead of looking for the convergence of the expansion we consider its truncation \equ(2.12.5) where the functions $f_n$ and $\phi_n$ are computed according to the procedure just explained, but setting $f_n=0$, $\phi_n=0$ for $n\ge K+1$. The remainders $R_f$ and $R_\phi$ have to be solutions of the equations $$\eqalign{D_t R_f + F\cdot\nabla_v R_f + W\cdot\nabla_v R_\phi=&\e^{-1}{\cal L}R_f+{\cal L}^{(1)}R_f+\e^{m-1}\big[ J(R_f,R_f)+ A_f \big], \cr D_t R_\phi+ W\cdot\nabla_v R_f+ F\cdot\nabla_v R_\phi=&\e^{-1}[\Gamma R_\phi +\tilde\Theta R_f]+\G^{(1)}R_\phi \cr&+\e^{m-1}\big[J(R_\phi,R_f)+A_\phi)\big]}\Eq(2.26)$$ where $F$ and $W$ are given by \equ(3.22), ${\cal L}^{(1)}$, $\G^{(1)}$ and $\tilde\Theta$ are given by \equ(3.21), while the expressions of $A_f$ and $A_\phi$ are slightly different from \equ(3.21.1), and are: $$\eqalign{&A_f=\e^{K-2m+1}\Big(\sum_{\st (h,h'):\ h, h'\ge 1\atop \st h+h'> K+1}\e^{h+h'-K-1}Q(f_h,f_{h'})-\cr&\sum_{\st (h,h'):\ h,h'\ge 0\atop h+h'> K}\e^{h+h'-K}\big[F_h\cdot \n_v f_{h'} +W_h \cdot \n_v\phi_{h'}\big]\cr& -D_t \bar f_{K}- {\cal P}[F_0\cdot \n_v f_K+W_0\cdot \n_v\phi_K]\Big),}\Eq(3.27.1)$$ $$\eqalign{&A_\phi=\e^{K-2m+1}\Big( \sum_{\st (h,h'):\ h, h'\ge 1 \atop \st h+h'> K+1}\e^{h+h'-K-1} J(\phi_h,f_{h'})\cr&-\sum_{\st (h,h'):\ h,h'\ge 0\atop h+h'> K}\e^{h+h'-K} \big[W_h\cdot \n_v f_{h'} +F_h \cdot \n_v\phi_{h'}\big]\cr&-D_t\bar \phi_{K}- {\cal K}[W_0\cdot \n_v f_K+F_0\cdot \n_v\phi_K]\Big).}\Eq(3.27.2) $$ %%\vfill\eject %\vskip.6cm %\numsec=5 %\numfor=1 %\head(5, Vlasov-Navier-Stokes equations) \vskip.3cm We now exploit the equations \equ(2.14.1)--\equ(2.15) in order to obtain the Navier-Stokes equations for the binary mixture with long range forces. Note that in \equ(2.14.1) the force terms do not contribute because ${\cal P}^\perp \n_v M_s=0$. Therefore $\bar f_1$ has the same expression as for the one component gas without self-interaction: recall that $${\cal P}^\perp(D_tf_0)=M\left[\sum_{i,j=1}^3 A_{i,j}\pt_i u_j +\sum_{i=1}^3 B_i \pt_i T\right],\Eq(5.1.3)$$ with $$A_{i,j}={1\over T}(\tilde v_i\tilde v_j -{\tilde v^2\over 3}\d_{i,j}), \quad B_i=\left({\tilde v^2\over 2} -{5\over 2}T\right){\tilde v_i\over T^2}$$ and $\tilde v=v-u$. Therefore $$\bar f_1=-\psi_1\sum_{i,j=1}^3 A_{i,j}\pt_i u_j -\psi_2\sum_{i=1}^3 B_i \pt_i T,\Eq(5.1.5)$$ with $\psi_1$ and $\psi_2$ non negative smooth functions of $|\tilde v|$. Moreover, ${\cal P}[\n_v \bar f_1]=0$ because $\bar f_1$ is orthogonal to the invariants. On the other hand $(M_s\chi_\a, \n_v \bar\phi_1)=0$ for $\a=0,\dots, 3$, because $\bar\phi_1$ is orthogonal to the constants. Furthermore $$\int dv [ F_0\cdot\nabla_v (f_0 )+W_0\cdot\nabla_v \phi_0]=0,\Eq(4.7.5)$$ $$\int dv \tilde v [ F_0\cdot\nabla_v (f_0 )+W_0\cdot\nabla_v \phi_0=-F_0\rho -W_0\varphi.\Eq(4.7)$$ Hence the mass equation is $$\pt_t \r +\n\cdot (\r u)=0,\Eq(4.7.8)$$ and the momentum equation is $$\rho D^u_t u +\nabla P= -\e\nabla \cdot\sigma +\r F_0 +\f W_0\Eq(4.8)$$ with $D^u_t=\pt _t+u\cdot\nabla$ and $$\sigma_{i,j}:=-\nu(\partial_ju_i+\partial_i u_j-{2\over 3}\delta_{i,j}\nabla\cdot u)$$ and $$\nu=\int dv \psi_1(|\tilde v|)A_{1,2}^2.$$ In order to compute the equation for the energy and for the concentration we need the expression of $\bar \phi_1$, which has to be computed using \equ(2.16). This implies $$\bar\phi_1=\Gamma^{-1}{\cal K}^\bot \Bigg[ D_t\phi_0 -\e{\f\over \r^2}{\cal P}[D_t \bar f_1]+ W_0\cdot \nabla_v f_0+ F_0 \cdot \nabla_v \phi_0 -J(\phi_0,\bar f_1)\Bigg]. \Eq(5.3)$$ We are interested in computing the component of $\n_v \bar \phi_1$ along $M\chi_4$ i.e., after an integration by parts: $$(M\chi_4, \n_v \bar \phi_1)=-\int d\tilde v \tilde v \bar \phi_1.\Eq(5.4). $$ Since $J(\phi_0,\bar f_1)={\f\over \r} {\cal L} \bar f_1 -\Gamma \bar f_1$, we get $$\Gamma^{-1}J(\phi_0,\bar f_1)={\f\over \r}\Gamma^{-1}{\cal L}\bar f_1 - \bar f_1= {\f\over \r}\Gamma^{-1}{\cal P}^\perp(D_tf_0) - \bar f_1,$$ after using \equ(2.14.1) to get the second equality The second term does not contribute to \equ(5.4) since $\bar f_1$ is orthogonal to $M\tilde v$. Hence, by \equ(5.1.5), $$\int dv\tilde v\Gamma^{-1}J(\phi_0,\bar f_1)={\f\over \rho}\int d\tilde v\tilde v\G^{-1}\big[\tilde v\cdot \nabla T {M\over 2T^2}(\tilde v^2 - 5 T)\big];\Eq(5.10)$$ the term proportional to $\n u$ is odd in $\tilde v$ and hence does not contribute to the previous expression. Now we compute $${\cal K}^\bot [D_t(\varphi M_s)+\rho W\cdot\nabla_vM_s+\phi F\cdot\nabla_vM_s] =$$ $$ \varphi D^u_t( M_s)+\varphi\tilde v\cdot \nabla M_s+M_s\tilde v\cdot\nabla\varphi +\rho W\cdot\nabla_vM_s+\f F\cdot\nabla_vM_s \Eq(4.3.9)$$ We have $$\varphi D^u_t(M_s)= {M_s\over 2T^2}\varphi D^u_tT(\tilde v^2 -3T]-\nabla_v M_s \varphi D^u_t u$$ Since the first term is even in $\tilde v$, only the second term contributes to $\int dv \tilde v\Gamma^{-1}_M$ in \equ(5.10). By using the equation for the momentum we get $$D^u_t u={1\over \rho}[-\nabla P +\rho F+\varphi W] -{\e\over \r}\n\cdot \s\Eq(4.3.11) $$ Moreover $$\big[{\f\over \rho}\nabla P-\f F-{\varphi^2\over \rho}W+\rho W+\varphi F\big]\cdot\nabla_v M_s= [{\f\over \rho}\nabla P+W{\rho^2-\f ^2\over\rho}]\cdot\nabla_v M_s.$$ Now $$\eqalign{-{\f\over \rho}\nabla P\cdot{\tilde v\over T}M_s+M_s\tilde v &\cdot\nabla\varphi=[-\f{1\over T}\nabla T-{\f\over\rho}\nabla \rho+\nabla\varphi]\cdot M_s\tilde v=\cr & [-\f{1\over T}\nabla T+\rho\nabla {\f\over\rho}]\cdot M_s\tilde v }$$ and $$\eqalign{-\f{1\over T}\tilde v\cdot \nabla T M_s& +\f\tilde v \cdot\nabla M_s=[-{1\over T}\nabla T +({\tilde v^2\over 2T^2}-{3\over 2 T})]\f\tilde v\cdot \nabla T M_s + {\f\over T}M_s \tilde v \otimes \tilde v\cdot \n u\cr &=\f\nabla T\cdot M_s\tilde v [{\tilde v^2\over 2T^2}-{5\over 2 T}]+ {\f\over T}M_s \tilde v \otimes \tilde v\cdot \n u. }\Eq(10)$$ The last term in \equ(10) does not contribute to \equ(5.4). Since the first term in the r.h.s of \equ(10) cancels out with the term in \equ(5.10) (remember that $M=\rho M_s$) we have $$\hbox{\rm r.h.s \equ(4.3.9)}=M\tilde v\Big[\nabla{\f\over\rho}- {W\over\rho^2 T}(\rho^2-\f^2) -\e{\f\over\rho^2}\n\cdot \s\Big].\Eq(11)$$ It is now easy to check that in the computation of the l.h.s. of \equ(5.3) the last term of \equ(11) is compensated by the term $\e{\f\over \r^2}{\cal P}[D_t \bar f_1]$. Collecting terms we get $$(M_s\chi_4,\n_v\bar\phi_1)=-D\Big[\nabla{\phi\over\rho}- {W\over \rho^2T}(\rho^2-\phi^2)\Big] \Eq(4.2)$$ where $$D:=\int dv M\tilde v_i\Gamma^{-1}\tilde v_i \Eq(4.3)$$ \vskip.5cm The other terms appearing in the equation for the energy are computed in the standard way: $$\int d\tilde v\chi_4\Bigg[\v\cdot \nabla \bar f_1 +F_0\cdot \nabla_v\bar f_1 +W_0\cdot\nabla_v \bar \phi_1\Bigg]=0 \Eq(4.5.1)$$ $$\nabla \cdot\int d\tilde v\chi_4 \bar f_1 =-\n\cdot [\kappa \n T],$$ with $$\kappa=\int d\tilde v \psi_2B_i^2.$$ Moreover $$\int dv\chi_4F_0\cdot\nabla_v\bar f_1=-\int dv \tilde v\cdot F_0\bar f_1=0$$ $$\int dv\chi_4W_0\cdot\nabla_v\bar\phi_1=-\int dv \tilde v\cdot W_0\bar\phi_1= -W_0\cdot DQ,$$ with $$Q:=\nabla{\f\over\rho} - {1\over \rho^2T} W (\rho^2-\f ^2)\Eq(4.5)$$ Therefore, the equation for the energy is $${3\over 2} \rho D_t T +p \nabla\cdot u=\e\nabla(\kappa\nabla T) -\e\sigma: \nabla u -\e W\cdot D Q \Eq(4.6)$$ \vskip.2cm Finally, to get the equation for the concentration we have to exploit the condition \equ(2.15). $$\eqalign{ {\cal K}&\Big[ D_t(\varphi M+\e \bar \phi_1)\Big]=\partial_t\varphi+\int dv v\cdot\nabla \varphi M +\e \int dv v\cdot\nabla \bar\phi_1 \cr= & \partial_t\varphi+ \nabla\cdot(u\f)+\e\nabla\cdot\int dv\tilde v\bar\phi_1 } \Eq(4.1.1) $$ The last term has already been computed in \equ(4.2). Therefore, the equation for $\f$ is $$\partial_t \varphi + \nabla\cdot(\varphi u) =\e \nabla\cdot(D Q)\Eq(4.4)$$ where $Q$ is given in \equ(4.5). Recalling the definitions of $F_0$ and $W_0$ we finally get the {\it Vlasov-Navier-Stokes equations} (VNS) for a binary mixture given by \equ(4.11). %in %the form %$$\eqalign{&\pt_t \r +\n\cdot [\r u]=0,\cr& %\partial_t \varphi + \nabla\cdot(\varphi u) =\e %\nabla\cdot(D Q), %\cr&\rho D_t u +\nabla P -{\rho\bf K}*\rho + \varphi{\bf K}*\varphi= %-\e\nabla \sigma, %\cr&{3\over 2} \rho D_t T +P \nabla\cdot u=\e\nabla(\kappa\nabla T) %-\e\sigma: \nabla %u -\e{\bf K}*\f\cdot %D Q.} %\Eq(4.11)$$ %with %$$Q:=\nabla{\v\over\rho} + {1\over \rho^2T} (\rho^2-\f ^2){\bf %K}*\f\Eq(4.11.1)$$ \vskip.3cm As for the Euler limit, the arguments in the Appendix A prove the following Proposition 5.1 which holds under the same assumptions as before Proposition 3.1: periodic boundary conditions and smoothness of the long range potential. \vskip.3cm \noindent{\bf Proposition 5.1}. {\it Suppose that for $\e>0$ small enough there is a solution $(\r^\e,u^\e,T^\e,\f^\e)$ to the Vlasov-Navier-Stokes equations \equ(4.11) sufficiently smooth in the time interval $[0,\bar t]$ independent of $\e$. Then there are positive constants $\e_0$ and $C$ such that, for $\e<\e_0$ an unique classical solution to the system \equ(2.26) with $m\ge 4$ exists and satisfies the bounds $$\eqalign{&\supt\|R_f(\,\cdot\, ,t)\|_{\a,\ell,s}\le C\e \supt\big[\|A_f(\,\cdot\, ,t)\|_{\a,\ell,s}+\|A_\phi(\,\cdot\, ,t)\|_{\a,\ell,s}\big],\cr& \supt\|R_\phi(\,\cdot\, ,t)\|_{\a,\ell,s}\le C\e \supt\big[\|A_f(\,\cdot\, ,t)\|_{\a,\ell,s}+\|A_\phi(\,\cdot\, ,t)\|_{\a,\ell,s}\big],}\Eq(5.26)$$ for any $\a<\bar T/2$, $\bar T\defi\sup_{x\in \O, t\in [0,\bar t]} T^\e(x,t)$, $\ell>3$, $s\ge 2$.} \vskip.3cm \noindent{\bf Corollary 5.2}. {\it Under the assumptions of Proposition 3.1, for $\e<\e_0$ there is a smooth solution $(f^\e_t,\phi^\e_t)$ to the rescaled Vlasov-Boltzmann equation \equ(2.7) and moreover, denoting by $M_t$ the Maxwellian with parameters evolving according to the Vlasov-Navier-Stokes equations, it satisfies: $$\sup_{0\le t\le \bar t}[\|f^\e_t-M_t-\e f_1\|_{\a,\ell,s}+\|\phi^\e_t-{\f_t\over \r_t}M_t-\e\phi_1\|_{\a,\ell,s}\le C\e^2]. $$} \vskip.8cm \numsec=6 \numfor=1 \head(6, Incompressible Navier-Stokes limit) \vskip.5cm In this Section we consider a different scaling limit such that one can get hydrodynamic equations with dissipative terms of order $1$ instead of order $\e$ as in the previous Section. We choose also in this case $\gamma=\e$ but we use the parabolic space time scaling $$t\to \e^{-2}t,\quad x\to \e^{-1}x. $$ After rescaling, eq.'s \equ(2.5') become: $$\pt_t f +\e^{-1}v\cdot \n_x f +\e^{-1}F\cdot \n_v f+ \e^{-1}W\cdot \n_v\phi = \e^{-2}J(f,f),\Eq(6.0)$$ % $$\pt_t \phi +\e^{-1}v\cdot \n_x \phi +\e^{-1}F\cdot \n_v \phi+\e^{-1} W\cdot \n_vf = \e^{-2}J(\phi,f),$$ with $F$ and $W$ given by \equ(2.8). We shall solve \equ(6.0) as in the Euler case in terms of a truncated Hilbert expansion of the form $$\eqalign{&f=\sum_{n=0}^K\e^n f_n+\e^m R_f,\cr&\phi=\sum_{n=0}^{K}\e^n \Phi_n+\e^m R_\Phi,}\Eq(6.1.1)$$ with suitably chosen positive integers $K$ and $m$, but in this case we choose in a different way the terms of order $0$ in the expansion. $$f_0= M_0, \quad \phi_0= 0$$ where $ M_0$ is a Maxwellian with parameters $\bar\rho$ and $\bar T$ some fixed constants and $u=0$. This implies the vanishing of the forces at the lowest order: $W_0=F_0=0$. We remark that choosing the first order term in the expansion to be a global Maxwellian is essential to the incompressible limit set up. The choice $\phi_0=0$ is made for simplifying the computations. By plugging \equ(6.1.1) in the rescaled Boltzmann equations \equ(6.0), one easily obtains the conditions for the higher order terms in the expansions: $${\cal L}f_1=0, \Eq(6.2)$$ $$\Gamma \phi_1=0,\Eq(6.2.1)$$ for $0\le n\le K-1$: $$\eqalign{&{\cal L}f_{n+2}+\sum_{\st (h,h'):\ h, h'\ge 1 \atop \st h+h'=n+2}Q(f_h,f_{h'})=\cr&\sum_{\st (h,h'):\ h>0,h'\ge 0\atop h+h'=n+1} \big[F_h\cdot \n_v f_{h'} +W_h \cdot \n_v\phi_{h'}\big]+v\cdot \n_x f_{n+1}+\pt_t f_{n},}\Eq(6.2.2)$$ $$\eqalign{&\G\phi_{n+2}+ \sum_{\st (h,h'):\ h, h'\ge 1 \atop \st h+h'=n+2}J(\phi_h,f_{h'})\cr&=\sum_{\st (h,h'):\ h,h'\ge 0\atop h+h'=n+1} \big[F_h\cdot \n_v \phi_{h'} +W_h \cdot \n_vf_{h'}\big]+v\cdot \n_x \phi_{n+1}+\pt_t \phi_{n}.}\Eq(6.2.3)$$ Moreover $$\eqalign{&\pt_t R_f +\e^{-1}\left[v\cdot R_f+ F\cdot\n_v R_f+W\cdot \n_v R_\phi\right] \cr&=\e^{-2} {\cal L}R_f+\e^{-1} {\cal L}^{(1)}R_f+{\cal L}^{(2)}R_f+\e^{m-2}[J(R_f,R_f)+A_f], \cr& \pt_t R_\phi +\e^{-1}\left[v\cdot R_\phi +F\cdot\n_v R_\phi+W\cdot \n_v R_f\right]=\e^{-2} \G R_\phi\cr& +\e^{-1}\G^{(1)}R_\phi+\e^{-1}\tilde\Theta^{(1)} R_f+\G^{(2)}R_\phi+\tilde\Theta^{(2)} R_f+ \e^{m-2}[J(R_\phi,R_f)+A_\phi],}\Eq(6.20)$$ where $$\eqalign{&{\cal L}^{(1)} g= J(f_1,g)+J(g,f_1),\quad{\cal L}^{(2)} g = \sum_{h=2}^K \e^{h-2}[J(f_h,g)+J(g,f_h)],\cr& \tilde\Theta^{(1)} g=J(\phi_1,g),\quad \tilde\Theta^{(2)} g= J(\sum_{n=2}^K\e^{n-2} \phi_n, g),\cr& \G^{(1)}g=J(g,f_1),\quad \G^{(2)} g = \sum_{h=2}^K \e^{h-2}J(g,f_h),}\Eq(6.21)$$ $$\eqalign{&A_f=\e^{K-2m+2}\Big( \sum_{\st (h,h'):\ h, h'\ge 1 \atop \st h+h'> K+2}\e^{h+h'-K-2} Q(f_h,f_{h'})\cr&-\sum_{\st (h,h'):\ h,h'\ge 0\atop h+h'> K+1}\e^{h+h'-K-1} \big[F_h\cdot \n_v f_{h'} +W_h \cdot \n_v\phi_{h'}\big]\cr& -\pt_t f_{K-1}-v\cdot \n_x f_K-\e\pt f_K\Big)\cr& A_\phi=\e^{K-2m+1}\Big( \sum_{\st (h,h'):\ h, h'\ge 1 \atop \st h+h'> K+2}\e^{h+h'-K-2} J(\phi_h,f_{h'})\cr&-\sum_{\st (h,h'):\ h,h'\ge 0\atop h+h'> K+1}\e^{h+h'-K-1} \big[W_h\cdot \n_v f_{h'} +F_h \cdot \n_v\phi_{h'}\big]\cr& -\pt_t \phi_{K-1}-v\cdot \n_x \phi_K-\e\pt \phi_K\Big),}\Eq(6.21.1) $$ and $F$ and $W$ are given by $$\eqalign{&F=\sum_{n=1}^K\e^n F_n+ \e^m{\bf K}\bstar R_f, \cr& W= \sum_{n=1}^K \e^n W_n +\e^m{\bf K}\bstar R_\phi,}\Eq(6.22)$$ with $F_n={\bf K}\bstar f_n$, $W_n={\bf K}\bstar\phi_n$. \vskip.3cm We now find the expressions of the first terms in the expansions $f_1,\phi_1,f_2,\phi_2$. \noindent From \equ(6.2.1) we get $$\phi_1=\f_1 M_0$$ and from \equ(6.2) we have $$f_1=\ M_0(v)\left(\rho_1+ u_1\cdot {v \over \bar T}+\theta_1{v^2-3\bar T\over \bar T^2}\right)$$ where $\rho_1, u_1,\theta_1$ are to be determined as functions of $x,t$. By \equ(6.2.2) with $n=0$ we obtain $${\cal P} ( v\cdot \n_x f_1 +F_1\cdot \n_v f_0 )=0.$$ But $${\cal P}[v\cdot\n_x f_1]= M_0\left[\left(1+{v^2-3\bar T\over 2 \bar T^2}\right)\n_x\cdot u_1 +{v\over \bar T}\cdot \n_x(\bar T\rho_1+\bar\rho\theta_1)\right],$$ while $${\cal P}[F_1\cdot \n_vf_0]= - M_0{v\over \bar T}\cdot F_1.$$ Hence we find the conditions $$\n_x\cdot u_1=0,\quad \n_x\Big[\rho_1+\theta_1+\int d\/y U(|x-y|)\rho_1(y)\Big]=0. \Eq(6.7)$$ On the other hand, $${\cal P}^\perp[F_1\cdot \n_v f_0 ]=0,$$ so $$f_2={\cal L}^{-1} \Big[{\cal P}^\perp v\cdot\n_x f_1\Big] -{\cal L}^{-1}J(f_1,f_1) +\hat f_2,$$ with $\hat f_2\in \hbox{\rm Null}{\cal L}$. Therefore $f_2$ has the usual expression $$f_2={1\over 2} \sum_{i,j=1}^3 A_{i,j}[ u_{1,i}u_{1,j}-\psi_1\pt_i u_{1,j}]+ \sum_{i=1}^3 B_i[\theta_1 u_{1,i}-\psi_2 \pt_i\theta_1]+ {1\over 2} M_0 \theta_1^2{\cal P}^\perp\Big[\Big({v^2-3\over 2}\Big)^2\Big].\Eq(6.99)$$ >From \equ(6.2.3) with $n=0$ we get the expression for $\phi_2$: $$\phi_2= \Gamma^{-1}\Big[ v\cdot \n_x \phi_1+W_0\n_v f_0-J(\phi_1,f_1)\Big]+\hat \phi_2,$$ with $\hat \phi_2 \in \hbox{\rm Null}{\G}$. Moreover, by \equ(6.2) $$\Gamma^{-1}J(\phi_1,f_1)=-\f_1\Gamma^{-1}\Gamma f_1= -\f_1 f_1.$$ Hence $$\phi_2= \f_1 f_1+ \Big[\n_x \f_1-{1\over \bar T} W_1\Big]\cdot \Gamma^{-1}[v M_0 ]+\hat \phi_2.$$ >From \equ(6.2.2) with $n=1$ we get $$\pt_t u_1+u_1\cdot \n_x u_1= -\n_x p + F_2+ \nu\Delta_xu_1+\rho_1 F_1 +\f_1 W_1$$ and $${5\over 2}[\pt_t \theta_1 +u_1\cdot\nabla_x\theta_1]=u_1\cdot F_1+{\kappa}\Delta_x\theta_1$$ Since $F_2=\n_x G$, with $G(x)=\int d\/y U(|x-y|)\rho_2(y)$, putting $\bar p=p-G$, the previous equation reduces to $$\pt_t u_1+u_1\cdot \n_x u_1= -\n_x \bar p+ \nu\Delta_xu_1+\rho_1 F_1 +\f_1 W_1,$$ which is the usual incompressible Navier-Stokes equation with the self-consistent force $$\rho_1 F_1 +\f_1 W_1=-\rho_1\n_x\int d\/y U(|x-y|)\rho_1(y) +\f_1\n_x\int d\/y U(|x-y|)\f_1(y).$$ Finally from \equ(6.2.1) with $n=1$ we get the equation for $\f_1$ $$\pt_t\f_1 +u_1\cdot \nabla_x \f_1= D\Big[{1\over\bar\rho}\Delta_x \f_1- {1\over\bar T}\Delta_x\int d\/y U(|x-y|)\f_1(y)\big]\Big],$$ with $$D= - \int d\/v v\cdot \Gamma^{-1}(vM).\Eq(9)$$ Summarizing, dropping the index $1$, the set of equations for $\rho, u, \theta,\phi, p$ is: $$\eqalign{&\pt_t u+u\cdot \n_x u= -\n_x p + \nu\Delta_xu+\rho F +\f W,\cr& {5\over 2}[\pt_t \theta +u\cdot\nabla_x\theta]=u\cdot F+k\Delta_x\theta,\cr& \pt_t\f +u\cdot\nabla_x \f= D\Big[{1\over\bar\rho}\Delta_x \f- {1\over\bar T }\Delta_x\int d\/y U(|x-y|)\f(y)\big]\Big], \cr&F=-\nabla_x\int dyU(|x-y|)\rho(y),\quad W= \nabla_x\int dyU(|x-y|)\f(y), \cr& \n_x\Big[\rho+\theta+\int d\/y U(|x-y|)\rho(y)\Big]=0,\cr&\nabla_x\cdot u=0.}\Eq(INS) $$ The equation for $\f$ is linear unlike the one we get in the VNS equations, but there is still a non linear term in $\f$ in the momentum equation. The equation for $\theta$ which corresponds to the deviation in the temperature decouples from the rest. In fact, if we consider a solution to the previous equation with an initial datum $\rho=const, \theta=const$ such conditions persist in time and $u$ and $\f$ have to solve the simplified set of equations $$\eqalign{&\pt_t u+u\cdot \n_x u= -\n_x p + \nu\Delta_xu +\f W, \cr& \pt_t\f +u\cdot\nabla_x \f= D\Big[{1\over\bar\rho}\Delta_x \f- {1\over\bar T }\Delta_x\int d\/yU(|x-y|)\f(y)\big]\Big], \cr&W=\nabla_x\int dyU(|x-y|)\f(y), \cr& \nabla_x\cdot u=0.}\Eq(10a)$$ \vskip.8cm In Appendix B there is the proof of the following proposition: \noindent{\bf Proposition 6.1}. {\it Suppose that for $\e>0$ small enough there is a solution $(\r,u,T,\f)$ to the incompressible Navier-Stokes equations \equ(INS) sufficiently smooth in the time interval $[0,\bar t]$ independent of $\e$. Then there are positive constants $\e_0$ and $C$ such that, for $\e<\e_0$ a unique classical solution to the system \equ(6.20) exists and satisfies the bounds $$\eqalign{&\supt\|R_f(\,\cdot\, ,t)\|_{\a,\ell,s}\le C\e \supt\big[\|A_f(\,\cdot\, ,t)\|_{\a,\ell,s}+\|A_\phi(\,\cdot\, ,t)\|_{\a,\ell,s}\big],\cr& \supt\|R_\phi(\,\cdot\, ,t)\|_{\a,\ell,s}\le C\e \supt\big[\|A_f(\,\cdot\, ,t)\|_{\a,\ell,s}+\|A_\phi(\,\cdot\, ,t)\|_{\a,\ell,s}\big],}\Eq(.26)$$ for any $\a<\bar T/2$, $\ell>3$, $s 0$, assume that there exists a sufficiently smooth solution of the Vlasov-Euler equations \equ(3.18) and \equ(3.19) in the time interval $[0,\bar t]$. Then for any $j>3$, $s0$ such that the terms in the expansion $f_i,\phi_i, \ i=1,...,K$, with $K=2m$, solutions of the equations \equ(2.17)--\equ(2.20) $$||f_i||_{\a,j,s}\le c,\quad ||\phi_i||_{\a,j,s}\le c \Eqa(A.1)$$ $$||\partial_vf_i||_{\a,j,2}\le c,\quad ||\partial_v\phi_i||_{\a,j,2}\le c\Eqa(A.1.1)$$ } \vskip.2cm We need bounds on the remainders $R_f$ and $R_\phi$. We set $$R^r\equiv R^{(1)}= R_f-R_\phi, \quad R^b\equiv R^{(2)}=R_f+R_\phi. \Eqa(RR)$$ The reason is that terms like $W\cdot \nabla_v R_\phi$ and $F\cdot \nabla_v R_f$ in \equ(3.20) are not well suited when some force term is present to represent the solutions of the equations in terms of characteristics, which is essential in the method we are going to use. The equations for the new variables are: $$\eqalign{ D_t R^{(1)}& + F^{(1)}\cdot\n_v R^{(1)}=\e^{-1} \Big[\rho^{(1)}\bar{\cal L}R^{(1)}+\rho^{(1)}\bar \Theta R^{(2)} +\rho^{(2)}\bar \Gamma R^{(1)}\Big] +{\cal L}_1 R^{(1)} \cr & + \Gamma_2 R^{(1)} + +\Theta_1 R^{(2)} + \e^{m-1}\big[J(R^{(1)},R^{(1)})+ J(R^{(1)},R^{(2)})+A^{(1)}\big],\cr % D_t R^{(2)}& + F^{(2)}\cdot\n_v R^{(2)}= \e^{-1} \Big[ \rho^{(2)}\bar{\cal L}R^{(2)}+\rho^{(2)}\bar \Theta R^{(1)} +\rho^{(1)}\bar \Gamma R^{(2)}\Big] + {\cal L}_2 R^{(2)}\cr & + \Gamma_1 R^{(2)} +\Theta_2 R^{(1)}+ \e^{m-1}\big[J(R^{(2)},R^{(2)})+ J(R^{(2)},R^{(1)})+A^{(2)}\big], }\Eqa(A.1.2)$$ where $\bar M$ is the Maxwellian $M$ with $\rho=1$ and $$\eqalign{ & f_j= {1\over 2}[f^{(1)}_j+ f^{(2)}_j],\quad \phi_j= {1\over 2}[-f^{(1)}_j+ f^{(2)}_j]\cr & \bar{\cal L} h=J( \bar M, h)+J(h,\bar M),\cr &\bar \Theta h=J( \bar M,h),\quad \bar\Gamma h= J(h,\bar M)\cr & {\cal L}_ih=J(\sum_{j=1}^K \e^{j-1}f^{(i)}_j,h)+J(h,\sum_{j=1}^K\e^{j-1} f^{(i)}_j)\cr &\Gamma_i h=J(h,\sum_{j=1}^K\e^{j-1} f^{(i)}_j), \quad \Theta_ih =J(\sum_{j=1}^K \e^{j-1}f^{(i)}_j,h)\cr & A^{(1)}=A_f+A_\phi, \quad A^{(2)}=A_f-A_\phi\cr } $$ Following Caflisch [Ca80] we now decompose the remainders in low and high velocity parts, by looking for solutions to equations \equ(A.1.2) in the form $$R^{(1)}=\sqrt{\rho^{(1)}\bar M} g^{(1)} +\sqrt{M^*} h^{(1)},\quad R^{(2)}=\sqrt {\rho^{(2)}\bar M} g^{(2)} +\sqrt {M^*} h^{(2)}$$ $M^*$ is a global Maxwellian with a temperature $T^*$. We have $$\eqalign{ D_t g^{(1)}& +F^{(1)}\cdot\nabla g^{(1)} =\e^{-1} \Big[\rho^{(1)}\bar L g^{(1)}+\sqrt{\rho^{(1)}}\sqrt{\rho^{(2)}}\bar T g^{(2)}+\rho^{(2)}\bar G g^{(1)} \Big] \cr & \hskip2cm + {\e}^{-1} \chi{\s} ^{-1} \Big[\sqrt{\rho^{(1)}}(K^* h^{(1)} + K^*_T h^{(2)})+ {\rho^{(2)}\over\sqrt{\rho^{(1)}}}K^*_G h^{(1)}\Big] % \cr D_t h^{(1)}& +F^{(1)}\cdot\nabla h^{(1)}=\sigma[\mu^{(1)} + F^{(1)} \cdot \mu'^{(1)} ] \sqrt{\rho^{(1)}}g^{(1)}+F^{(1)} \cdot \mu'_* h^{(1)}\cr &+ {\e}^{-1}\rho^{(1)}\Big[-\nu + \bar\chi (K^* h^{(1)}+ K^*_T h^{(2)}) +{\rho^{(2)}\over\rho^{(1)}}(-\nu_G+K^*_G h^{(1)})\Big] \cr & +L_1\big(\s \sqrt{\rho^{(1)}}g^{(1)}+h^{(1)}\big)+G_2\big(\s \sqrt{\rho^{(1)}}g^{(1)}+h^{(1)}\big)+T_1\big(\s\sqrt{\rho^{(2)}} g^{(2)}+h^{(2)}\big)\cr & +\e^{m-1}\Big[\nu^*Q^*\big(\s \rho^{(1)}g^{(1)}+h^{(1)},\s\rho^{(1)} g^{(1)}+h^{(1)}\big)\cr &+ \nu^*Q^*\big(\s \rho^{(1)}g^{(1)}+h^{(1)},\s \rho^{(2)}g^{(2)}+h^{(2)}\big)+A^{(1)}\Big] } \Eqa(A.4)$$ The equation for $g^{(2)},h^{(2)}$ is obtained by the exchange $1\to 2$. where $$\chi(v)=\cases{1,\quad |v|\le \gamma\cr \hskip3cm\bar\chi=1-\chi\cr 0,\quad \hbox{ otherwise}} \Eqa(A.5)$$ $$\eqalign{ \mu^{(i)}&={1\over 2} D_t (\log \bar M),\quad \mu'^{(i)}={1\over 2}\rho^{(i)}\nabla_v \log\bar M, i=1,2\cr& \mu'_*={1\over 2}\nabla_v \log M_*,\quad \s=\sqrt{\bar M\over M^*} } \Eqa(A.6)$$ $$\eqalign{ &\bar L f={1\over\sqrt {\bar M}}\bar {\cal L}\sqrt {\bar M} f)=(-\nu +K) f, \quad L^* f={1\over\sqrt {M^*}}{\cal L}\sqrt {M^*} f)=-\nu^* +K^* f\cr &\bar T f={1\over\sqrt {\bar M}}\bar \Theta \sqrt {\bar M} f,\quad \bar G f={1\over\sqrt {\bar M}}\bar \Gamma\sqrt {\bar M}f=(-\nu_G +\bar K_G) f,, \quad K^*_T={1\over\sqrt {M^*}}\bar \Theta \sqrt {M^*} f\cr &G^* f={1\over\sqrt M^*}\bar \Gamma\sqrt {M^*} f=(-\nu^*_G +K^*_G)f, \quad G_i f={1\over\sqrt {M^*}}\Gamma_i\sqrt {M^*} f, i=1,2\cr & L_i f={1\over\sqrt {M^*}}{\cal L}_i\sqrt {M^*} f, i=1,2, \quad T_i f={1\over\sqrt {M^*}}\Theta_i\sqrt {M^*} f, i=1,2\cr } \Eqa(A.7)$$ $$ \nu^* Q^*(f,\ell)={1\over\sqrt {M^*}}Q(\sqrt {M^*}f,\sqrt {M^*}\ell),\quad \nu^* J^*(f,\ell)={1\over\sqrt {M^*}}J(\sqrt {M^*}f,\sqrt {M^*}\ell)\ \Eqa(A.8)$$ \vskip.7cm {\it Linear problem} \vskip.2cm We solve first the linear problem for $g_i, h_i, i=1,2$, assuming that $F^{(i)}$ are given functions such that $$\|F^{(i)}\|_\infty+\|\n_x F^{(i)}\|_\infty<\a_F.\Eqa(bound)$$ We consider the linear system $$\eqalign{ D_t g^{(1)}& +F^{(1)}\cdot\nabla g^{(1)} =\e^{-1} \Big[\rho^{(1)}\bar L g^{(1)}+\sqrt{\rho^{(1)}}\sqrt{\rho^{(2)}}\bar T g^{(2)}+\rho^{(2)}\bar G g^{(1)} \Big] \cr & + {\e}^{-1} \chi{\s} ^{-1} \Big[\sqrt{\rho^{(1)}}(K^* h^{(1)} + K^*_T h^{(2)})+ {\rho^{(2)}\over\sqrt{\rho^{(1)}}}K^*_G h^{(1)}\Big] % \cr D_t h^{(1)}& +F^{(1)}\cdot\nabla h^{(1)}=\sigma[\mu^{(1)} + F^{(1)} \cdot \mu'^{(1)} ] \sqrt{\rho^{(1)}}g^{(1)}+F^{(1)} \cdot \mu'_* h^{(1)}\cr &+ {\e}^{-1}\rho^{(1)}\Big[-\nu + \bar\chi (K^* h^{(1)}+ K^*_T h^{(2)}) +{\rho^{(2)}\over\rho^{(1)}}(-\nu_G+K^*_G h^{(1)})\Big] \cr & +L_1\big(\s \sqrt{\rho^{(1)}}g^{(1)}+h^{(1)}\big)+G_2\big(\s \sqrt{\rho^{(1)}}g^{(1)}+h^{(1)}\big)+T_1\big(\s\sqrt{\rho^{(2)}} g^{(2)}+h^{(2)}\big)\cr & +\e^{m-1}D^{(1)} } \Eqa(A.9)$$ and the equation for $g^{(2)},h^{(2)}$ obtained by the exchange $1\to 2$. Here $ F^{(1)}$ has to be considered as a given force. We use the integral form of \equ(A.9) [ELM98]: $$\eqalign{ g^{(i)}(t,x,v)&= \int_{t^-}^t ds H^{(i)}\big(s,\f^{(i)}_{s-t}(x,v)\big)\exp\Big\{-\int_{s}^t ds'{1\over \e} \tilde\nu^{(i)}\big(\f^{(i)}_{s'-t}(x,v)\big)\Big\}. }\Eqa(A.10) $$ where $\tilde\nu^{(i)}= \r^{(i)}\nu+ \r^{(j)}\nu_G$, $\f_t^{(i)}(x,v)$ the characteristics of the equation $$\pt_t f +v\cdot \nabla_x f + F^{(i)}\cdot\nabla_{v}f=0 \Eqa(A.11)$$ and $$\eqalign{H^{(1)}=&\e^{-1} \Big[\rho^{(1)}K g^{(1)}+\sqrt{\rho^{(1)}}\sqrt{\rho^{(2)}}\bar T g^{(2)}+\rho^{(2)}K_G g^{(1)} \Big] +\cr& {\e}^{-1} \chi{\s} ^{-1} \Big[\sqrt{\rho^{(1)}}(K^* h^{(1)} + K^*_T h^{(2)})+ {\rho^{(2)}\over\sqrt{\rho^{(1)}}}K^*_G h^{(1)}\Big]} \Eqa(A.12)$$ and $H^{(2)}$ is given by the same expression after the exchange $1\to 2$. $$\eqalign{ h^{(i)}(t,x,v)&= \int_{t^-}^t ds H'^{(i)}\big(s,\f{(i)}_{s-t}(x,v)\big)\exp\Big\{-\int_{s}^t ds'{1\over \e} \hat\nu^{(i)}\big(\f{(i)}_{s'-t}(x,v)\big)\Big\}. } \Eqa(A.13) $$ with $$\hat\nu^{(i)}= \tilde \nu^{(i)}- \e \mu'_*\cdot F^{(i)}$$ (which is positive for $\e$ sufficiently small, depending on $\a_F$, since $\tilde \nu^{(i)}$ grow linearly at high velocities) and $$\eqalign{ H'^{(1)}&= {\e}^{-1}\rho^{(1)}\Big[ \bar\chi (K_* h^{(1)}+ K^*_T h^{(2)}) +{\rho^{(2)}\over\rho^{(1)}}(K^*_G h^{(1)})\Big]\cr & +{ L}^{(1)}(\s\sqrt{\rho^{(1)}} g^{(1)}+h^{(1)})+G^{(2)}(\s\sqrt{\rho^{(1)}} g^{(1)}+h^{(1)})+T^{(1)}(\s \sqrt{\rho^{(2)}} g^{(2)}+h^{(2)})\cr & +\sigma[\mu^{(1)} + F^{(1)} \mu'^{(1)} ] \sqrt{\rho^{(1)}}g^{(1)}+\e^{m-1}D^{(1)} } \Eqa(A.14)$$ We do not write explicitly the equations for $g^{(2)},h^{(2)}$ in integral form. In the following we use the compact notation: $g=\{g^{(1)},g^{(2)}\}$ and $h=\{h^{(1)},h^{(2)}\}$. Below we use the notation $\|\,\cdot\,\|_{\ell,s}=\|\,\cdot\,\|_{0.\ell,s}$ and $\|\,\cdot\,\|_{\ell}=\|\,\cdot\,\|_{0.\ell,0}$. Generalizing the method by Caflisch [Ca80]to our case, we get bounds for the norms $\|\,\cdot\,\|_{\ell}$ of $g^{(i)}$ $,h^{(i)}$ in terms of the $L_2$ norm of $g^{(i)}$ in the form $$\eqalign{ ||h||_r&\le\e(1+\a_F) ||g||_{L_2}+\e^m|D|_{r-1}\cr ||g||_r&\le ||g||_{L_2}+\e^{m+1}|D|_{r}, } \Eqa(A.16) $$ provided that $\e,\e_0$ for some suitable $\e_0$ positive and finite for any finite $\a_F$. To conclude the argument we need a bound for $||g||_{L_2}=\sum_{i=1}^2||g^{(i)}||_{L_2}$ in terms of the $L_2$ norm of $D$. This last step is not standard so that we give a sketch of the proof. To estimate $||g||_{L_2}$, we multiply the first equation in \equ(A.9) by $g^{(i)}$, $i,j=1,2, i\ne j$, respectively, integrate over $x,v$ and finally sum over $i=1,2$ $$\eqalign{ {1\over 2} {d\over dt} & \Big[||g^{(1)}||_{L_2}^2+||g^{(2)}||_{L_2}^2\Big] =\e^{-1} \Big[\big<\sqrt{\rho^{(1)}}g^{(1)}, L \sqrt{\rho^{(1)}}g^{(1)}\big>+ \big<\sqrt{\rho^{(2)}}g^{(2)}, L \sqrt{\rho^{(2)}}g_2\big> \Big]\cr & +\e^{-1} \Big[ \Big( \big<\sqrt{\rho^{(2)}}g^{(1)}, {\bar M}^{-1/2}J(\bar M,\sqrt{\bar M} \sqrt{\rho^{(1)}}g^{(2)})\big> +\cr& \big<\sqrt{\rho^{(1)}}g^{(2)},{{\bar M}}^{-1/2}J(\bar M,\sqrt{\bar M} \sqrt{\rho^{(2)}}g^{(1)}\big>\Big)+ \cr& +\big<\sqrt{\rho^{(2)}}g^{(1)},{{\bar M}}^{-1/2}J(\sqrt{\bar M}\sqrt{\rho^{(2)}} g^{(1)},\bar M)\big>+\cr& \big<\sqrt{\rho^{(1)}}g^{(2)},{{\bar M}}^{-1/2}J(\sqrt{\bar M} \sqrt{\rho^{(1)}}g^{(2)},\bar M)\big>\Big]+\cr & {\e}^{-1}\big< \chi{\s} ^{-1} \big[\sqrt{\rho^{(1)}}(K^* h^{(1)} + K^*_T h^{(2)})+ {\rho^{(2)}\over\sqrt{\rho^{(1)}}}K^*_G h^{(1)}\big],g^{(1)}\big> + \cr& {\e}^{-1}\big< \chi{\s} ^{-1} \big[\sqrt{\rho^{(2)}}(K^* h^{(2)} + K^*_T h^{(1)})+ {\rho^{(1)}\over\sqrt{\rho^{(2)}}}K^*_G h^{(2)}\big],g^{(2)}\big>. } \Eqa(A.16.1)$$ Here $\big< f,g\big>$ denotes the $L_2(\O\times \Bbb R^3)$ scalar product. The operator $L$ is symmetric with respect to $\big< \cdot,g\cdot\big>$. The terms in the first square brackets are non positive by the non positivity of the operator $L$. It is easy to see, by using the symmetry properties of the Boltzmann kernel [CC], that also the contribution coming from the terms in the second square bracket are non positive. Therefore we have $$ {1\over 2}{d\over dt }||g||_{L_2}^2\le C \ {\e}^{-1} \|h^{(2)})\|_r \cdot ||g||_{L_2} +\beta ||g||_2{L_2} ^2, \Eqa(A.18)$$ with $r\ge 3$. The final estimate is $$||g||_{L_2}\le C \e^{m-1}|| \nu^{-1} D||_r$$ The same kind of arguments provides the bound for the derivatives of $g$ with respect to $x$. Because of the force terms the argument differs from the one given in [Ca80] in the fact that we have to control at the same time the derivatives with respect to velocity and space. We sketch the proof for the derivatives of $g$. Differentiating the first equation in \equ(A.9) we get two coupled equations for $\nabla_v g$ and $\nabla g$ $$\eqalign{ \partial_t \nabla g^{(i)} &+(v\cdot \nabla)\n g^{(i)} +\nabla F^{(i)}\cdot \nabla_v g^{(i)} +F^{(i)}\cdot \nabla_v(\nabla g^{(i)})=\nabla N^{(i)}(g,h)\cr \partial_t \nabla_v g^{(i)} &+v\cdot \nabla (\nabla_v g^{(i)}) +\nabla g^{(i)} +F^{(i)}\cdot \nabla_v(\nabla_v g^{(i)})=\nabla_v N^{(i)}(g,h) }$$ where $N(g,h)$ is the r.h.s. of \equ(A.9) for $g$. Proceeding as before in getting \equ(A.16.1) we obtain $${d\over dt} (||\nabla g||_{L_2}+ ||\nabla_v g||_{L_2})\le c\a_F (||\nabla g||_{L_2}+ ||\nabla_v g||_{L_2}) + ||\nabla N||_{L_2}+ ||\nabla_v N||_{L_2}$$ The derivatives of $N$ with respect to velocity can be estimated by the methods in [ELM94], where it is proven the identity $${\pt \over \pt v}Q(f,g)=Q (f,{\pt g\over \pt v}) +Q (g,{\pt f \over \pt v})$$ where ${\pt \over \pt v}$ stands for the partial derivative with respect to any of the components of $v$. The final result is \goodbreak {\bf Lemma A.2} {\it There is an $\e_0>0$ finite for each finite $\a_F$ such that any solution to the linear problem \equ(A.9), with $D$ and $F_i$ given, satisfies for $j>3$, $s\le 3$ and any $0<\e\le \e_0$ $$|| g^{(i)}||_{j,s}+|| \partial_v g^{(i)}||_{j,s-1}\le C(1+\a_F)\e^{m-s}\Big[||\tilde\nu^{-1} D^{(i)}||_{j+2,s} +||\tilde\nu^{-1} \partial_v D^{(i)}||_{j+2,s}\Big] \Eqa(A.19)$$ $$|| h^{(i)}||_{j,s}+|| \partial_v h^{(i)}||_{j,s-1}\le C(1+\a_F)\e^{m-s+1}\Big[||\tilde\nu^{-1} D^{(i)}||_{j,s} +||\tilde\nu^{-1} \partial_v D^{(i)}||_{j,s}\Big] \Eqa(A.20)$$ } \vskip.4cm \noindent{\it Non-linear problem and fixed point argument.} \vskip.4cm The nonlinear equations \equ(A.4) are solved by a fixed point method. This method works if there is some small parameter in front of the non linear terms. In \equ(A.4) there are two kinds of non linear terms: the usual Boltzmann non linear term, which is multiplied by a power of $\e$ and the Vlasov term involving the forces which gives rise to linear and quadratic terms in the remainders: in fact the forces are given by expressions of the type $$F^{(1)}={\bf K}\bstar f^{(2)}={\bf K}\bstar [\sum_{n=0}^K\e^n f^{(2)}_n]\Big) +\e^m{\bf K}\bstar R^{(2)} \Eqa(A.21.0)$$ Hence the non linear term due to the force is small and we can apply the recursive argument. The Boltzmann terms are dealt with as in Caflisch [Ca80]. The control of the force term requires the boundedness of the gradient of the Kac potential. The result is \vskip.2cm \goodbreak {\bf Theorem A.3} {\it There is an $\e_0>0$ such that the remainders satisfy for $j>3$, $z=s-(d-1)$ with $s< m$ and any $0<\e\le \e_0$ $$|| g^{(i)}||_{j,z}\le c \e^{m-s+1},\quad || h^{(i)}||_{j,z}\le c \e^{m-s+1}, \quad i=1,2 \Eqa(A.21)$$ } \vskip.5cm {\it Proof}. \ Let $R^{(i)}_k$ be the solution of \equ(A.1.2) with force $$\eqalign{F^{(i)}_k &=\Big({\bf K}\bstar [\sum_{n=0}^K\e^n f^{(j)}_n)]\Big) +\e^m\Big({\bf K}\bstar R^{(j)}_{k-1}\Big) \cr & := F_\e^{(i)} +\e^m\tilde F^{(i)}_k, \quad j=i+1\, \hbox{\rm mod }2, } \Eqa(A.21.1)$$ and the collision integrals computed with $R^{(i)}$ replaced by $R^{(i)}_{k-1}$; moreover, we put $R^{(i)}_0 =0, i=1,2$. Lemma A.2 and an inductive argument assure that the sequences $R^{(i)}_k$ are uniformly bounded for $\e$ sufficiently small. In fact, setting $\bar\a=\a_{F_\e^{(i)}}$, we have $\a_k=\a_{F^{(i)}_k}\le\bar\a + C\e^m\|R_{k-1}\|_1$. By a standard argument $\|R_{k-1}\|_1$ is bounded by a constant $\lambda(\a_{k-1})$ with $\l(\,\cdot\,)$ some monotone function. Hence, $\a=\sup\a_k$ satisfies the inequality $\a\le\bar\a +\e^m\lambda(\a)$. By setting $\e^m\lambda(2\bar\a)\le \bar\a$, we conclude $\a<2\bar\a$ and hence the uniform boundedness of the sequences $R^{(i)}_k$ for $\e$ sufficiently small. The differences $\d R^{(i)}_k:=R^{(i)}_k-R^{(i)}_{k-1}$ can be decomposed again in high and low velocity parts $\d g_k, \d h_k$, with $\d g_k=(\d g^{(1)}_k,\d g^{(2)}_k)$ and $\d h_k=(\d h^{(1)}_k,\d h^{(2)}_k)$ solve the equations $$D_t \d g_k +(\e^m\tilde F_{k-1}+F_\e)\cdot\nabla_v\d g_k = N(\delta g_k ,\d h_k)$$ $$D_t \d h_k +(\e^m\tilde F_{k-1}+F_\e)\cdot\nabla_v\d h_k = N'(\delta g_k ,\d h_k)$$ where $N=(N^{(1)},N^{(2)})$, $N'=(N'^{(1)},N'^{(2)})$ and $ N^{(1)}$, $N'^{(1)}$ are the r.h.s. in \equ(A.9) with $$\eqalign{&D^{(i)}=-\e\d \tilde F_k\cdot\nabla_v ( g_{k-1}+ h_{k-1})+\cr&[J(R^{(i)}_{k-1},R^{(i)}_{k-1})-J(R^{(i)}_{k-2},R^{(i)}_{k-2})]+ [J(R^{(i)}_{k-1},R^{(j)}_{k-1})-J(R^{(i)}_{k-2},R^{(j)}_{k-2})].}$$ with $\d \tilde F_k^{(i)}:=(\tilde F^{(i)}_k-\tilde F^{(i)}_{k-1})={\bf K}\bstar[R^{(j)}_{k-1}-R^{(j)}_{k-2}]$. By Lemma A.2 the solutions satisfy \equ(A.19) and \equ(A.20), so that $$|| \delta g_k||_{j,s}\le C \e^{m-s+1}||\tilde\nu^{-1} [\d \tilde F_k)\cdot\nabla_v ( g_{k-1}+ h_{k-1})]||_{j+2,s} \Eqa(A.19.1)$$ $$|| \d h_k||_{j,s}\le \e^{m-s+2}||\tilde\nu^{-1} [\d \tilde F_k\cdot\nabla_v ( g_{k-1}+ h_{k-1})]||_{j,s} \Eqa(A.20.1)$$ We have $$\eqalign{ ||\d \tilde F_k\cdot \nabla_v& ( g_{k-1}+ h_{k-1})||^2_{j0}\le C\sup_{x,v}(1+|v|^2)^j|\nabla_v (\d g_k+\d h_k)| \sup_v\int dx|\d \tilde F_k|^2 \cr & \le C\int dx\Big|\int dv\int dy K(x-y) (R_{k-1}-R_{k-2})\Big|^2\cr &\le C\int dx\Big|\sup_v(1+|v|^2)^j\int dy K(x-y) (\d g_{k-1}+\d h_{k-1})\Big|^2\cr &\le C \Big|(\int dy |K(x-y)|^2)^{1/2}\sup_v(1+|v|^2)^j \big(\int dy|\d g_{k-1}+\d h_{k-1}|^2\big)^{1/2}\Big|^2\cr & \le C [||\d g_{k-1}||^2_{j0} +||\d h_{k-1}||^2_{j0}] }$$ To get the second inequality we have used that the norms $||\nabla_v\d g_k||_{j,s}$ and $||\nabla_v\d h_k||_{j,s}$ are finite by Lemma A.2, so that the supremum over $x$ in the first row exists finite. The last inequality is a consequence of the fact that the Kac potential is bounded and that the space integration is on a torus. Finally, by \equ(A.19.1) and \equ(A.20.1) we have $$|| \delta g_k||_{j,0}\le c\e^{m-s+1}||\d g_{k-1}||_{j,s} +||\d h_{k-1}||_{j,0}] $$ $$|| \d h_k||_{j,0}\le c\e^{m-s+1}||\d g_{k-1}||_{j,s} +||\d h_{k-1}||_{j,0}]$$ We remark that it is possible to prove that the norm $||\cdot||_{j,z}$ for the remainders $g^{(i)}, h^{(i)}$ are bounded with $z=s-(d-1)$ and $s< m$. \qed \vskip.4cm \numfor=1 {\bf Appendix B} \vskip.3cm In this Appendix we show how to bound the remainders which are solutions of \equ(6.20). The method we use is different from the one in Appendix A. In fact in this case the lowest order is a global Maxwellian and we do not need to introduce the decomposition into low and high velocity. Also in this case we need a Theorem on the regularity of the terms of the expansion analogous to Theorem A.1 {\bf Theorem B.1} {\it Given $\e>0$, assume that there exists a sufficiently smooth solution of the incompressible Navier-Stokes equations \equ(10) in $(0, t_0]$. Then there is a constant $c>0$ and $s$ depending on the smoothness of the solution of INS such that the terms in the expansion $f_i,\phi_i, \ i=2,...,K$ solutions of the equations \equ(6.2.2) and \equ(6.2.3) satisfy $$||f_i||_{j,s}\le c,\quad ||\phi_i||_{j,2}\le c \Eqb(B.0)$$ $$||\partial_vf_i||_{j,s}\le c,\quad ||\partial_v\phi_i||_{j,2}\le c\Eqb(B.1.1)$$ for any $j$.} \vskip.2cm We write \equ(6.20) for the variables $R^{(i)}, 1=1,2$ defined in \equ(RR) $$\eqalign{ \pt_t R^{(1)}& + \e^{-1}\big[ v\cdot R^{(1)}+F^{(1)}\cdot\n_v R^{(1)}\big]=\e^{-2} \Big[{\cal L}R^{(1)}+ \Gamma R^{(1)}+\Theta R^{(2)} \Big]\cr & +\e^{-1}\Big[ {\cal L}_1 R^{(1)} + \Gamma_2 R^{(1)} +\Theta_1 R^{(2)} \Big]+ \Big[{\cal L}'_1 R^{(1)} + \Gamma'_2 R^{(1)} +\Theta'_1 R^{(2)} \Big] \cr & + \e^{m-2}\big[J(R^{(1)},R^{(1)})+ J(R^{(1)},R^{(2)})+A^{(1)}\big],\cr % \pt_t R^{(2)}& +\e^{-1}\big[ v\cdot R^{(2)}+ F^2\cdot\n_v R^{(2)}]= \e^{-2} \Big[ {\cal L}R^{(2)}+ \Gamma R^{(2)}+\Theta R^{(1)}\Big] \cr & +\e^{-1}\Big[ {\cal L}_2 R^{(2)} + \Gamma_1 R^{(2)} +\Theta_2 R^{(1)}\Big]+\Big[ {\cal L}'_2 R^{(2)} + \Gamma_1' R^{(2)} +\Theta_2' R^{(1)}\Big]\cr &+ \e^{m-2}\big[J(R^{(2)},R^{(2)})+ J(R^{(2)},R^{(1)})+A^{(2)}\big], }\Eqb(B.1)$$ $\Theta, {\cal L}, \Gamma$ are defined as $\bar\Theta, \bar{\cal L}, \bar\Gamma$ in the list after \equ(A.1.2) after substituting the global Maxwellian $M_0(\bar\rho,\bar T)$ to $\bar M$. Finally, $${\cal L}_i g = J(f^{(i)}_1,g)+J(g,f^{(i)}_1)], \quad \Theta_i g= J( \f^{(i)}_1, g),\quad \G _i g = J(g,f^{(i)}_1),$$ $${\cal L}'_i g = \sum_{h=2}^K \e^{h-2}[J(f^{(i)}_h,g)+J(g,f^{(i)}_h)],$$ $$\Theta'_i g= J(\sum_{n=2}^K\e^{n-2} \f^{(i)}_n, g),\quad \G '_i g = \sum_{h=2}^K \e^{h-2}J(g,f^{(i)}_h),$$ The first step is to consider the linear problem associated to \equ(B.1), namely to study \equ(B.1) with the last terms $D^{(i)}:=J(R^{(i)},R^{(i)})+ J(R^{(i)},R^{(j)})+A^{(i)}, i=1,2, i\ne j$ given and $F^{(1)}, F^{(2)}$ fixed, independent of $R^{(1)}, R^{(2)}$. Moreover remembering that the forces vanish to the lowest order in $\e$, we assume that the $L_\infty$ norms of $F^{(i)}$ and their gradients are bounded by some constant $\e \a_F$. The role of the constant $\a_F$ is similar to the one discussed in the previous appendix and we do not repeat the iterative argument in this case. We will just provide an estimate for the $L^2$ norm in $(x,v)$ $||R||_2:=||R^{(1)}||_2 +||R^{(2)}||_2$ for the solution $R=(R^{(1)},R^{(2)})$ of this problem, the rest of the argument being standard (see for example [ELM98]). We put $R^{(i)}=\sqrt{M_0} \Psi^{(i)}$ so that $$\eqalign{ \pt_t \Psi^{(1)}& + \e^{-1}\big[ v\cdot \n \Psi^{(1)}+F^{(1)}\cdot\n_v \Psi^{(1)}-{1\over 2}\Psi^{(1)}F^{(1)}\cdot v \big]=\cr &\e^{-2} \Big[ L\Psi^{(1)}+ G \Psi^{(1)}+T \Psi^{(2)} \Big] +\e^{-1}\Big[ L_1 R^{(1)} + G_2 \Psi^{(1)} +T_1 \Psi^{(2)} \Big]+ \cr &\Big[ L'_1 \Psi^{(1)} + G'_2 \Psi^{(1)} +T'_1 \Psi^{(2)} \Big] + \e^{m-2}{D^{(1)}\over \sqrt{M_0}} } \Eqb(B.9)$$ where the relation between the old operators ${\cal L}, \Theta, \Gamma, {\cal L}_i, \Theta_i, \Gamma_i, {\cal L}'i, \Theta'_i, \Gamma'_i$ and the new ones $L,T, G, L_i,T_i,G_i, L'_i,T'_i,G'_i$ is of the form $${\cal L}f={1\over \sqrt{M_0}}{ L}\sqrt{M_0}f.$$ It is easy to see that, setting $\|\Psi\|_{L_2}^2=\|\Psi^{(1)}\|_{L_2}^2+\|\Psi^{(2)}\|_{L_2}^2$, $$\eqalign{ {1\over 2}{d\over dt} ||\Psi||_{L_2}^2&=\e^{-2} \Big[\big<\Psi^{(1)},L \Psi^{(1)}\big>+\big<\Psi^{(2)}, L\Psi^{(2)}\big>-{1\over 2}\e\sum_{i=1}^2 F^{(i)}\cdot\big<\Psi^{(i)}, v\Psi^{(i)}\big>\Big]\cr & + \e^{-2}\Big[\big< \Psi^{(1)}, M_0^{-1/2}J(M_0,\sqrt{M_0}\ \Psi^{(2)})\big>+\big< \Psi^{(1)}, M_0^{-1/2}J(\sqrt{M_0}\ \Psi^{(1)},M_0)\big> +\cr & \big<\Psi^{(2)}, M_0^{-1/2}J(M_0,\sqrt{M_0}\ \Psi^{(1)})\big>+\big<\Psi^{(2)}, M_0^{-1/2}J(\sqrt{M_0}\ \Psi^{(2)},M_0)\big>\Big]\cr & +\e^{-1}\Big[\big<\Psi^{(1)},L_1 \Psi^{(1)}\big>+\big<\Psi^{(2)}, L_2\Psi^{(2)}\big> +\big<\Psi^{(1)},(G_2 \Psi^{(1)} +T_1 \Psi^{(2)})\big>+\cr & \big<\Psi^{(2)},(G_1 \Psi^{(2)} +T_2 \Psi^{(1)})\big>\Big] + \Big[\big<\Psi^{(1)},L'_1 \Psi^{(1)}\big>+\big<\Psi^{(2)}, L'_2\Psi^{(2)}\big> \cr &+ \big<\Psi^{(1)},(G'_2 \Psi^{(1)} +T'_1 \Psi^{(2)})\big>+ \big<\Psi^{(2)},(G'_1 \Psi^{(2)} +T'_2 \Psi^{(1)})\big>\Big]\cr & +\e^{m-2}\sum_{i=1}^2\big<\Psi^{(i)},{D^{(i)}\over \sqrt{M_0}}\big> } \Eqb(B.10)$$ First of all, we observe that the terms in the second square bracket are non positive [CC]. To estimate the other terms we will use the strict negativity of the operator $L$ (see \equ(3.14)) on the space orthogonal to the collision invariants and the following estimate on the operator $J(f,h)$ (see for example [GPS]): for any Maxwellian $M$ and for any $y\in[-1,1]$ $$\int_{\Bbb R^3}\ dv {|J (\sqrt M f,\sqrt M h)|^2\over \nu M} \le \int_{\Bbb R^3}\ dv \nu |f|^2\ \int_{\Bbb R^3}\ dv \nu|h|^2 \Eqb(4.24.1)$$ This inequality and the bounds on the $f_n$'s imply the following bounds: $$|\big<\sum_{i=1}^2\Psi^{(i)} { L}_i\Psi^{(i)}\big>|\le C\pa \sum_{i=1}^2\sqrt{\nu }\bar \Psi^{(i)}\pa_{L_2} \ \pa \Psi^{(i)}\pa_{L_2} \pa \ M_0^{-1/2}f_1^{(i)}\pa_{j,s},\Eqb(BL1)$$ $$|\big<\sum_{i=1}^2\Psi^{(i)} { L}'_i\Psi^{(i)}\big>|\le C\pa \sqrt{\nu} \bar\Psi\pa_{L_2} \ \pa \Psi\pa_{L_2} \pa M^{-1/2}_0\sum_{n=2}^7 f_n\pa_{j,s}\Eqb(BL2)$$ where $\bar g$ denotes the projection of a function $g$ on the orthogonal to the invariant space of $L$, while $\hat g$ is the projection on the invariant space. Note that the presence of the product $\pa\sqrt{\nu}\bar\Psi\pa_2\/\/ \pa\Psi\pa_2$ depends on the fact that $ L_i$ and $ L'_i$ are both orthogonal to the collision invariants. Similar estimates hold for the terms involving the other operators. By using the bounds on the $f_n$'s and $\phi_n$'s and after some algebra, the terms in the forth, fifth and sixth rows are bounded by $$C\pa \sqrt{\nu} \bar\Psi\pa_{L_2} \ \pa \Psi\pa_{L_2} $$ To bound the last term in the first square bracket of \equ(B.10), we note that $$\e^{-1}|\sum_{i=1}^2 F^{(i)}\cdot\big<\Psi^{(i)}, v\Psi^{(i)}\big>|\le \a_F[\|\sqrt{\nu}\bar\Psi\|^2_{L_2}+ C\|\hat\Psi\|_{L_2}^2,$$ and we assume $\e$ so small that $\e^2\a_F<1/2$. We integrate \equ(B.10) in time between $0$ and $ t_0$. With the notation $\Psi_t(\,\cdot\,)=\Psi(\,\cdot\,,t)$, we get $$\eqalign{&{1\over 2} \pa \Psi_{t_0} \pa^2_{L_2} \le C\int_0^{t_0}d\/t\Bigg\{-\e^{-2}\Big[{1\over 2}\pa \sqrt\nu\bar \Psi_t\pa^2_{L_2} +C_F\pa \sqrt\nu \Psi_t\pa^2_{L_2}\Big]\cr & + C(\e^{-1}+1)\pa \sqrt\nu \bar \Psi_t\pa_{L_2} \ \pa \Psi_t\pa_{L_2} +\e^{m-2}\pa D(\,\cdot\,,t)\pa^2_{L_2}\Bigg\} }\Eqb(B.3) $$ The first term in the second line is due to the bounds \equ(BL1) and \equ(BL2). Moreover \noindent$\pa M^{-1/2}_0 f_1\pa_{j,s}$ and $ \pa M^{-1/2}_0 f_1\pa_{j,s}$ are bounded by the regularity of the solutions of the macroscopic equations for $00$ such that any solution to the linear problem \equ(B.9), with $D$ and $F_i$ given, after choosing $K=2m$ satisfies for $j>3$, $s0\},$$ $$(\un z_{j_r+1}^r)'= (z^r_1,\dots (z^r_i)', \dots, z^r_{j_r},(z^r_{j_r+1})'),$$ $$\overline{\un z^r_{j_r+1}}= (z^r_1,\dots \bar z^r_i, \dots,z^r_{j_r}, \bar z^r_{j_r+1})$$ and, for any $z$, $z_*$, the phase points $z'$, $z'_*$, $\bar z$ and $\bar z_*$are defined by $$q'=q, \quad q'_*=q+\d \o,\quad v'=v-\o(\o\cdot(v-v_*),\quad v'_*=v_*+\o(\o\cdot(v-v_*),$$ $$\bar q=q, \quad \bar q_*=q-\d \o,\quad \bar v=v,\quad \bar v_*=v_*.$$ Moreover, $$\eqalign{&({\cal B}^{b,r}_{\d,i}r_{j_r,j_b+1})(\un z^r_{j_r};\un z^b_{j_b})=\cr&\int_{\IR^3} dv^b_{j_r+1}\int_{S^2_+} d\o(v^b_{j_b+1}-v^r_i)\cdot \o \Big [r_{j_r,j_b+1}((\un z_{j_r}^r)';(\un z^b_{j_b+1})')-r_{j_r,j_b+1}(\overline{\un z^r_{j_r}};\overline{\un z^b_{j_b+1}})\Big],}$$ where $$(\un z_{j_r}^r)'=(z^r_1,\dots (z^r_i)', \dots, z^r_{j_r}),\quad (\un z^b_{j_b+1})'= (z^b_1,\dots,z^b_{j_b}, (z^b_{j_b+1})'),$$ $$\overline{\un z_{j_r}^r}=(z^r_1,\dots \bar z^r_i, \dots, z^r_{j_r}),\quad \overline{\un z^b_{j_b+1}}= (z^b_1,\dots,z^b_{j_b}, \bar z^b_{j_b+1}).$$ The collision terms ${\cal B}^{b,b}_{\d,i}$ and ${\cal B}^{r,b}_{\d,i}$ are defined in a similar way. Furthermore $$\eqalign{&({\cal V}^{b,r}_{i}r_{j_r,j_b+1})(\un z^r_{j_r};\un z^b_{j_b})= \cr&-\int_{\IR^3} dv^b_{j_r+1}\int_{\L} dq^b_{j_b+1}\gamma^3\n_{q^r_i}U_\g(|q_i^r-q^b_{j_b+1}|)\n_{v^r_i} r_{j_r,j_b+1}(\un z_{j_r}^r;\un z^b_{j_b+1})} $$ and a similar expression for ${\cal V}^{r,b}_{i}$. Taking formally the limit $\d\to 0$, the limiting correlations satisfy the following Vlasov-Boltzmann hierarchy: $$\eqalign{&\pt_\t r_{j_r,j_b}+\sum_{i=1}^{j_r}v^r_i\cdot\n_{q^r_i}r_{j_r,j_b}+ \sum_{i=1}^{j_b}v^b_i\cdot\n_{q^b_i}r_{j_b,j_b}=\cr& \sum_{i=1}^{j_r}\Big[{\cal B}^{r,r}_{i}r_{j_r+1,j_b}+ {\cal B}^{b,r}_{i}r_{j_r,j_b+1}+{\cal V}^{b,r}_{i}r_{j_r,j_b+1} \Big]+\cr& \sum_{i=1}^{j_b}\Big[{\cal B}^{b,b}_{i}r_{j_r,j_b+1}+ {\cal B}^{r,b}_{i}r_{j_r+1,j_b}+ {\cal V}^{r,b}_{i}r_{j_r+1,j_b}\Big],}\Eqc(BHier)$$ where, $$\eqalign{&({\cal B}^{r,r}_{i}r_{j_r+1,j_b})(\un z^r_{j_r};\un z^b_{j_b})=\cr&\int_{\IR^3} dv^r_{j_r+1}\int_{S^2_+} d\o(v^r_{j_r+1}-v^r_i)\cdot \o \Big [r_{j_r+1,j_b}(\un z_{j_r+1}^r)';\un z^b_{j_b})-r_{j_r+1,j_b}(\un z^r_{j_r+1};\un z^b_{j_b})\Big]}$$ and, for any $z$, $z_*$, the phase points $z'$, $z'_*$ are defined by $$q'=q, \quad q'_*=q,\quad v'=v-\o(\o\cdot(v-v_*),\quad v'_*=v_*+\o(\o\cdot(v-v_*).$$ Similar modifications provide the other terms of the Vlasov-Boltzmann hierarchy. It is easy to see that if the initial condition of {\it molecular chaos} $$r_{j_r,j_b}(\un z^r_{j_r};\un z^b_{j_b};0)=\prod_{i=1}^{j_r}f^{r}(z^r_i,0) \prod_{k=1}^{j_b}f^{b}(z^b_k,0)$$ \noindent is satisfied, then the correlation functions stay factorized at positive times $\t$ and $f^{r}(q,v,\t)$ and $f^{b}(q,v,\t)$ are the solutions of the coupled Vlasov-Boltzmann equations $$\eqalign{ \partial_\t &f^r(q,v,\t) +v\cdot\nabla_q f^r(q,v,\t)+F^r \cdot \n_v\ f^r(q,v,\t) = J(f^r, f^r+f^b),\cr \partial_\t &f^b(q,v,\t) +v\cdot\nabla_q f^b(q,v,\t)+F^b \cdot \n_v\ f^b(q,v,\t) = J(f^b, f^r+f^b),}\Eqc(0.7)$$ where $$F^{r}(q,\t)= -\n_q\int_\O d\/q' \gamma^3(\n U_\g)(|q-q'|)\int_{\IR^3}dv f^{b}(q',v,\t),\Eqc(0.22)$$ $$F^{b}(q,\t)= -\n_q\int_\O d\/q'\gamma^3(\n U_\g)(|q-q'|)\int_{\IR^3}dv f^{r}(q',v,\t),\Eqc(0.33)$$ and $$J(f,g)=\int_{\IR^3} d\/v_*\int_{S^2_+}d\/\omega (v-v_*)\cdot\omega [f(v')g(v'_*)-f(v)g(v_*)]. \Eqc(0.44)$$ \vskip.3cm Summarizing, we obtained formally the Vlasov-Boltzmann equations for a binary mixture, where the Boltzmann collision kernel terms are due to the short range interaction while the Vlasov self consistent force is due to the repulsive weak long range interaction. If, instead of the hard core interaction the short range force is given by a finite range potential, we would get formally the same equations but with a different cross section. We want to stress that an important step is missing in order to make the above derivation rigorous. The first rigorous result on the derivation of the Boltzmann equation has been given by Lanford [Lan] where the convergence of the correlation functions is proven in $L_\infty$-norms. On the other hand, the derivation of the Vlasov equation is based on the use of the variation norm and we have not been able to find a norm suited for both terms. The only related result, as far as we know, has been obtained in [GM] and is about a stochastic particle systems converging to a Vlasov-Boltzmann equation with a modified Boltzmann kernel (Povzner). The proof is based on martingale methods. In the linear case of a Lorentz gas with a Kac potential term it is possible to prove the convergence to a Boltzmann equation with a linear collision term and a non-linear self-consistent force term [MR]. \vskip 1cm {\bf Acknowledgements}: The work on this paper was partially done during visits of R. Esposito and R. Marra to DIMACS and Math. Department, Rutgers University, and of R.Esposito, J.L.Lebowitz and R. Marra to IHES. We thank those institutions for their hospitality. Research at Rutgers was partially supported by NSF Grant DMR-9813268, AFOSR Grant F49620-98-1-0207 and by DIMACS and its supporting agencies, the NSF under contract STC-91-1999 and the NJ Commission on Science and Technology. Supported also by the Italian MURST and INFM. Part of this work (S.B.) was performed under the auspices of the U. S. 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