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%\leftheadtext{R. Esposito, J. L. Lebowitz and R. Marra}
%\rightheadtext{Hydrodynamic Limit of the Stationary
%Boltzmann Equation in a Slab}
%\pretitle{\centerline{Draft February 15, 1993}}
%\vskip .5cm
\topmatter
\title
Hydrodynamic Limit
of the Stationary\\
Boltzmann Equation
in a Slab
\endtitle
\author
R. Esposito$^1$,
J. L. Lebowitz$^2$,
R. Marra$^3$
\endauthor
\affil
$^1$Dipartimento di
Matematica,
Universit\`a di Roma
Tor Vergata.\\
$^2$ Mathematics and
Physics
Departments,
Rutgers
University.\\
$^3$Dipartimento di
Fisica, Universit\`a
di Roma Tor Vergata.
\endaffil
%\leftheadtext %\nofrills {R. Esposito{\it etal.}}
%\leftheadtext {}
%\rightheadtext{}
%\address{ }
%\endaddress
\abstract We study the stationary solution of the
Boltzmann equation in a slab with a constant
external force parallel to the boundary and complete
accommodation condition on the walls at a specified
temperature. We prove that when
the force is sufficiently small there exists a solution which
converges, in the hydrodynamic limit, to a local
Maxwellian
with parameters given by the stationary solution of the
corresponding compressible Navier-Stokes equations with no-slip
boundary conditions. Corrections to this Maxwellian are obtained
in powers of the Knudsen number with a controlled remainder.
\endabstract
%\thanks Research supported
%in part by AFOSR Grant 91-0010, MURST and GNFM
%\endthanks
\endtopmatter
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\heading
1. Introduction.
\endheading
\sectno=1
\vskip
.5cm
In this paper we continue our study of the derivation of
hydrodynamic equations from the Boltzmann equation (BE),
a problem which goes back to Hilbert [\rcite{H}].
The BE is
believed to accurately describe the time evolution of
rarefied gases on a \lq\lq kinetic\rq\rq scale intermediate
between the microscopic and macroscopic [\rcite{S}].
To go from the BE to the macroscopic (hydrodynamic)
descriptions the locally conserved
density fields have to be slowly varying on the kinetic
(to which we shall refer from now on as microscopic) scale
but have sensible space variations over macroscopic
distances. Let $\e$ be the ratio between
microscopic and macroscopic space units (usually called
the Knudsen number). It can be shown that the conserved
densities, {\it observed at microscopic times of order}
$\e^{-1}$, converge, as $\e \to 0$, to macroscopic fields
whose time evolution is given by the solution of the Euler
equations (EE) (at least when the latter have a
smooth solution) [\rcite{N}], [\rcite{Ca}],
[\rcite{U1}]. This derivation of
the EE in the above hydrodynamical (Euler) scaling limit is
consistent with (indeed made possible by) the fact that the
EE are themselves invariant under uniform space and
time scaling.
Unfortunately there is no such scale invariance (and thus no such
scaling limit) for the
Navier-Stokes equations (NSE). The NSE are usually deduced,
via the Chapman-Enskog expansion (see [\rcite{Ce1}]), as
corrections to the EE on the Euler time
scale $\e^{-1}$ with viscosity coefficient and thermal conductivity
of order $\e$. To describe situations which discriminate
between the two equations when the Knudsen number $\e$ goes to
zero we need to consider longer time scales, i.e.
microscopic times of order $\e^{-2}$. On this time scale
viscosity and thermal conductivity have finite effects, but the
effect of the Euler term may now become very large making
any rigorous mathematical analysis very difficult.
This problem can be controlled when the velocity field
itself is of order $\e$ -- in which case a kind of
scale invariance is recovered for the NSE. It is then
possible to get the incompressible NSE as a
scaling limit from the BE (see
[\rcite{DEL}], [\rcite{BU}], [\rcite{A}], [\rcite{BGL1}],
[\rcite{BGL2}]). To deal with the
case of macroscopic velocities with non vanishing Mach number it
is necessary to consider situations in which some of the
non-linear terms, which prevent invariance under scaling, are
absent from the NSE due to the symmetry of the problem. Some
partial results in this direction are given in [\rcite{ELM}].
Alternatively we can consider stationary situations in the
presence of boundaries in which the time independent
NSE have suitable scaling behavior.
In this paper we consider such an example and show that
the NS description is obtained as the scaling limit of the
stationary solution of the BE. More precisely we
consider a fluid in a channel with planar
parallel boundaries subject to a constant
force parallel to the boundary. Using
the Euler equations the fluid velocity would keep on
increasing linearly with time. For the NSE with no-slip
boundary condition, on the other hand, the effect of the
viscosity counteracts the force field and a stationary
situation is reached, with the heat production
balanced by a flow of energy to the thermal walls.
Experiments, as well as molecular dynamics
simulations [\rcite{HLC}], show that this situation can be
reasonably described by the NSE.
The kinetic description of this problem in terms of a
BE requires boundary conditions; a
natural choice is to assume a complete
accommodation coefficient. This means that
any particle hitting the boundary is reflected
with a velocity chosen at random
according to a Maxwellian with variance
corresponding to the fixed temperature of the walls.
We confine ourselves to the fully symmetric solutions
of the hydrodynamic and kinetic equations.
The solution to the rescaled BE
is then found like in [\rcite{Ca}] in terms of a
truncated expansion in the Knudsen parameter $\e$
whose leading term is a Maxwellian with
parameters given by the solution of the NSE.
The corrections to the leading order are given in the bulk
by a {\it modified} truncated Hilbert expansion
with a remainder.
It is also necessary to introduce
boundary layer terms to accommodate the
expansion to the boundary conditions. This is
accomplished along the lines of [\rcite{BCN2}], [\rcite{Ma}],
[\rcite{BCN1}], [\rcite{Ce2}], but there are
some extra difficulties arising from the force field.
Most of the paper is devoted to the proof of a bound on
the remainder, which is obtained for a sufficiently
small force field; this implies that
the solution will match the
expansion to any order, although, of course, no
convergence of the expansion is provided or expected.
The outline of the paper is as follows.
In Section 2 we present the problem and the basic
results. Section 3 is devoted to
the bulk and boundary layer expansions. The control of
the boundary layer expansion requires the use of some well
known results about the Milne problem (see [\rcite{Ma}],
[\rcite{BCN1}], [\rcite{Ce2}]) and some estimates on the
velocity derivatives of the boundary layer terms; these
are given in Appendix B. There are many technical difficulties
in estimating the remainder, which satisfies
a weakly non linear equation, due to the fact that for the
stationary problem the only available a priori bound
on the solution comes from the non
positivity of the quadratic form corresponding to the production
of entropy in the linearized regime.
Unfortunately the linearized Boltzman operator has a
non trivial kernel so the control of
the hydrodynamic component of the solution is very
inefficient. This difficulty, combined with the high
velocity difficulty already
present in [\rcite{Ca}] and the low
velocity difficulty typical of the
stationary case make the estimates rather
intricate. In Section 4 we construct the solution of the
remainder equation as the limit of a sequence of solutions
of approximating linear problems. The estimates for
these solutions are obtained, like in [\rcite{Ca}],
by decomposing them into a low and high velocity
part, but the poor control of the hydrodynamic part at
low velocities allows only $L_2$ bounds on the solution.
The proof of these bounds is given in Section 5. We then
take advantage of the structure of the equation to improve
the bounds to pointwise estimates on the solution which
can be used to handle the non linear terms. This is
discussed in Section 6. Appendix A contains a short
discussion of the solution of the macroscopic equations.
\vskip 1.cm
%
%
%
\heading
2. Statement of problem and results.
\endheading
\resetall
\vskip.5cm
\noindent
a) $\underline{\text{Hydrodynamical Description}}$
We consider the flow of a compressible viscous
fluid in a three dimensional slab
infinite (or periodic) in the $x$ and $z$ directions with
planar boundaries perpendicular to the $y$ direction:
top and bottom planes will correspond to $y=1$ and
$y=-1$ respectively. The fluid is subject to a constant
force parallel to the $x$-axis. A stationary
state will be reached when the heat
produced by the friction due to the relative
motion of the gas under the action
of the force is balanced by the energy loss to the
walls maintained at a given temperature $T_0$.
Considering only
situations which have the full symmetry of
the problem the Navier-Stokes
equations for this system, with a perfect gas
equation of state, reduce to
$$\align&{d\over dy}(\rho T) =0\(2.1)\\&
{d\over dy}(\eta (T) {du \over dy} ) + \rho F
=0\(2.2)\\&
{d\over dy}(\kappa (T) {dT \over dy}) +
\eta (T) ({du \over dy} )^2 =0
\(2.3)\endalign$$
\noindent where $u(y)$ is the $x$-component of the
velocity field
(the only non vanishing one), $T(y)>0$ is
the temperature field,
$\rho(y)~>~0$ is the density of the fluid,
$F>0$ is the constant
intensity of the external field; $\eta(T)$ and
$\kappa(T)$ are the viscosity and the
thermal conductivity, which for gases depend only on
$T$. The hydrostatic condition
\(2.1), stating the constancy of the pressure $\Cal P$
given by the perfect gas law $\Cal P=\rho\/T$, ensures
that there is no flow in the $y$ direction.
We complete the system \(2.1)--\(2.3)
with a no-slip boundary condition on the thermal
walls at temperatures $T_0$, that is we assume
$$u(-1)=u(1)=0; \phantom{.....} T(-1)= \ T(1)\ =T_0>0
\(2.4)$$ Once the total mass per unit area
$m=\int_{-1}^1 \rho(y) dy$ is given, the problem is
completely specified.
\noindent The functions $\eta(T)$ and $\kappa(T)$
are assumed to be smooth and satisfy the conditions
$$\eta(T)\ge
\eta_0>0; \phantom{.....} \kappa(T)\ge\kappa_0>0
\(2.5)$$ at least for $T\ge T_0$.
Under this conditions it is not difficult to
prove the following
\proclaim {Proposition 2.1} The system
\(2.1)--\(2.5) has solutions in $C^{\infty}([-1,1])$,
such that $T(y) > T_0$ for $y\in(-1,1)$.
Moreover there is an $F_0>0$ such that, if
$F\le F_0$ the solution is unique and for any positive $n$
there is a $c>0$ such that the following estimate holds:
$$\sup_{y\in[-1,1]}\Big[|u(y)| +
\sum_{k=1}^n|u^{(k)}(y)| + |T(y)-T_0|+
\sum_{k=1}^n|T^{(k)}(y)|\Big] 0\(2.9)\\f(1,v)&=\alpha_+ \overline
M(v), \phantom{....} v_y<0 \endalign$$
with
$${\overline M}(v) = {1 \over 2\pi T_0^2}
\text{\/e}^{-v^2/2T_0} \(2.10)$$
normalized so that $\int _{v_y \lessgtr 0} |v_y|
{\overline M}(v) dv =1$.
In \(2.9) the coefficient
$\alpha_{\pm}$ are determined by the condition of
impermeability of the walls, i.e.
$$ \langle v_y f
\rangle= 0 \phantom{....} \text{ for } y=\pm 1 \text{ ,}
\(2.11)$$
\noindent and we use the notation
$$\langle g\rangle \equiv \int_{\Bbb R^3} g(v) dv
\(2.12)$$
\noindent Hence $\alpha_\pm$ are given by
$$\a_\pm=\pm\int _{v_y \gtrless 0} v_y
f(\pm 1,v)dv \(2.10.1)$$
and can be interpreted as the flux of mass outgoing
from the fluid toward the walls.
We now make explicit our restriction to flows which
are symmetric
under reflection of the $y$-axis. Denote by
$\Cal R v$ the reflected velocity, $\Cal R v =
(v_x,-v_y,v_z)$, and by $\Cal R ^* f(y,v) = f(-y, \Cal R v)$;
we consider only solutions of (2.7)
such that $\Cal R ^* f = f$. As a consequence of this
reflection symmetry we have: $\int_{v_y>0}dv f(1,v)
v_y = - \int_{v_y<0}dv f(-1,v) v_y$, so that
$\alpha_+=\alpha_-=\alpha$.
Finally we require that
the kinetic mass be equal to the hydrodynamic one, that
is we assume the normalization condition $$\int_{-1}^1 dy
\langle f \rangle=m \(2.11.1)$$
The collision operator $Q(f,g)$ has the
following properties:
\item{i)}
$$\langle \hat\chi_iQ(f,g)\rangle = 0
\phantom{....}\text{for } i=0,\dots,4\(2.13)$$
where $$\align &\hat\chi_0(v)=1;\phantom{....}
\hat\chi_4(v)={1\over 2}v^2 \\&
\hat\chi_1(v)=v_x, \phantom{....}\hat\chi_2(v)=
v_y, \phantom{....} \hat\chi_3(v)=v_z \(2.14)
\endalign$$
\item{ii)}
$$ Q(f,f)=0 \phantom{....}\text{iff }
f(y,v)=M_{\rho,U,T}(v)\equiv{\rho\over (2\pi
T)^{3\over2}} \text{\/e}^{-(v-U)^2/2T}\(2.15)$$
with $\rho$ and $T$ positive and $U \in \Bbb
R^3$.
The function $M_{\rho,U,T}$ is called a local
Maxwellian with parameters $\rho$, $T$ and $U$ depending
on position.
\vskip 1cm
\noindent Integrating (2.7) over velocities, it follows
from \(2.13) and \(2.11) that any solution of (2.7) with
the property \(2.11) satisfies
$$\langle v_y f \rangle =0 \phantom{....} \text{for
any } y\in [-1,1] \(2.16)$$
We denote by $M$ the
Maxwellian $M_{\rho,U,T}$ with $T=T(y)$,
$\rho=\rho(y)$ and $U=(u(y),0,0)$ given
by the solution of (2.1)--(2.4)
and write, following [\rcite{Ca}], [\rcite{DEL}], the solution to
\(2.7)--\(2.9) as
$$f=M+\sum_{n=1}^6 \e^n\/f_n + \e^3\/f_R
\(2.17)$$
The functions $f_n$'s will be
specified in next section as a combination
of terms of a bulk expansion
and of a boundary layer expansion. The remainder
$f_R$, defined by \(2.17), then has to solve the {\it weakly}
non linear equation $$v_y {\pa f_R\over \pa y } +\e F {\pa
f_R\over \pa v_x}= {1\over \e}\Cal L f_R + \Cal L^1 f_R +
\e^2 Q(f_R,f_R) + \e^3 A \(2.18)$$
with
$$\Cal L f \equiv 2 Q(M,f) \(2.19)$$
$$\Cal L^1 f_R = 2Q(\sum_{n=1}^6 \e^{n-1}f_n,f_R)
\(2.20)$$
and $A$ a given function, specified in terms of the
functions $f_n$ and their derivatives.
\noindent Moreover, to satisfy the normalization condition
\(2.11.1), we must have
$$\int dv dy f_R =0\(2.29)$$
We summarize our results in the following
\proclaim {Theorem 2.2} Given
$\rho$, $T$ and $u$ satisfying
\(2.1)--\(2.4) ans the force field $F$
small enough, then it is possible
to determine uniquely functions
$f_n$, $n=1,\dots, 6$ and a remainder $f_R$ such that $f$,
given by \(2.17), is a solution to the BE \(2.7), \(2.9)
and \(2.11.1) for $\e$ sufficiently small. Moreover
$$||f_n||_{\infty}0\\ f_n(1,v)&=\a_n M(1,v) +
\ga^+ _{n,\e} (v), \phantom{....}\ v_y<0 \(2.25)
\endalign$$
\item{v)}
There is for any positive $r$, a
constant $c$ such that:
$$|M^{-{1\over 2}}\/f_n|_r0 \(3.32)$$
$$\langle v_y f(\tau,v) \rangle =0 \phantom
{...}\text{for } \tau\in \Bbb R^+ \(3.33)$$
$$\lim_{\tau\to \infty} f=f_\infty \in L_\infty (\Bbb
R^3) \(3.34)$$
for a given distribution $\vt$ of
incoming particles at the boundary $\tau=0$ and a given
source of particles $s$ such that
$$\langle s \rangle =0\(3.35)$$
This problem has been studied by several authors (see
for example [\rcite{BCN1}], [\rcite{Ma}],
[\rcite{Ce2}] and references quoted therein).
\proclaim {Theorem 3.2}
1) Suppose that for $r>3$ and some $\sigma' >0$
there are finite constants $c_1$ and $c_2$ such that
$$\sup_{v \in \Bbb R^3} (1+ |v|)^r |M_0 ^{-{1\over2}}
\vt (v)|3$, $\ell\ge1$ and some $\s'>0$
$$\sup_{v \in \Bbb R^3} (1+|v|)^r
|M_0 ^{-{1\over2}}{\pa^\ell \vt \over \pa{v_x^\ell}}|+
\sup_{\tau \in \Bbb R^{+}}e^{\sigma'\tau}\sup_{v \in
\Bbb R^3} (1+|v|)^r |M_0 ^{-{1\over2}}{\pa ^\ell s
\over \pa{v_x ^\ell}}|0) = t_1(-1,v) +
b^+_1(2\e^{-1},v)-q_1^-(v) \(3.48.1)$$
and at $y=1$
$$f_1(1,v_y<0) = t_1(1,v) + b^-_1(2\e^{-1},v)-q_1^+(v)
\(3.48.2)$$ where the function $t_1$ will be dermined later.
Notice that by \(3.38) $q_1^{\pm}(v)$ and $t_1(\pm 1,v)$
are both in $\text{Null}\/\Cal L_0$, so that we can use
$t_1(\pm 1,v)$ to compensate $q_1^{\pm}(v)$. The terms
$b^+_1(2\e^{-1},v)$ are exponentially small in $\e^{-1}$ by
\(3.39) and in fact they are the terms
denoted by $\ga^{\pm}_{1,\e}$ in \(2.25).
By \(3.33) $\langle v_y b_1^{\pm}\rangle=0$
and, of course, $\langle v_y q_1^{\pm}\rangle=0$ too.
Therefore to satisfy \(2.22) we have to choose
$$p_2^{(1)}=0 \(3.48.3)$$
To determine completely the hydrodynamic part of $B_1$,
that is the remaining coefficients $p^{(1)}_i,
i\ne 2$, we use the linear differential equations obtained
for them by means of the solvability
conditions. First we solve \(3.2) and obtain
$B _2$ up to $t_2 =\sum_{1=0} ^4 p_i ^{(2)} (y)
\psi_i M^{1/2}$.
Then the solvability condition \(3.44)
with $n=3$ gives a system of three second order
linear non homogeneous differential equations for the
unknown functions $p_i^{(1)}$ for $i=1,3 \ \text{and }
4$. The coefficients $\Lambda_{ij}$ of the
derivatives are given by $\Lambda_{ij}= (v_y
\psi_i, L^{-1} v_y \psi_j)$. Since $L^{-1}$ is a strictly
non positive operator on $\Cal W$, $||\Lambda_{i,j}||$ are the
coefficients of a $3\times 3$ non singular matrix.
Hence the functions $p_i^{(1)}$ for $i=1,3
\text{ and } 4$ are completely determined up to the
values on the boundary. We write
$q_1^{\pm}=\sum_{i=0}^4
q_{1,i}^{\pm}\psi_0^i M_0^{1\over 2}$ and, for
$i=1,3,4$, we choose the boundary values of $p_i^{(1)}$
as
$$p_i^{(1)}(\pm 1) = q_{1,i}^{\pm} \(3.44.1)$$
Then $p_0^{(1)}$ is determined by the first equation of
\(3.48) up to an additive constant that is chosen
so that \(2.24) is satisfied for $n=1$.
Finally we obtain
$$f_1(\pm1,v_y>0) = \alpha^\pm_1
M_0 + \ga^\pm_{1,\e} \(3.44.2)$$
with $\alpha^\pm_1= p_0^{(1)}(\pm 1) - q^\pm_{1,0}$ and
$\ga^\pm_{1,\e}= b^\mp_1(2\e^{-1})$. In fact, by
reflection symmetry $f_1=\Cal R f_1$ and, in
particular, $\a^+_1=\a^-_1$.
Now that $B_1$ and the non-hydrodynamical part of
$B_2$ are completely determined, we can solve
\(3.6). To do that we have to use Theorem 3.2 with
$$s= 2Q(B_1,b_1^-) + Q(b_1^-,b_1^-) +2Q(\Delta M,b_1^-)
+ Q(b_1^-,b_1^+) \(3.44.3)$$
and $\vt$ given, as before, by the difference between the
value of the non hydro\-dy\-na\-mi\-cal part of $B_2$ in $y=-1$
and the limit value $q_2^-$ of $\tilde b_2^-$.
Reserving to Appendix B the check of the conditions of
applicability of Theorem 3.2, we proceed as before to
the construction of $b_2^\pm$ which vanish for $y'\to
\infty$. The determination of $t_2$ follows the same
lines as the determination of $t_1$, and, in fact, we
can repeat the above procedure step by step for the functions
$f_n$.
The only difference in the $n$-th step, for $n>2$,
is due to the presence of the $v_x$-derivative of
$b_{n-2}^\pm$ in the sources, arising from the presence
of the external field. Their control is guaranteed by
part 2 of Theorem 3.2.
We note that $t_5$ and $t_6$ are not completely
determined by the above considerations. In fact \(3.44)
with $n=6$ only implies the conditions $${d\over
dy}[p_0^{(5)}+{5\over 2\sqrt 3}\/p_4^{(5)}]=0,
\phantom{...} {d\over dy}p_2^{(5)} = 0 \(3.48.99)$$
Since there is no compatibility condition to satisfy
for $n>6$, there is also no differential equation
to be satisfied by $t_6$. Nevertheless we still need to
assume $$p_2^{(5)} = 0,\phantom{...} p_2^{(6)} = 0
\(3.48.100)$$ to satisfy \(2.22) for $n=5,6$.
Moreover we have to satisfy boundary conditions of
the form
$$p_i^{(5)}(\pm 1,v) = q_{5,i}^{\pm}(v) \(3.44.88)$$
$$p_i^{(6)}(\pm 1,v) = q_{6,i}^{\pm}(v) \(3.44.89)$$
for $i=1,3,4$.
Therefore we choose $p_i^{(5)}$ and $p_i^{(6)}$,
$i=1,3,4$, as constants matching the
prescribed boundary values: they coincide by reflection
symmetry. Moreover we use the
arbitrary constant arising from the first equation in
\(3.48.99) to satisfy \(2.24) with $n=5$. Finally we
use choose $p_0^{(6)}$ as a constant such that \(2.24)
is satisfied for $n=6$. The condition \(2.23) is
automatically satisfied by \(3.8.1) since we have
already chosen $p_2^{(6)}=0$. The estimate \(2.26)
easily follows by combining Proposition 2.1 with
\(3.30), \(3.30.1), \(3.39) and \(3.41).
Finally we get the estimate \(2.27) for $A$. The first
three terms in \(3.8.1) are controlled using
\(3.30), \(3.30.1), \(3.39) and \(3.41).
To bound the last term of \(3.8.1) we use \(3.39) and the
following estimate which has been proved by Grad
[\rcite{G3}] for any Maxwellian:
$$|M^{-{1\over 2}}Q(R,S)|_{r-1} \le c
|M^{-{1\over 2}}R|_{r}|M^{-{1\over 2}}S|_{r} \(2.47)$$
We use again \(2.47) to estimate the remaining
terms:
$$|M^{-{1\over2}} 2Q(\Delta M,b^{\pm}_6)|_{r-1} \le
|M^{-{1\over2}}\Delta M e^{-\s \tau}|_r |M
^{-{1\over2}} b^{\pm}_6 e^{\s \tau}|_r \le cF
\sup_{\tau\in \Bbb R^{+}}|\tau e^{- \s \tau) }| \le cF
\(3.48)$$
where the second inequality is due to
\(3.39) and \(3.8), since $M^{-{1\over 2}} \le cM_0^{-{1\over
2}}$.
This concludes the proof of Proposition 3.1.\qed
%
\vskip.5cm
\heading 4. The remainder. \endheading
\vskip .5cm
\resetall
In this section we construct the remainder $f_R$ and prove
the crucial estimate \(2.31).
\noindent To fulfill the condition \(2.29) it is convenient to
put
$$f_R = \a_R M + R \(2.30)$$
so that
\(2.29) is satisfied if $\a_R$ is given by
$$ {\a}_R = -{1 \over m} \int dv dy R \(2.31)$$
\noindent In order to satisfy the boundary conditions
\(2.9) for $f$, taking into account the conditions
\(2.25) verified by the $f_n$'s, we complete the
non linear problem \(2.18) with the following boundary
conditions:
$$\align f_R(-1,v)&= \a_R M(-1,v)
-\sum_{n=1}^6 \e ^{n-3} \ga^- _{n,\e}
\phantom{....}
v_y>0\\ f_R(1,v)&=\a_R M(1,v)
-\sum_{n=1}^6 \e ^{n-3} \ga^+_ {n,\e} \phantom{....}
v_y<0\(2.32) \endalign$$
The outgoing mass flux
at the walls is thus determined
in terms of the constants we have
already fixed and $\a_R$ as
$$\a = \rho_0 ( T_0/2
\pi)^{1/2} \big (1+ \sum_{n=1}^6 \e ^n \a_n + \e^3 {\a}_R
\big )\(2.33)$$
Using \(2.18)
we obtain the following equation for $R$, defined by \(2.30)
$$v_y {\pa R\over \pa y } +\e F {\pa R\over
\pa v_x}={1\over \e}\Cal L R
+ \Cal N R + \e^2 \ti {Q}(R,R) + \e^3 A \(2.35)$$
\noindent where the linear operator $\Cal N R$
is the following modification of $\Cal L^1$:
$$\Cal N R= \Cal { L}^1 R -{1 \over m}
\Big[\Cal L^1\/ M -\e F {\pa M\over \pa
v_x} - v_y {\pa M\over \pa y}\Big]
\int dv dy R\(2.36)$$
The non linear term is given by
$$\ti {Q}(R,R) = {Q}(R,R) + {2 \over m}\Cal L R
\int dv dy R \(2.37)$$
The boundary conditions for $R$ are
$$\align R(-1,v)&= -\sum_{ n=1} ^6 \e ^{n-3}
\ga^- _{n,\e} \phantom{....}
v_y>0\\ R(1,v)&=-\sum_{ n=1} ^6 \e ^{n-3}
\ga^+ _{n,\e} \phantom{....}
v_y<0\(2.38) \endalign$$
The reflection symmetry of $R$ implies that
$R(0,v)$ is an even function of $v_y$ and, as a
consequence, $\langle v_yR(0,v)\rangle = 0$.
Therefore, integrating \(2.35) on velocities, we see by
\(2.23) that $R$ satisfies the vanishing mass flux
condition
$$\langle v_y R\rangle=0 \(2.39)$$
We prove the existence of the solution
of a suitable integral
form of equations \(2.35), \(2.38) and
\(2.39) that will be specified in Section 6.
It is constructed as a limit of the sequence
$\{R_k,k\in \Bbb N\}$ of solutions of
the approximate equations
$$v_y {\pa R_k\over \pa y } +\e F {\pa R_k\over
\pa v_x}={1\over \e}\Cal L R_k + \Cal N R_k +
\e^2 \ti { Q}(R_{k-1},R_{k-1}) + \e^3 A \(2.40)$$
with boundary conditions
$$\align R_k(-1,v)&= -\sum_{ n=1} ^6 \e ^{n-3}
\ga^- _{n,\e}\phantom{....}
v_y>0\\ R_k(1,v)&=-\sum_{ n=1} ^6 \e ^{n-3}
\ga^+ _{n,\e} \phantom{....}
v_y<0\(2.41) \endalign$$
for $k\ge1$, with $R_0=0$.
To prove the convergence of the sequence
$\{R_k,k\in \Bbb N\}$ we have to deal with a preliminary
linear problem. We consider a function $D$ such that
$$\Cal R^* D = D, \phantom{...}\langle D\rangle=0 \(2.42)$$
and a function $\z_-$ of velocity,
defined for $v_y>0$. We look for
the solution of the following linear
boundary value problem:
$$v_y {\pa R\over \pa y }
+\e F {\pa R\over \pa v_x}={1\over \e}\Cal L R +
\Cal N R + \e^2D \(2.43)$$
$$\align R(-1,v)&=\z^-\phantom{....}
v_y>0\\ R(1,v)&=\z^+\phantom{....}
v_y<0\(2.43.1) \endalign$$
with $\z^+(v) = \z^-(\Cal R v)$.
The solution $R$ of \(2.43), \(2.43.1) will be estimated
using the following norm:
$$|f|_{r,\beta}= \sup_{y\in [-1,1]}\sup_{v\in\Bbb
R^3}(1+|v|)^r |f(y,v)|\text{e}^{\beta\/v^2}. \(2.44)$$
and the same notation will be used also for the norm
of functions of velocity even if defined for incoming
velocities only. We shall prove the following
\proclaim {Proposition 4.1} There are $\e_0>0$, $F_1>0$ and
$\beta_0>0$ such that for any $\e<\e_0$, $F0$ and for any
$r\ge 3$ we have the estimate
$$|R_k|_{r,\beta} \le c\e^{1\over
2}(|R_{k-1}|_{r,\beta})^2 + c \e^{3\over 2}
|A|_{r,\beta} +c\/\e^{-2}e^{-c'\e^{-1}} \(2.48)$$
This implies that, for $\e$ small enough, uniformly
in $k$, for any $c'' 1$,
$W_k$ satisfies the equation
$$v_y {\pa W_k\over \pa y } +\e F {\pa W_k\over \pa
v_x}={1\over \e}\Cal L W_k + \Cal N W_k + \e^2
\ti Q (R_{k-1}+R_{k-2},W_{k-1})\(2.50)$$
$$\align W_k(-1,v)&=0\phantom{....}
v_y>0\\ W_k(1,v)&=0\phantom{....}
v_y<0\(2.50.1) \endalign$$
\noindent In \(2.50) we used the notation
$\ti Q(R,S)= Q(R,S) + \a _S \Cal L R + \a _R \Cal L S$.
Hence, by choosing
$D=\ti Q (R_{k-1}+R_{k-2},W_{k-1})$
and $\z^{\pm}=0$, using again
Proposition 4.1, \(2.47)
and \(2.49), it follows that
$$|W_k|_{r,\beta} \le c\e^2 |W_k|_{{r-1},\beta}
\(2.51)$$
Therefore for $\e \le \e_0$ the sequence $R_k$
converges and the
limit satisfies \(2.49).
Let $R_1$ and $R_2$ be two solutions of \(2.35), \(2.38)
with uniformly bounded $|\ .\ |_{r,\beta}$. Then
$W=R_1-R_2$ satisfies
$$v_y {\pa W\over \pa y } +\e F {\pa W\over \pa
v_x}={1\over \e}\Cal L W + \Cal N W + \e^2
\ti Q (R_1+R_2,W)\(2.50.2)$$
By Proposition 4.1 it follows that
$|W|_{r,\beta}\le c\e^{{1\over 2}}|W|_{r,\beta}$,
which implies uniqueness for $\e$ small enough.
We have thus proved the
following
\proclaim {Theorem 4.2} There are
$\e_0>0$, $F_1>0$ and $\beta_0>0$ such that for
any $\e<\e_0$, $F\sup_ {y\in[-1,1]}T(y) $ which is
finite by Proposition 2.1. In this way $M_* \ge c M $ for
all $(y,v)$ and some positive $c$.
We look for a solution of Eq.\(2.43) in the form
$$R=\sqrt M g + \sqrt M_* h \(4.2)$$
where the {\it low velocity} part $g$ and the
{\it high velocity} part $h$ are defined as the solutions
of the following system of coupled equations
$$v_y {\pa g \over{\pa y }} + \e F {\pa g \over {\pa {v_x}}} +
(\mu +\e F \mu' )\h g ={\e}^{-1} Lg + {\e}^{-1} \chi_\ga
{\s} ^{-1} K_* h + L^1 \h g \(4.3)$$
$$ \aligned v_y {\pa h \over {\pa y }}+ \e F {\pa h
\over {\pa {v_x}} } +
\e F \mu'_* h &+ (\mu +\e F \mu' ) \s (\bar g + g_2) = \\
{\e}^{-1}(-\nu &+ \bar\chi_\ga K_*) h +L_*^1[\s (\bar g +
g_2) + h] +\e^2 d \endaligned \(4.4)$$
The notation is:
$$\mu = v_y {1 \over 2} \pa_{y}\log M, \phantom{...}
\mu' = {1 \over 2} \pa_{v_x}\log M, \phantom{...}
\mu'_* = {1 \over 2} \pa_{v_x}\log M_*,\phantom{...}
\s =\sqrt {M\over M_*}\(4.5)$$
$$ \chi_\ga(v) = \Bigg\{ \aligned &1, \ |v|\le \ga \\ &0,
\ |v|\ge \ga \endaligned \(4.6)$$
\noindent $\bar\chi_\ga= 1- \chi_\ga$,
$$ L_* f = M_*^{-1/2} 2Q(M,M_*^{1/2}\ f )=(-\nu + K_*)f
\(4.7)$$
$K_*$ is an integral operator analogous to $K$, for which the
same properties \(3.25)--\(3.27) hold (see [\rcite{Ca}]).
$$L^1 f = M^{-1/2} \Cal N (M^{1/2}\ f ) \phantom {....} L_*^1 f =
M_*^{-1/2} \Cal N (M_*^{1/2}\ f ) \(4.8)$$
$$ d=M_*^{-1/2} D \(4.9)$$
The low velocity part $g$ has been decomposed into a
hydrodynamical part $\hat g +g_2$ and a non hydrodynamic
part $\bar g$
$$g= \h g + g_2 + \bar g, \phantom{...}\text{with}\
g_2=p_2 (y) \psi_2 \/, \text{ } \/
\h g =\sum_{j \ne 2} p_j (y) \psi_j \(4.10)$$
We choose the following boundary conditions for $g$ and $h$
in equations \(4.3) and \(4.4)
$$ \Bigg \{ \aligned g(1,v) &=0 \hskip .5cm v_y <0 \\
g(-1,v) &=0 \hskip .5cm v_y >0 \endaligned \hskip 1cm
\Bigg\{ \aligned h(1,v) &=\z^+
M_*^{-1/2}\equiv h_+ \hskip 1cm v_y <0\\
h(-1,v) &=\z^- M_*^{-1/2}\equiv h_-
\hskip 1cm v_y >0\endaligned \(4.11)$$
Of course $h_+(v)=h_-(\Cal R v)$.
It is also convenient to
consider $h_{\pm}$ extended to $\Bbb R^3$ putting
it equal to 0 for $v_y \gtrless 0$.
The norm we are interested in is
$$||f|| = \Big (\int_{[-1,1]\times \Bbb R^3} dy\ dv
(1+|v|) f^2 (y,v) \Big )^{1\over 2}\(4.12)$$
We remark that in \(4.4) the unbounded terms
$\mu$ and $\mu'$ are compensated by the factor $\si$
for large velocities while in \(4.3)
they appear as multipliers of $\h g$: $\h g$ has a good
behavior for large velocities, but has a bad
estimate in $\e$. This is the reason why we chose
our decomposition in such a way that $\h g$ does not appear in \(4.4).
Finally, the factor $\mu_0'$ is also unbounded for
large velocities but is a
polynomial of degree $1$, so that it can be dominated in the
norm \(4.12). (For cross sections softer than hard
spheres this would not be true because the natural
norm to be used would include a power of $|v|$ less than $1$.)
\proclaim {Theorem 5.1} There exist positive $\e_0$,
$F_1$ and $c>0$ such that the solutions to Eq. \(4.3), \(4.4)
and \(4.11) satisfy the bounds
$$ ||\bar g || \le \e^2 c ||(1+|v|)^{-1} d||
+c\e^{-1/2}||h_-||\(4.14)$$
$$||\h g ||\le \e c ||(1+|v|)^{-1}d||
+c\e^{-3/2}||h_-||\(4.15)$$
$$|| g_2 || + ||h || \le\e^3 c||(1+|v|)^{-1}d||
+c\e^{1/2}||h_-||\(4.16)$$
\endproclaim \vskip 1cm
The proof is organized as follows:
First we will obtain a bound for $||g||$ in terms of
$||h||$. Using such a bound it will be possible to
estimate $||h||$ in terms of $||d(1+|v|)^{-1}||$ and $||h_-||$.
To estimate the operators $L^1$ and
$L^1_*$ we will use the following estimate on the
collision operator $Q$ proven in [\rcite{GPS}]: for any Maxwellian
$M$ and for any $y\in[-1,1]$
$$\int_{\Bbb R^3}\ dv
{|Q (\sqrt M f,\sqrt M g)|^2\over (1+|v|) M}
\le \int_{\Bbb R^3}\ dv (1+|v|)|f|^2\ \int_{\Bbb R^3}\ dv (1+|v|)|g|^2
\(4.17)$$
\noindent (A) \underbar {Estimates on $g$}.
The condition \(2.39) together with \(4.2) and the fact
that $\bar g + \hat g$ is orthogonal to $\psi_2$, implies that
$$\int dv (\sqrt M g_2 + \sqrt M_*h) v_y = 0. \(4.18)$$
As a consequence
$$\sqrt{\rho T} p_2 =-\int dv \ \sqrt M_* h v_y \(4.19)$$
\noindent Therefore, by Proposition 2.1 and the Schwarz
inequality, we conclude that
$$||g_2|| \le C ||h||\(4.20)$$
We now give estimates on $\bar g$ and on $\h g$
separately. \vskip 1cm a) Bound on $\bar g$.
Multiplying Eq.\(4.3) by $g$ and integrating it on $\Bbb
R^3_v$ we have.
$$\aligned {d \over dy}\langle v_y {1\over 2} g^2\rangle &
+\langle (\mu + \e F \mu')g \h g\rangle =\\ &\e^{-1} (
\bar g, L \bar g) + \e^{-1}( \la, g) + (g, L^1 \h g )
\endaligned \(4.21)$$
\noindent where $ \la =\chi_\ga {\s} ^{-1} K_* h$.
\noindent
Let us observe that
$$\langle \mu \h g ^2\rangle =\sum_{i,j \ne 2} p_i p_j\int
dv \mu \psi_i \psi_j =0 \(4.23)$$
\noindent since, for $i,j \ne 2$, $\mu \psi_i \psi_j$, is an odd
polynomial in $v_y$ times the Maxwellian, which is even
in $v_y$.
\noindent Moreover, using \(3.1) in \(2.36) we get
$$\Cal N R= \Cal { L}^1 R -{1 \over m}
\Big[ 2Q(\sum_{n=2}^6 \e^{n-1}f_n,M) + \Cal L(b^+_1 + b^-_1)
-\e F {\pa M\over \pa v_x}\Big]\int dv dy R\(4.23.2)$$
from which we see that the only term in $\Cal N f $ that is not
orthogonal to the collision invariants is $\e F {\pa_{v_x} M}$.
Hence we have
$$|(\hat g, L^1\hat g)| \le c\e F ||\hat g||^2 \(4.23.1) $$
We remark that \(4.23) and \(4.23.1) are crucial
to get an estimate on
$g$.
Integrating Eq.\(4.21) on $[-1,1]$ we get
$$\Cal I + \int dy dv (\mu + \e F
\mu')g \h g = \int dy dv [\e^{-1} \bar g L \bar g
+ \e^{-1} \la g + g L^1 \h g] \(4.24)$$
\noindent where
$$\Cal I ={1 \over 2} [\langle v_y g^2 (1,v) \rangle -
\langle v_y g^2 (-1,v) \rangle] \ \ge 0 \(4.25)$$
using the boundary conditions \(4.11) for $g$.
Let us examine more closely the term involving $L^1$. By
\(4.17), \(4.23.2) and \(4.8), we have:
$$|| (1+|v|)^{-1}L^1 \h g|| \le \Big [ \sum_{n=1}^6 \e^{n-1}
|M^{-1/2}f_n|_3 \ + c\e F \Big ]||\h g||\(4.26)$$
Hence, using \(2.26), it follows that
$$|(\bar g, L^1 \h g)| \le c F ||\bar g|| \ ||\h g|| \(4.27)$$
To conclude the analysis of Eq. \(4.24) consider the term
$\langle(\mu +\e F\mu')\h g g)\rangle$. Taking into account
that $\mu$ is a function of the derivatives of
$\rho,T,u$ it follows by Proposition 2.1 and \(4.10) that
$$|| \mu \h g|| \ \le cF \big ( \int dy \sum_{j\ne 2}
p_j^2\big)^{1\over 2}\le cF||\h g||\sum_{j\ne 2}||\psi_j||^2 \le
cF||\h g|| \(4.28)$$
using the decay of $\h g$ in $v$ to get the first inequality and the
relation $p_i =(\hat g,\psi_i)$ to obtain the second inequality. In
the same way it follows that $||\mu'\h g||\le c ||\h g||$.
Hence, using \(4.23) we have the following bound
$$|((\mu +\e F\mu')\h g, g)| \le c
F ||\h g|| ( ||\bar g|| +||g_2||) +c\/\e F ||\h g||^2 \(4.29)$$
\noindent Since $\la=\chi_\ga { \s }^{-1} K_* h$ we have,
by the $L_2$-boundedness of $K_*$ (see [\rcite{Ca}]),
$$||\la|| \le C_\ga ||h|| \(4.30)$$
Finally, by \(4.29), \(4.30), \(4.23), \(4.23.1), \(4.27) and
\(3.23) we have:
$$\e \Cal I + \nu_0 ||\bar g||^2 \le
C_\ga ||h||\ ||g|| + \e c F ||\h g|| \ ||\bar g|| +\e
^2 F ||\h g||^2 \(4.31)$$
\vskip 1 cm
b) Bound on $\h g$
Multiplying Eq. \(4.3) by $v_y \psi_i,\ i=0,1,3,4$ and
integrating on $[-1,y]\times \Bbb R^3$, we have
$$\aligned \Phi _i (y) = \Phi_i (-1) +& \int_{-1}^ydy'
\int_{\Bbb R^3} dv v_y \psi_i \big [- (\mu +\e F \mu' )\h
g- \e F {\pa g \over {\pa {v_x}}} + \\&{\e}^{-1} L \bar g +
{\e}^{-1} \la + L^1 \h g + v_y^2 g
\pa_y \psi_i\big ] \endaligned \(4.32)$$
\noindent where $\Phi_i (y) =\langle v_y ^2 \psi_i
g\rangle$.
First we give the estimate for $\Phi (-1)$. By the Schwarz
inequality
$$|\langle v_y ^2 \psi_i g (-1,v)\rangle | \le c
\langle |v_y| g ^2 (-1,v)\rangle ^{1\over2}\ \langle
|v_y|^{3} \psi_i^2 \rangle ^{1\over2} \le c \Cal I
^{1/2} \(4.33)$$
because $g(-1, v)=0$ for $v_y>0$.
Therefore, taking into account the bound \(4.31)
for $\Cal I $, we have
$$|\Phi (-1)| \le c \big [{\e}^{-1} C_{\ga}
||h|| \ ||g|| + F ||\bar g|| \ ||\h g ||
+ {\e F} ||\h g||^2 \big ]^{1\over2} \(4.34)$$
When the term involving the $v_x$-derivative in \(4.32) is
integrated by parts, then all the terms in Eq. \(4.32) can
be estimated using the Schwarz inequality and the bounds \(4.26),
\(4.27), \(4.29) and \(3.24). The result is
$$\aligned |\Phi_i (y)| \le & c \big [{\e}^{-1}
C_{\ga} ||h|| \ ||g|| + F ||\bar g||\
||\h g || + {\e F} ||\h g||^2\big ]^{1\over2} \\ & +
c {\e}^{-1} ||\bar g|| + {\e}^{-1} C_{\ga} ||h|| + c F
||g||\big ]\endaligned \(4.35)$$
\noindent By the definition of $\Phi_i (y)$ we have:
$$\Phi_i (y) =\langle v_y ^2 \psi_i g \rangle = \sum_{j
\ne 2}A_{ij} p_j + \langle v_y^2 \psi_2^2 \rangle p_2
\de_{i,2} +\langle v_y ^2 \psi_i \bar g \rangle\(4.36)$$
where $\de_{i,j}$is the Kronecker delta.
The matrix $A$ whose elements are $A_{ij}=\langle
v_y^2\psi_i \psi_j\rangle$, $i,j=0,1,3,4$ is non singular.
Hence the ${p_i}, \phantom{..} i=0,1,3,4 $ are
determined by
$$ p_i= \sum_{j\ne 2} A^{-1}_{ij} \big [ \Phi_j -
\langle v_y^2 \psi_2^2 \rangle p_2
\de_{j,2} - \langle v_y ^2 \psi_j \bar g \rangle \big ] \(4.37)$$
Therefore
$$||\h g||^2 \le c \sum_{i\ne 2} \int_{-1}^1 dy |p_i(y)|^2
\le c\int_{-1}^1 dy \big[\sum_{i\ne 2} [ \Phi_i ^2 + C p_2
^2\/\big] + c||\bar g||^2 \(4.38)$$
\noindent Hence, using the bound \(4.35) for the $\Phi_j$'s and
\(4.20) for $||g_2||$, we obtain
$$\aligned ||\h g|| \le c \big [{\e}^{-1} &
C_{\ga} ||h|| \ ||g|| + F ||\bar g|| \
||\h g || + \e F ||\h g ||^2 \big ]^{1\over 2} +\\&
c{\e}^{-1} ||\bar g|| + {\e}^{-1} C_{\ga} ||h|| + c F ||g||
\endaligned \(4.39)$$
\noindent We simplify the right hand side of Eq. \(4.39)
using the inequality
$$ |ab| \le \kappa a^2 + ( 4\kappa)^{-1} b^2\(4.40)$$
for any $\kappa >0$, to bound
the product ${\e}^{-1} ||h|| \ ||g||$ with $\kappa ||g||
+ (4\kappa)^{-1}\e^{-2}||h||$. So we get
$$||\h g|| \le c
{\e}^{-1} ||\bar g|| + {\e}^{-1} C_{\ga}(1+ {1 \over 4
\kappa}) ||h|| +(2 F + \ka)||g|| \(4.41)$$
Finally, choosing $\ka$ and $F$ sufficiently
small, we get
$$||\h g|| \le c {\e}^{-1} ||\bar g|| + {\e}^{-1}
C_{\ga} ||h||\(4.42)$$
Using \(4.20) in \(4.31) we have
$$||\bar g||^2 \le C_\ga ||h||\big ( ||\bar g|| +||\h
g|| +||h|| \big ) + c\e F ||\bar g||\ ||\h g|| + c\e^2 F
||\h g||^2\(4.43)$$
Substituting \(4.42) in \(4.43), using again \(4.40) we get
$$||\bar g||^2 \le (\sigma C_\ga + cF) ||\bar g||^2 +
{c\/C_\ga\over 4 \sigma \e^2} ||h||^2 \(4.44) $$
Hence, choosing $\sigma$ and $F$ sufficiently small, we have
$$||\bar g || \le {\e}^{-1} C_\ga ||h|| \(4.45)$$
\noindent In conclusion we have the following estimates
for the hydrodynamic and non hydrodynamic parts of
$g$
$$||g_2|| \le C_\ga ||h||\(4.46)$$
$$||\bar g|| \le {\e}^{-1} C_\ga ||h||\(4.47)$$
$$|| \h g|| \le {\e}^{-2} C_\ga ||h|| \(4.48)$$
Estimates \(4.46) and \(4.47) will be used below to estimate
$||h||$.
\vskip 1cm
\noindent (B) \underbar {Estimates on $h$}.
The way to get bounds on $h$ is analogous to the one
followed for $g$ but simpler since we do not need to
control separately the hydrodynamic and the kinetic part.
In fact in \(4.4) the operator $L$ has been replaced by
$(-\nu + \bar\chi_\ga K_*)$ which has a trivial null
space.
\noindent Multiplying Eq. \(4.4) by $h$ and
integrating on
$[-1,1]\times \Bbb R^3$ we have
$$ \aligned \Cal J +
\int dy & \langle \e F \mu'_* h^2\rangle +\int dy
\langle(\mu +\e F \mu' ) h \s (\bar g + g_2) \rangle = \\
& \int dy dv\ h \ \big [{\e}^{-1} (-\nu + \bar\chi_\ga
K_*) h +
L_*^1[\s (\bar g + g_2) + h] +\e^2 d \big ] \endaligned
\(4.49)$$
\noindent where $ \Cal J = \langle v_y h^2 (1,v)
\rangle - \langle v_y h^2 (-1,v)\rangle \ge -\langle
|v_y|h_-^2\rangle$ by the boundary conditions \(4.11).
\noindent We observe that, by the $L_2$-boundedness
of the operator $K_*$ it follows that
$$|\int dy
\langle \bar \chi_\ga h\/K_* h\rangle| \le ||h||\Big(
\int dy \langle \bar \chi_\ga (K_* h)^2 (1 +
|v|)^{-1}\rangle\Big)^{1/2}\le c||h||^2
(1+\ga)^{-{1\over 2}}
\(4.50)$$
Hence
$$ \aligned {\e}^{-1} \nu _0 ||h||^2 \le
& c
\big [ \e F ||h||^2 + F ||h|| \ (||\bar g || +||g_2||) +\
F ||h||^2 + \\& \e^2
||h||\ ||d(1+|v|)^{-1}|| +{\e}^{-1} (1 + \ga)^{-1/2}
||h||^2 + \langle |v_y|h_-^2\rangle\big ]\endaligned \(4.51)$$
In fact the first integral in the l.h.s of \(4.49) has been
bounded by $\e F ||h||^2$, the second using the estimate $|(\mu
+ \e F \mu')\si| \le c F$. Moreover we used \(2.26)
\(4.17), \(4.23.2), \(4.8) to get the bound
$$||(1+|v|)^{-1}L_*^1(h+\si(\bar g + g_2))||<
cF(||\bar g|| +||g_2|| + ||h||)
\(4.51.1)$$
Then, using \(4.46) and \(4.47), we have
$$||h||^2 \le (C_\ga F + c(1+\ga)^{-{1\over2}} )
||h||^2 + \e^3 ||h||\
||d(1+|v|)^{-1}|| + c\e\langle |v_y|h_-^2\rangle \(4.52)$$
We fix $\ga$ large enough to make
$c(1+\ga)^{-{1\over2}}\le {1\over3}$ and then F
sufficiently small to make $C_\ga F \le {1\over3}$, so
that we can conclude that
$$||h|| \le c\e^3 ||d(1+|v|)^{-1}|| +\e^{1/2}||h_-||\(4.53)$$
\noindent The proof of Theorem 5.1 then follows by combining
\(4.46)--\(4.48) and \(4.53). \qed
\vskip 1cm
%
%
%
\heading 6. Pointwise estimates for the linear
problem. \endheading
\resetall
Pointwise estimates of the solution of the
linear problem are necessary to deal with the non
linear term of the Boltzmann operator. To get
them we have to use an integral form of the
equations for $g$ and $h$. This will give
estimates of $L_\infty$-norms in terms of the
norms used in the previous section.
Let us consider the equation
$$v_y {\pa f \over \pa y} + \e F {\pa f \over \pa v_x}+{\e}^{-1}
\nu \ f = {\e}^{-1}G \(5.1) $$
\noindent with the boundary conditions
$$f(-1,v)= f_- , \ v_y > 0;\phantom{...} f(1,v)= f_+ ,
\ v_y < 0 \(5.2)$$
We use the following notation:
$$\Phi_{y, y'} =\int_{y'} ^y dz \ \nu (z,v_x +
{\e F \over v_y} (z-y),v_y, v_z )\(5.3)$$
$$U_\e G(y,v)= {1 \over \e v_y}\int_{-1} ^y dy' G (y',v_x +
{\e F \over v_y} (y'-y),v_y, v_z )\/\exp\big[- {\Phi_{y, y'}\over {\e
v_y}}\big] \(5.4)$$
\noindent for $\ v_y > 0$ and
$$U_\e G(y,v)= -{1 \over \e v_y}\int_{y} ^1 dy' G (y',v_x +
{\e F \over v_y} (y'-y),v_y, v_z )\/\exp\big[ {\Phi_{y', y}\over {\e
v_y}}\big] \(5.5)$$
\noindent for $\ v_y < 0$.
$$V_\e^- f^- = \chi(v_y>0)f^- \/\exp\big[ -{\Phi_{y, -1}\over {\e
v_y}}\big]\(5.5.1)$$
$$V_\e^+ f^+ = \chi(v_y<0)f^+ \/\exp\big[ {\Phi_{1,y}\over {\e
v_y}}\big]\(5.5.2)$$
The solution of Eq. \(5.1), \(5.2) can be written as
$$f= V_\e^+f^+ + V_\e^- f^- + U_\e G \(5.5.3)$$
\vskip .5cm
\proclaim {Proposition 6.1} For any integer
$r\ge 0$ there is a constant $c$ such that the
integral operator $U_\e $ satisfies the following
inequality, uniformly in $\e$
$$|U_\e G|_r \le c \Big | {G \over \nu}\Big | _r
\(5.7)$$
\endproclaim
\underbar{Proof}:
It is sufficient to prove \(5.7) for $r=0$.
In fact, let $G_r = G\ (1 + |v|)^ {r}$
and $f_r = U_\e G\ (1 + |v|)^ {r}$. Then the $f_r$
satisfy equations of the same type as Eq.\(5.1),
$$v_y {\pa f_r \over \pa y} + \e F {\pa f_r \over \pa v_x}
+{\e}^{-1}(\nu - \e\ d_r) f_r = {\e}^{-1}G_r \(5.8) $$
\noindent where $d_r = \e \/r\/F\/v_x [|v| (1+|v|)]
^{-1},\ |d_r| \le \e F\/ r$, with vanishing boundary conditions.
\noindent Since for $\e$ small enough $\nu - \e\
d_r \ge c \nu$ the proof for $r=0$ can be applied.
\noindent Now for $r=0$ we have ($v_y > 0$)
$$\aligned |(U_\e G)(y)| \le &
{1 \over \e v_y} \int_{-1} ^y dy' |G (y',v_x +
{\e F \over v_y} (y'-y),v_y, v_z )\ |\ \exp\Big[ -
{1 \over \e v_y } \Phi _{y,y'}\Big]\\
\le &\ \Big|{G \over \nu} \Big | _0
\int_{-1} ^y dy'{1 \over \e v_y} \ \nu (y',v_x +
{\e F \over v_y} (y'-y),v_y, v_z )
\exp\Big[ -{1 \over \e v_y } \Phi _{y,y'}\Big]\\
\le &\ \Big|{G \over \nu} \Big | _0
\endaligned \(5.9)$$
\noindent because $\int_{-1} ^y dy'
(\e v_y)^{-1} \ \nu \ \exp \{- (\e v_y)^{-1}
\Phi_{y,y'} \} <1$
The estimate for the case $v_y < 0$ is obtained in the same way.\qed
In the next Proposition we use the following norm
$$N(f) = \sup_{y\in [-1,1]} \big ( \int_{\Bbb R^3} dv |f(y,v)|^2 \
\big )^{1/2}\(5.6) $$
\vskip 1cm
\proclaim {Proposition 6.2} For any $\de >0$ and for any
$r\ge2$ there is a constant $C_\de$ such that
$$N(U_\e G) \le {1 \over \sqrt \e } C_{\de} ||\nu ^{-1/2}G || +
\de |G|_r \(5.10)$$
\endproclaim
\vskip 1cm
\underbar{Proof}:
We give explicitly the proof only for $v_y >0$.
Let us consider
$$\aligned \int_{v_y > 0}& dv |U_\e G|^2 =\\
& \int dv \Big[ \int_{0} ^{1+y} dt
{1 \over \e v_y} G (y-t,v_x - { \e F \over v_y}t,v_y, v_z ) \
\exp \big[ - {1 \over \e v_y }
\Phi_{y,y-t}\big] \Big]^2\endaligned \(5.11)$$
In analogy to [\rcite{GPS}] we estimate the r.h.s of \(5.11) by
decomposing it in the three parts
$I_1, I_2, I_3 $ defined as
$$I_1 = \int_{ v_y \ge \ell} dv |U_\e G|^2 \(5.12)$$
$$I_2 = \int_{0< v_y \le \ell} dv \Big [ \int_{\s}^{1+y} {1
\over \e v_y} G (y-t,v_x - {\e F \over v_y}t,v_y, v_z ) \
\exp \big[ - {1 \over \e v_y} \Phi_{y,y-t }\big] \Big ]^2
\(5.13)$$
$$I_3 = \int_{0< v_y \le \ell} dv \Big [ \int_0^{\s} {1 \over
\e v_y} G (y-t,v_x - {\e F \over v_y}t,v_y, v_z )\ \exp
\big[- {1 \over \e v_y} \Phi_{y,y-t} \big] \Big ]^2 \(5.14)$$
\noindent By the Schwarz inequality we get
$$\aligned I_1 \le & \int_{ v_y \ge \ell} dv
\int_{0} ^{1+y} dt {1 \over \e v_y}\ {G^2 \over \nu}((y-t,v_x -
{\e F \over v_y}t,v_y, v_z ) \\
& \int_{0} ^{1+y} dt\/ {\nu \over \e v_y} \ \exp
-\{ {2 \over {\e v_y}} \Phi _{y,y-t}\} \le c (\e \ell
)^{-1}||\nu ^{-1}G ||^2 \endaligned \(5.15)$$
\noindent The bound on $I_2$ is obtained as follows.
We observe that by Eq.\(3.17)
$${1 \over \e v_y}\exp -\{ {2 \over
{\e v_y}} \Phi _{y, y-\s}\} \le c \s^{-1}\(5.16)$$
\noindent Setting $t' = t - \s$ in \(5.13) we are left with a
term like $I_1$ times the l.h.s. of \(5.16).
Hence, using
\(5.16) we have
$$ I_2 \le c {\s}^{-1} ||\nu ^{-1}G ||^2
\(5.17)$$
\noindent The third term $I_3$ can be handled
by noting that for $0 \le \beta \le 1$,
$$\aligned {1 \over \e v_y} \exp - \{{1 \over \e v_y}
\Phi_{y,y- t }\} \le & c
(\e v_y)^{\beta -1}\Phi_{y,y-t }^{- \beta}\\ & \le c(\e
v_y)^{\beta -1} (\nu_0 t)^{- \beta} \endaligned \(5.18)$$
\noindent Hence, by the Schwarz inequality,
$$\aligned & I_3 \le c {1 \over \e^{2- 2 \beta }}
\int _0 ^\s dt_1 {1 \over (\nu_0 t_1)^{ \beta}}
\int _0 ^\s dt_2 {1 \over (\nu_0 t_2)^{ \beta}}
\int_{0< v_y \le \ell}{dv_y \over v_y^{2-2 \beta }}
\Big [ \int dv_xdv_z \\ & G^2 (y-t_1,v_x -
{\e F \over v_y}t_1,v_y, v_z ) \Big ]^{1/2}
\Big [ \int dv_xdv_z G^2 (y-t_1,v_x -
{\e F \over v_y}t_2,v_y, v_z ) \Big ]^{1/2}
\endaligned \(5.19)$$
\noindent Then
$$I_3 \le c |G|^2 _r({\s \over \e})^{2- 2 \beta }
\int_{v_y \le \ell}{dv_y \over v_y^{2-2 \beta }}
\int dv_xdv_z
{1 \over {(1+v_x ^2 +v_z ^2)}^{r}} \(5.20)$$
\noindent where we have fixed $\beta <1$ to make finite the
integral over $t$. If we choose also $\beta > 1/2$
and $r > 1$ all the integrals in Eq. \(5.20) are finite.
Finally, since $\s$ is arbitrary, we can
take $\s = \e$ getting
$$I_3 \le C \de ^2 |G|^2 _r \(5.21)$$
\noindent with $\de ^2=\ell^{2 \beta -1}$.
\noindent Combining \(5.15), \(5.17) and \(5.21) we get the result.\qed
\vskip 1cm
The regularizing properties of the operator $U_\e$ suggest to
consider the following integral form of equation \(2.35)
$$\align R = &U_\e\Big[ {1\over \e}(\Cal L R -\nu R) +
\Cal N R + \e^2
\ti {Q}(R,R) + \e^3 A\Big ]
\\& - V_\e^-\big[\sum_{ n=1} ^6 \e ^{n-3}
\ga^- _{n,\e}\big] - V_\e^+\big[\sum_{ n=1} ^6 \e ^{n-3}
\ga^+ _{n,\e}\big]\(5.21.1)\endalign$$
In fact we define a {\it solution} of \(2.35), \(2.38)
as any solution of \(5.21.1).
We consider now the integral versions of \(4.3) and \(4.4)
that allow to prove estimates for the norm $|\cdot|_r$ of
$g $ and $h$.
\vskip 1cm
\proclaim {Proposition 6.3} There exist positive constants $c$
and $H_{\ga}$ such that for any $r \ge 2$ the solution $g$ of
Eq.\(4.3) verifies
$$|g|_r \le c\sqrt \e ||d(1+|v|)^{-1}|| + H_{\ga} |h|_r +c
\e^{-2} |h_-|_r \(5.22)$$
\endproclaim
\vskip 1cm
\underbar{Proof}:
\noindent We write Eq.\(4.3) as
$$v_y {\pa g \over \pa y} + \e F {\pa g \over \pa
v_x}+{\e}^{-1} \nu \ g = {\e}^{-1}(Kg +S) \(5.23) $$
\noindent with
$$S= -\e (\mu + \e F \mu')\h g + {\chi}_\ga {\s}^{-1}
K_* h +\e L^1 \h g \(5.25)$$
Most of the proof is devoted to getting an estimate for $N
(g)$ in terms of $N(h)$ and
the $ L_2$ norms of $g$ and $h$,
whose control is assured by the results of Section 4.
Then we conclude by relating $N(h)$
to the $L_\infty$ norm of $h$.
By Proposition 6.2 the solution of Eq. \(5.23) verifies,
for any $r\ge 2$, the inequality
$$N(g) \le {1 \over \sqrt \e } C_{\de} ||\nu ^{-1}Kg || +
\de |Kg|_r + {1 \over \sqrt \e } C_{\de} ||\nu ^{-1}S || +
\de |S|_r \(5.25)$$
The first and third terms on the right hand side are easily
estimated using Theorem 5.1 and the inequalities
$$||\nu ^{-1}Kg || \le ||g ||;\phantom {...}||\nu ^{-1}S || \le \e
F ||\h g|| +C_\ga ||h||\(5.26)$$
The second term in \(5.25), is controlled by using
the estimates \(3.25) and \(3.26), which allow to relate the supremum
on velocities of $Kf$ with the $L_2 (dv)$ norm of $f$. In fact
$$|Kg|_r \le \sup_{y\in [-1,1]}\sup_{v\in \Bbb R^3} (1+ |v|)^{r-1} |g
(y,v)| = c |g|_{r-1} \(5.27)$$
\noindent by \(3.25) and
$$|Kg|_0^2 \le c \sup_{y\in [-1,1]} \int dv g^2 (y,v) =
c\/N(g)^2 \(5.28)$$
\noindent by \(3.26).
Furthermore, since $g= U_\e Kg + U_\e S$, we get by Proposition
(6.1) $$|g|_r \le |Kg|_r + |S|_r \le |g|_{r-1} + |S|_r \le
...\le |Kg|_0 +\sum_{k=0} ^r |S|_k \le N(g)+\sum_{k=0} ^r |S|_k
\(5.29)$$ where we have used \(5.28) to get last inequality.
Thus to complete the proof we have to estimate
$S$ in the norm $|\/.\/|_r$. We have
$$|S|_r \le \e|(\mu + \e F \mu')\h g|_r +
|{\chi}_\ga {\s}^{-1} K_* h|_r + \e |L^1 \h g|_r
\(5.30)$$
\noindent By the exponential decay of $\h g$ in velocities and
\(5.28) we have
$$|(\mu + \e F \mu')\h g|_r \le cF |\h g|_0 \le cFN(\h
g) \(5.31)$$
By the analogue of \(5.28) for $K_*$ (see [\rcite{Ca}]) we have
$$|{\chi}_\ga {\s}^{-1} K_* h|_r \le \sup_{y\in [-1,1]}
\sup_{v\in \Bbb R^3}
[(1+ |v|)^r {\chi}_\ga
{\s}^{-1}] \sup_{y\in [-1,1]}
\sup_{v\in \Bbb R^3} | K_* h|\le H_{\ga} N(h) \(5.32)$$
\noindent The operator $L^1$ satisfies the estimate
$$|\nu^{-1} L^1 f|_r \le c|f|_r \(5.33)$$
analogous to the one proved in [\rcite{Ca}].
Using it we can estimate the last term:
$$|L^1 \h g|_r \le c|\h g|_{r+1} \le c\sup_{y\in [-1,1]}|\sum_{i \ne
2} p_i(y)| \le c|\h g|_0 \le c N(\h g) \(5.34)$$
\noindent Collecting together \(5.30), \(5.31) and \(5.34) we
have the following bound for $S$
$$|S|_r \le c \e N(\h g) + H_{\ga} N(h) \(5.35)$$
\noindent By \(5.27), \(5.29) and \(5.30), we have
$$|Kg|_r \le c N(g) + H_{\ga} N(h) \(5.36)$$
\noindent Now we can estimate $N(g)$ by combining \(5.25),
\(5.26), \(5.35) and \(5.36):
$$N(g) \le c[ {1 \over \sqrt \e } \ C_{\de} ||g || +
\de N(g) + H_{\ga} N(h)+ {1 \over \sqrt \e } C_{\de} H_{\ga} ||h||\/ ]
\(5.37)$$
Finally, taking into account estimates \(4.46)--\(4.48) and
\(4.53), we get, for $\de$ small enough,
$$N(g) \le c [\ \sqrt \e ||d(1+|v|)^{-1}|| +
\e^{-2}|h_-|_3] + H_{\ga} N(h)\/]
\(5.38)$$
\noindent On the other hand Proposition 6.1 implies
$$\aligned |g|_r \le c |\nu^{-1} Kg|_r + &|\nu^{-1} S|_r
\le c [N(g) + H_{\ga} N(h)] \\ &\le c[\sqrt\e ||d(1+|v|)^{-1}|| +
\e^{-2}|h_-|_3]+ H_{\ga} N(h)\/ ] \endaligned\(5.39)$$
\noindent where the second inequality comes from \(5.35) and \(5.36)
and the third one from \(5.38).
\noindent For $r \ge 2$ we have
$$[N(h)]^2 \le \sup_{y\in [-1,1]}\sup_{v\in \Bbb R^3}
[ h^2 (y,v) (1+|v|)^{2r}] \int_{\Bbb R^3} {dv \over {(1+|v|)^{2r}}}
\le c |h|_r ^2.\(5.40)$$
\noindent Putting together estimates \(5.39) and \(5.40) we get the
result \(5.22).\qed
Finally, we prove
\proclaim {Proposition 6.4} For any $r \ge 3$
the solution $h$ of Eq.\(4.4) verifies
$$|h|_r \le c[\e^{3\over 2} ||d(1+|v|)^{-1}|| +
\e^3 |\nu ^{-1} d|_r + \e^{-1}|h_-|_r]\(5.41)$$
\endproclaim
\vskip 1cm
\underbar{Proof}:
Let us write Eq. \(4.4) in the form
$$ h = U_\e ({\bar \chi}_{\ga} K_* h) + U_\e Z +V_\e^+h_+
+V_\e^-h_-\(5.42)$$
where
$$Z= - \mu'_0 \e^2 F h -\e (\mu + \e F \mu') \s
(\bar g + g_2 ) +\e L^1 _* (h +\s (\bar g + g_2 ))
+\e^3 d \(5.43)$$
Since $|V^{\pm}_\e h_{\pm}|_r\le |h_-|_r$, we have by Proposition 6.1
$$\aligned |h|_r & \le c [|U_\e ( \nu ^{-1} {\bar \chi}_{\ga}
K_* h)|_r + |\nu ^{-1} Z|_r + |h_-|_r]\\ & \le c [{1 \over {1+ \ga}}|
K_* h)|_r + \e^2 F |h|_r + \e F (|\bar g|_r + |g_2 |_r) +\e F |h|_r
+ \e^3|\nu ^{-1} d|_r \\ & +|h_-|_r]\le c [ {1 \over {(1+
\ga)}}|h|_r + \e |g|_r +\e^3|\nu ^{-1} d|_r + |h_-|_r] \endaligned
\(5.44)$$
Choosing $\ga$ in such a way that ${1 \over {(1+ \ga)}} <
{1\over 3}$ we have, for $\e$ small enough,
$$|h|_r \le c[ \e |g|_r +\e^3 |\nu ^{-1} d|_r + |h_-|_r]\(5.45)$$
Finally, substituting the estimate for $g$ given by
Proposition 6.3 in \(5.45) and taking $\e$ small enough, we
complete the proof of Proposition 6.4.\qed
By using \(5.45) to estimate $|h|_r$ in \(5.22), the estimate
$||d(1+|v|)^{-1}|| \le c |{d/\nu}|_3$ and taking $\e$ small
enough we then get the proof of Proposition 4.1.
\vskip2cm
%
%
%\input appendixA.tex
%
\heading
Appendix A. \endheading
\vskip .5cm
\resetall
\redefine \firstpart {A}
To prove Proposition 2.1 we first consider Eqs.
(2.1)--(2.4) with fixed pressure $\Cal P$ instead of fixed total
mass, by eliminating the density $\rho$ from \(2.2) using
\(2.1). We then introduce the following sequence of approximate
solutions:
$$\align&{d\over dy}(\eta (T_{n-1}) {du_n \over dy} ) +
\Cal P F T_{n-1}^{-1} =0 \(A.1))\\&
{d\over dy}(\kappa (T_{n-1}) {dT_n \over dy}) + \eta (T_{n-1})
({du_n \over dy} )^2 =0\(A.2)\endalign$$
for $n\ge1$, $T_0$ being the value on the boundary and $u_0$=0.
The boundary conditions are given by \(2.4). By using induction, the
{\it maximum principle } implies that
$$T_n(y)>T_0 \(A.3)$$
for $y\in(-1,1)$. Hence
$$\eta(T_n)>\eta_0 \phantom{....} \text{ and }
\kappa(T_n)>\kappa_0\(A.4)$$
for $y\in(-1,1)$.
Multiplying \(A.1) by
$u_n$ and \(A.2) by $T_n$ and integrating
on $[-1,1]$ one gets, by
the lower bounds \(A.3) and \(A.4),
the uniform boundedness of $T_n$ and
$u_n$ in the Sobolev norm of order
$1$ on $[-1,1]$, hence the
compactness in $C([-1,1])$. Therefore the existence of solutions
follows by choosing subsequences.
The estimates \(2.6) follow using the
smoothness of $\eta$ and $\kappa$
and the lower bounds \(A.3) and \(A.4)
in the {\it explicit} form of the solution:
$$\align& u_n(y)= -\Cal P
F \int_{-1}^ydy' \eta_{n-1}^{-1}(y')\int_{-1}^{y'}dy''
T_{n-1}^{-1}(y'') + C_1\int_{-1}^ydy'
\eta_{n-1}^{-1}(y') \(A.5) \\&
T_n(y)=T_0 -\int_{-1}^ydy'
\kappa_{n-1}^{-1}(y')\int_{-1}^{y'}dy''
\eta_{n-1}(y'') \big[{du_n\over dy}(y'')\big]^2 \\&
\phantom{............}+C_2\int_{-1}^ydy'
\kappa_{n-1}^{-1}(y') \(A.6) \endalign$$
with $C_1$ and $C_2$ chosen so that the boundary
conditions are satisfied also at $y=1$.
$\eta_{n}$ and $\kappa_{n}$ denote here
$\eta(T_n)$ and $\kappa(T_n)$.
To show the uniqueness, let $(u^{(1)},
T^{(1)})$ and $(u^{(2)},
T^{(2)})$ be two solutions of \(2.2),
\(2.3) with the given pressure
$\Cal P$, the boundary conditions \(2.4),
the lower bounds \(A.3) and
\(A.4) and satisfying the estimates \(2.6).
Denoting by $w$ and $\si$ the
differences $u^{(1)}-u^{(2)}$ and $T^{(1)}-
T^{(2)}$ respectively,
using above properties it is easy to prove that
$$\int_{-1}^1\Big\{ [{dw \over dy}]^2 +
[{d\sigma\over dy}]^2\Big \}dy \le c F
\int_{-1}^1 \Big\{[{dw \over dy}]^2 +
[{d\sigma\over dy}]^2\Big\}dy \(A.7)$$
for a suitable constant $c$. This of
course implies the uniqueness
for small $F$. Finally we note that,
to give a solution of the
problem with fixed mass $m>0$, one
has to solve the equation in $\Cal
P$
$$\Cal P \int_{-1}^1dy\/ T_{\Cal P}^{-1} =m \(A.8)$$
where $T_{\Cal P}$ denotes the solution
with fixed pressure. Using
equations \(A.5) and \(A.6) it is not
difficult to prove the solvability
of this equation for $F$ small enough.
\vskip 2cm
%
%
%\input appendixB.tex
%
%
\heading {Appendix B}\endheading
\vskip .5cm
\resetall
\redefine \firstpart {B}
In this Appendix we prove the second part of Theorem 3.2.
Moreover we check the conditions on the sources terms in
equations \(3.6) and \(3.7) and on the boundary values that are
needed in proving Proposition 3.1.
First we state the following
\proclaim
{Lemma B.1} The following identity
holds for the Boltzmann collision
operator \(2.8)
$${\pa \over \pa v}Q(f,g)=Q (f,{\pa g\over \pa
v}) +Q (g,{\pa f \over \pa v})\(B.1)$$
where $\dsize{\pa \over \pa v}$ stands for the partial
derivative with respect to any of
the components of $v$.
\endproclaim
\underbar{Proof}: We observe that for any derivative, say
$\pa_{v_x}$,
$$\align {\pa \phantom{..} \over \pa v_x}&\int_{\Bbb R^3 \times
S} d v_* d \om (v_*-v)\cdot \om \ \chi ((v_*-v)\cdot \om \ge 0)
[f'g_* '+g'f_* '-fg_*-gf_*]=\\
&\int_{\Bbb R^3 \times S} d v_* d \om \chi ((v_*-v)\cdot \om
\ge 0)\big[{\pa \phantom{..} \over \pa v_x}+{\pa \phantom{...}
\over \pa v_{*x}}\big] [f'g_* '+g'f_* '-fg_*-gf_*] \(B.2)
\endalign$$ $S$ being the unit sphere, since the function
$(v_*-v)\cdot\om \chi ((v_*-v)\cdot\om \ge 0)$
depends on the difference $v_*-v$. In \(B.2) the notation is, as
usual, $f$, $f_*$, $f'$, $f'_*$ for $f(v)$, $f(v_*)$, $f(v')$ and
$f(v'_*)$ respectively.
Using the relations
$$v' =v + \om [\om \cdot (v_* -v)];\ v' _*
=v_* - \om [\om \cdot (v_* -v)]
\(B.3)$$
we get, for any function $h(v)$
$$\big[{\pa\phantom{..}\over \pa v_x}+{\pa \phantom{...} \over
\pa v_{*x}}\big] h' = ({\pa h\over \pa v_x})' \(B.4)$$
$$\big[{\pa\phantom{..}\over \pa v_x}+{\pa \phantom{...} \over
\pa v_{*x}}\big] h'_* =({\pa h\over\pa v_x})'_*\(B.5)$$
The result then follows by a straightforward calculation.\qed
\vskip .3cm
In order to prove part 2) of Theorem 3.2 we
differentiate \(3.31) to get, by Lemma B.1
$$v_y {\pa \phantom{.} \over \pa \tau}
\pa_v f =\Cal L_0 \pa_v f +2Q(f,
\pa_v M_0) + \pa_v s \(B.6)$$
where $\pa_v f$ stands for $\dsize{\pa f\phantom{..} \over \pa
v_x}$.
We claim that $\ti s =2Q(f, \pa_v M_0) + \pa_v s$ verifies the
condition \(3.37). In fact $\pa_v s$ satisfies \(3.37) by the
hypothesis \(3.40). On the other hand
$$\align \sup_{\tau\in \Bbb R^+} \sup_{v \in \Bbb R^3}&\/
e^{\sigma\/ \tau }(1+|v|)^{r+1} |M_0^{-{1\over2}}Q(f,\pa_v M_0)|
\le
\\ & \sup_{v \in \Bbb R^3} |(1+|v|)^r
M_0^{-{1\over2}}\pa_v M_0| \
\sup_{\tau\in \Bbb R^+} \sup_{v \in \Bbb R^3} e^{\sigma
\/\tau}(1+|v|)^r |M_0^{-{1\over2}}f| \(B.8) \endalign$$
The second factor in the r.h.s. of \(B.8) is bounded by
Theorem 3.2, part 1),
while the first factor is obviously bounded. Besides, $\pa_v
\vartheta (v)$ verifies \(3.36) by the hypothesis \(3.40).
Therefore we can apply the first part of Theorem 3.2 to \(B.6)
and conclude that $\pa_v f$ exists and satisfies \(3.41).
The proof of Theorem 3.2 is completed by the observation that the
result for the derivative of any order is achieved following the same
procedure, i.e. differentiating the equation for the derivative at
the preceding order and controlling by direct inspection
that the resulting equation has a source that satisfy condition
\(3.35) and \(3.37) so that it is possible to apply part 1) of
Theorem 3.2 to get the result.
The same argument applies to $\dsize{\pa f\over
\pa v_z}$ but not to $\dsize{\pa f\over \pa v_y}$ because in the
latter case an extra term $\pa_y f$ would arise.
\vskip .3cm
Now we verify that the source terms, that we denote by $s_n$,
in \(3.6) and \(3.7)
satisfy conditions \(3.35), \(3.37) and \(3.41) in Theorem 3.2.
The $s_n$'s are defined by
$$s_2 = \big [2Q(\Delta M ,b^- _1) + 2Q(B _1,b^- _1)
+ Q(b^-_1,b^-_1) + Q(b^-_1,b^+_1)\big ] \(B.9) $$
and for $3 \le n \le 6$
$$ s_n =- F{\pa \over \pa {v_x} }b^-_{n-2}+ 2Q(\Delta M,
b^-_{n-1}) + \sum \Sb i,j\ge 1\\i+j=n \endSb \big [2Q(B_i,b^- _j) +
Q(b^- _i,b^- _j) + Q(b^+ _i,b^- _j)\big ] \(B.10)$$
Property \(3.35) is true for any $n$ due to the property \(2.13) of
$Q$. Since in $s_2$ all the terms are of the form $Q(f,g)$, they can
be estimated using the Grad estimates \(2.47). For the first
term, for any $\s'<\s$
$$\align &\sup_{y'\in \Bbb R^+} \sup_{v \in \Bbb R^3} e^{\s' y'}
(1+|v|)^{r-1} |M_0 ^{-{1\over 2}} 2Q(\Delta M ,b^-_1)| \(B.11)\\&\le
c \sup_{y'\in \Bbb R^+} \sup_{v \in \Bbb R^3} |M_0^{-{1\over 2}}
\Delta M|(1+|v|)^{r}e^{-(\s-\s') y'}
\sup_{y'\in \Bbb R^+} \sup_{v \in \Bbb R^3} e^{\s y'} | M_0
^{-{1\over 2}} b^-_1|(1+|v|)^{r}| \\& \le {cF\over(\s-\s')}
\endalign$$
where the second inequality is obtained, as in \(3.48), by means of
the bound
$$\sup_{y'\in \Bbb R^+} \sup_{v \in \Bbb R^3} e^{\s y'}
(1+|v|)^{r} |M_0 ^{-{1\over 2}} b^-_1|