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Carlen\footnote{$^1$}{School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332--0160}, R. Esposito\footnote{$^2$}{Dipartimento di Matematica, Universit\`a di L'Aquila, Coppito---67100 L'Aquila---Italy}, J.L. Lebowitz\footnote{$^3$}{Department of Mathematics and Physics, Rutgers University, New Brunswick, NJ 08903}, R. Marra\footnote{$^4$}{Dipartimento di Fisica, Universita' di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy} and A. Rokhlenko{$^3$}} \bigskip \bigskip \bigskip {\baselineskip = 12pt\narrower{\noindent {\bf Abstract:}\/} We study, globaly in time, the velocity distribution $f(v,t)$ of a spatially homogeneous system that models a system of electrons in a weakly ionized plasma, subjected to a constant external electric field $E$. The density $f$ satisfies a Boltzmann type kinetic equation containing a full nonlinear electron-electron collision term as well as linear terms representing collisions with reservoir particles having a specified Maxwellian distribution. We show that when the constant in front of the nonlinear collision kernel, thought of as a scaling parameter, is sufficiently strong, then the $L^1$ distance between $f$ and a certain time dependent Maxwellian stays small uniformly in $t$. Moreover, the mean and variance of this time dependent Maxwellian satisfy a coupled set of nonlinear ODE's that constitute the ``hydrodynamical'' equations for this kinetic system. This remain true even when these ODE's have non-unique equilibria, thus proving the existence of multiple stabe stationary solutions for the full kinetic model. Our approach relies on scale independent estimates for the kinetic equation, and entropy production estimates. The novel aspects of this approach may be useful in other problems concerning the relation between the kinetic and hydrodynamic scales globably in time. \bigskip \bigskip \bigskip} \noindent{\bf 1. Introduction} \numsec= 1 \numfor= 1 \vskip.2cm The mathematical understanding of equilibrium phenomena has greatly advanced in the past few decades. The elegant and precise theory of Gibbs measures provides a direct bridge between the microscopic and macroscopic descriptions of such systems. This includes a general conceptual framework as well as nontrivial explicit examples of the coexistence of multiple equilibrium phases for certain values of the macroscopic control parameters. There is no comparable general theory for nonequilibrium systems and the microscopic study of phase transition phenomena in such situations appears to be far beyond our mathematical grasp at the present time. Our mathematical understanding of the great variety of nonequilibrium phase transitions observed in fluids, plasmas, lasers, etc., therefore depends entirely on the study of bifurcations and other singular phenomena occurring in the nonlinear equations describing the {\it macroscopic} time evolution of such systems. While there has been much progress recently in deriving such equations from simple microscopic and even realistic mesoscopic model evolutions the passage to the macroscopic scale is well understood {\it only over time intervals in which the solutions of the macroscopic equations stay smooth}. This is true for example in the passage from kinetic theory, where the evolution is described by the Boltzmann equation, to hydrodynamics, where it is described by either the compressible Euler or Navier-Stokes equations, depending on how we choose our macroscopic time scale [\rcite{N}, \rcite{Ca}, \rcite{U}, \rcite{DEL}, \rcite{ELM1}, \rcite{ELM2}]. These derivations, which are based on Chapman-Enskog type expansions, require for their validity the uniqueness and smoothness of the solutions of the hydrodynamic equations. The reason for this is that control of the error terms in the estimates depends on {\it a-priori} smoothness estimates for solutions of the macroscopic equations. Thus, they shed no light on the actual behavior of the mesoscopic description when the solution of the hydrodynamical equations develop singular behavior. To overcome this problem it is clearly desirable to develop methods in which one does not use any {\it a-priori} smoothness estimates for solutions of the macroscopic equations, but instead uses scale independent estimates on the mesoscopic equation. This is what we do here for a simple model inspired by plasma physics [\rcite{R}, \rcite{B}, \rcite{F}]. Our starting point is a kinetic theory description of the system. Grave difficulties are posed by the fact that as of yet, not very much is known in the way of {\it a-priori} regularity estimates for solutions of the spatially inhomogeneous Boltzmann equation. This is quite different however, from the lack of estimates for the macroscopic equations -- there it is clear that in the interesting cases the desired estimates just don't exist. Shock waves do form. In the Boltzmann case however, it is likely that {\it a-priori} regularity estimates in the velocity variables, say, invariant under the Euler scaling, are there, but simply have not yet been discovered. Still, the lack of such estimates is a grave difficulty in the way of rigorous investigation of the problem at hand. We sidestep this difficulty by considering a spatially homogeneous system, but one that is driven by an electric field, and coupled to heat reservoirs. In this case the usual hydrodynamic moments are not conserved and the system will have non-equilibrium stationary states. We prove then in a certain simplified, but still recognizable physical situations, that the kinetic description closely tracks the macroscopic description even when the driving is sufficiently strong for the latter to undergo phase transitions. More precisely, we show that the velocity distribution function is close to a Maxwellian parametrized by a temperature $T$ and mean velocity $u$ which satisfy certain non-linear equations, which are the macroscopic equations for this system. Moreover, it does so globally in time, even when the stationary solutions of these macroscopic equations are nonunique. We are in fact particularly concerned with the stability of of these stationary solutions -- the existence of multiple stationary states being analogous to the coexistence of phases in equilibrium systems. For such questions we need results that guarantee that a solution of the kinetic equations will stay near a solution of the macroscopic equations globaly in time. This seems to be difficult to accomplish by standard expansion methods, at least in the range of driving field strengths where the macroscopic equations have the most interesting behavior. Instead of expansion methods, we use entropy production [\rcite{CC1}, \rcite{CC2}] to show that the solution of the kinetic equations must stay close to {\it some} Maxwellian, globally in time. Then, we show that the moments of this Maxwellian must nearly satisfy the macroscopic equations. In this way we get our results. The next section specifies the model more closely, and states our main results. A preliminary account of this work in which the Boltzmann collisions were modeled by a BGK collision kernel was presented in [\rcite{CELMR}]. \bigskip \noindent{\bf 2. The model and the results} \bigskip \numsec= 2 \numfor= 1 \vskip.2cm Our formal set up is as follows: We consider a weakly ionized gas in ${\IR^3}$ in the presence of an externally imposed constant electric field $E$. The density of the gas, the degree of ionization and the strength of the field are assumed to be such that: (i) the interactions between the electrons can be described by some nonlinear, Boltzmann type collision operator, and (ii) collisions between the electrons and the heavy components of the plasma, ions and neutrals, are adequately described by assuming the latter ones to have a spatially homogeneous time independent Maxwellian distribution with an a proiri given temperature [\rcite{R}]. Under these conditions the time evolution of the spatially homogeneous velocity distribution function $f(v,t)$ will satisfy a Boltzmann type equation $$ {\partial f(v,t) \over \partial t} = -E \cdot \nabla f + Lf + \epsilon^{-1} Q(f),\Eq(2.1)$$ where $\nabla$ is the gradient with respect to $v$ in ${\IR^3}$, $E$ is a constant force field and $Q$ is a nonlinear collision term which will take either the form of the Boltzmann collision kernel for Maxwellian molecules, or the one corresponding to the BGK model. We treat both cases here because it is possible to provide a little more detail concerning the nature of the equilibria in the BGK case. The parameter $\epsilon> 0$ is thought of as a scaling parameter that goes to zero in the hydrodynamical limit. The linear operator $L$ represents the effect of collisions with reservoir particles. It will be assumed to have the form: $$Lf(v) = L_1f(v) +L_2f(v),\Eq(2.1.3)$$ with $$L_1f(v)=\nabla\cdot\biggl(D(v)M(v)\nabla\biggl({f(v)\over M(v)} \biggr)\biggr),\Eq(2.2)$$ a Fokker-Planck operator, representing energy exchanges with the reservoir assumed to be at temperature $T = 1$, so that $$M(v) = (2\pi )^{-3/2}\exp(-|v|^2/2)$$ and $$D(v) = a\exp(-b|v|^2/2) + c\Eq(2.3)$$ for some strictly positive constants $a$, $b$ and $c$: the symbols $a$, $b$, $c$ shall henceforth always refer to these parameters wherever they appear. The specific form \equ(2.3) of the velocity space diffusion coefficient is not important. We specify it for sake of concreteness. The properties we really need for $D(v)$ will be clear from the proofs. The operator $L_2$ represents momentum exchanges with the heavy reservoir particles and is given by $$L_2f(v)= \nu[\bar f(v) -f(v)],\Eq(2.3.3)$$ with $\nu$ a positive constant and $\bar f(v)$ the sphericalized average of $f(v)$. For any probability density $f$, we shall let $M_f$ denote the Maxwellian density with the same first and second moments as $f$. Explicitly, $$M_f= (2\pi T)^{-3/2} \exp[- {(v-u)^2\over 2T}],\Eq(2.3.5)$$ with $$u := \int_{\IR^3}vf(v)\d,\Eq(2.4)$$ $$e := {1 \over 2}\int_{\IR^3}v^2f(v)\d ,\Eq(2.5)$$ and $T = {2 \over 3} (e - {1 \over 2} u^2)$. In the BGK model the collision kernel is $$Q_{BGK}(f) = M_f - f\ .$$ The Boltzmann collision kernel is given by $$Q_B(f)(v)= \int_{\IR^3}d\/v_*\int_{S_2^+}d\/\o B(|v-v_*|,\o)\big[f(v')f(v'_*)-f(v)f(v_*)\big].\Eq(2.3.6)$$ Here $S_2^+=\{\o\in \IR^3: \o^2=1, \o\cdot (v-v_*)\ge 0\}$, and $$\eqalign { v'= &v-\o\cdot(v-v_*)\o,\cr v'_*=&v_*+\o\cdot (v-v_*)\o, }\Eq(2.3.61) $$ are the outgoing velocities in a collision with incoming velocities $v$ and $v_*$ and impact parameter $\o$, $B(|v-v_*|,\o)$ is the collision cross section, depending on the intermolecular interactions. For Maxwellian molecules, with a Grad angular cut-off [Gr], $$B(|v-v_*|,\o)= h(\theta),\Eq(2.3.7)$$ with $\theta$ the azimuthal angle of the spherical coordinates in $S_2$ with polar axis along $v-v_*$ and $h(\theta)$ a smooth non negative bounded function. Thus, for any normalized $f$ we can write $$Q_B(f)=\ell\bigl(f\circ f - \/\/f\bigr)\Eq(2.3.8)$$ with $$\eqalign { f\circ f(v)=&{1\over 2\ell}\int_{\IR^3}d\/v_*\int_{0}^{2\pi}d\/ \varphi\int_0^\pi d\/\theta\/ h(\theta)\/ |\sin\/\theta| f(v')f(v'_*),\cr \ell=& \pi \int_0^\pi d\/\theta\/ h(\theta)\/ |\sin \/\theta|>0, } \Eq(2.3.9)$$ With either $Q_{BGK}$ or $Q_B$ for the collision kernel in \equ(2.1), this term tends to keep $f$ close to $M_f$, and one could certainly expect this effect to dominate for small values of $\epsilon$. Thus, formally in the limit as $\epsilon$ vanishes, $f$ will actually equal $M_f$ for all time $t$, and to keep track of its evolution, we need only keep track of $u(t)$ and $e(t)$. Using the prescription $f= M_f$, in the right side of (2.1), one easily evaluates the time derivative of the first two moments of the so modified (2.1), to obtain formally $${{\rm d}\over {\rm d}t}\left(\matrix{\tilde u(t)\cr \tilde e(t)\cr}\right) = \left(\matrix{F(\tilde u(t), \tilde e(t))\cr G(\tilde u(t), \tilde e(t))\cr}\right),\Eq(2.5.3)$$ The functions $F$ and $G$ are given explicitly by $$ F(u,e) = E - u[\nu + c + a \exp(-w) {1+b \over (1+b/\beta)^{5/2}}], \Eq(2.5.5) $$ % $$ G(u,e) = Eu -c\Big[2e(1-\beta)+\beta u^2\Big] - {a \exp(-w) \over (1 + b/\beta)^{{5 \over 2}}} \Big[2e(1-\beta) + u^2(\beta - b {1-\beta \over b+\beta})\Big], \Eq(2.5.7) $$ where $\beta=T^{-1}$ and $w = bu^2/2(1+bT)$. The tildes in (2.14) are to remind us that the equation is valid only when $f = M_f$. Equations \equ(2.5.3) represent the hydrodynamical description of the gas. Our primary goal here is to show that such a description actually does hold for small, but positive, values of $\epsilon$, i.e. that the interaction between the hydrodynamic and the non--hydrodynamic modes does not destroy the picture involving only the hydrodynamic modes. The following theorem enables us to do this. \bigskip \noindent{\bf Theorem 2.1.} {\it Let $f$ be a solution of \equ(2.1) with $$f(\cdot,0) = M_{f(\cdot,0)}\ .$$ Then, for any fixed integer $k_0>0$, there is an $\e_0>0$ and functions $\delta_1(\e)$, $\delta_2(\e)$, going to zero as $\e\to 0$, depending only on $a$, $b$, $c$, $|E|$ and $e(0)$, such that for $\e<\e_0$ the solution of \equ(2.1) satisfies $$\sup_{t\in \IR^+}\|f(\cdot,t) - M_{f(\cdot,t)}\|_{L^1(\IR^3)} \le \delta_1(\e)\Eq((2.7)$$ and $$\sup_{t\in \IR^+}\|\nabla^k\bigl(f(\cdot,t) - M_{f(\cdot,t)}\bigr)\|_{L^2(\IR^3)} \le\delta_2(\e)\Eq((2.8)$$} for $k\le k_0$. \bigskip \noindent{\bf Remark:} The assumption on $f(\cdot,0)$ is not essential: our methods allow us to easily modify the result to take into account an initial layer. \bigskip We shall use Theorem 2.1, in the proof of Theorem 2.2 below, that if we compute $u(t)$ and $e(t)$ for a solution $f$ of \equ(2.1) satisfying the conditions of Theorem 2.1, then the moments of $f$ will satisfy the equation, $${{\rm d}\over {\rm d}t}\left(\matrix{u(t)\cr e(t)\cr}\right) = \left(\matrix{F(u(t), e(t))\cr G(u(t), e(t))\cr}\right) + \delta(\e)^{1/2} \left(\matrix{\gamma(t)\cr \eta(t)\cr}\right)\Eq(2.9)$$ where $F$ and $G$ are the non--linear functions of $u$ and $e$ given by \equ(2.5.5) and \equ(2.5.7), that arise in the $\e = 0$ limit, and $\gamma(t)$ and $\eta(t)$ are bounded uniformly in $t$ with a bound independent of $\e$. Of course they depend on the full solution $f$ of \equ(2.1), but such estimates and a simple comparison argument will then lead to the following theorem, which says that the system (2.14) obtained in the $\epsilon \to 0$ limit does give an accurate picture of the small $\epsilon$ regime. \bigskip \noindent{\bf Theorem 2.2 } {\it Let $(u^*,e^*)$ be a stable fixed point of the system \equ(2.5.3) and let $M_{(u^*,\e^*)}$ be the corresponding Maxwellian density, with moments $u^*$ and $e^*$. Then given any $\delta>0$, there is an $\epsilon$ greater than zero such that if $f_\e(\cdot,t)$ solves \equ(2.1) with this value of $\e$ and $$\|f_\epsilon(\cdot,0) - M_{(u^*,e^*)}(\cdot)\|_{L^1(\IR^3)} \le \epsilon\Eq(2.10)$$ then $$\|f_\epsilon(\cdot,t) - M_{(u^*,e^*)}(\cdot)\|_{L^1(\IR^3)} \le \delta\Eq(2.11)$$ for all $t\ge 0$. If, however, $(u^*,e^*)$ is not stable, then there exist a $\delta >0$ so that for every $\epsilon >0$, there is a solution $f_\epsilon(\cdot,t)$ of \equ(2.1) with Maxwellian initial data satisfying $$\|f_\epsilon(\cdot,0) - M_{(u^*,e^*)}(\cdot)\|_{L^1(\IR^3)} \le \epsilon\ ,\Eq(2.12)$$ but such that for some finite $t>0$ $$\|f_\epsilon(\cdot,t) - M_{(u^*,e^*)}(\cdot)\|_{L^1(\IR^3)} \ge \delta\ .\Eq(2.13)$$} \bigskip The proof of these theorems, which are fairly complicated even for the simple BGK model will be given in the next sections. The above theorems allow us to rigorously prove that our kinetic system has multiple equilibria in certain ranges of the parameters that specify it. This is because of the following result concerning the ``hydrodynamic'' system (2.14). \bigskip \noindent{\bf Proposition 2.3 } {\it There are choices of the parameters $a$, $b$, $c$ and $\nu$ for which there are nonempty intervals $(E_0,E_1)$ such that, if $|E|$ is outside of the closed interval $[E_0,E_1]$, then there is unique stable fixed point for the system \equ(2.5.3), while, if $|E|\in (E_0,E_1)$ then there are three fixed points for the system \equ(2.5.3). Moreover two of them are stable and one is unstable. }\bigskip Stability here is meant in the sense that the eigenvalues of the differential have a strictly negative real part. The proof of Proposition 2.3 is an explicit calculation which we omit (see [7]). However, to gain an intuitive understanding of why there should be multiple stable equilibria for certain parameter ranges, think of the $\e=0$ limit of \equ(2.1) as a constrained motion on the ``manifold of Maxwellians''. Without this constraint, which is generated by the collision kernel, the evolution would be the one ruled by the electric field and the linear operator $L$. Clearly, this evolution has an unique attracting fixed point, which, for $a>0$, is not Maxwellian. What happens is that there are one or more places on the {\it non-linear} constraint manifold that are {\it locally closest} to the attracting point of the unconstrained system. Each of these is a stable equilibrium for the constrained evolution. As the parameters are varied, the position of the unconstrained fixed point relative to the manifold of Maxwellians varies, and with this variation in geometry, the number of locally closest points varies. The main physical issues regarding this model are sttled at this point: We have proved the existence of the multiple stable equilibria at the kinetic level -- $\epsilon$ small, but positve -- that had been found and investigated in [7] at $\epsilon = 0$. Moreover, we remind the reader that we do not know how to establish such a result using conventional expansion methods: the difficulty being that if $E$ is not small, and hence possibly out of the range where multiple equilibria exist for (2.14), we only know how to prove (2.21) {\it locally} in time. This is insufficient to show that for $\epsilon$ small enough, one {\it never} wanders far from any $M_{(u^*,e^*)}$ with $(u^*,e^*)$ stable for (2.14). While the entropy methods we use do let us do this for arbitrary $E$, there are several finer question that one could ask, but for which we have only incomplete answers. First, one can ask whether or not there is an actual stationary solution inside the invariant neighborhoods that we have found of the $M_{(u^*,e^*)}$, and second, once one knows that stationary solutions exist, one can ask whether or not solutions actually tend to converge to one of these stationary solutions as $t$ tends to infinity. The first question we can answer comletely only in the BGK case. The positive answer is given by: \medskip \noindent{\bf Theorem 2.4} {\it Let $Q=Q_{BGK}$ and $a>0$. Then for each fixed point of \equ(2.5.3) there is exactly one stationary solution of eq. \equ(2.1). This solution lies in a suitably smalll neighborhood of $M_{u^*,e^*}$, the Maxwellian corresponding to $(u^*,e^*)$, and it inherits the stability properties of the hydrodynamical fixed point. } \medskip For the Boltzmann kernel we have only a partial result that is reported in Section 8. Also on the question of convergence we have only very partial results. These are reported in Section 9, where the difficulties are explained as well. But though it would be desirable to have a more complete resolution of these issues, they are not central to establishing that the kinetic systems does actually have the several stable regimes that one sees in the limiting ``hydrodynamic'' equations. The proofs are organized as follows. In Section 3 we prove moment bounds. Section 4 contains the proof of two ``interpolation inequalities''. The first of these will be used to obtain {\it a--priori} smoothness bounds in Section 5. The second will be used to transform the smoothness bounds of Section 5 into a lower bound on the variance of our density. The smoothness requires Sobolev estimates for the collision kernel, which are straightforward for $Q_{BGK}$, while for $Q_B$ they rely on some recent results [\rcite{CGT}], which we simply state here. Having assembled these moment, smoothness and interpolation bounds, we can use a key entropy production inequality for $Q_B$ proved in [\rcite{CC2}]. The analogous inequality for $Q_{BGK}$ is proven in a simpler way in Section 6. This is used to get quantitative bounds on the tendency of the collision operator to keep the density nearly Maxwellian. What we obtain directly is $L^1$ control on the difference between $f$ and $M_f$, but the smoothness bounds together with the interpolation bounds allows us to obtain control in stronger norms. Section 7 contains the proofs of the Theorems 2.1 and 2.2, which, given the lemmas, are quite short. Section 8 is devoted to the proof of existence of stationary solutions, In section 9 we discuss the tendancy toward these stationary solutions. \bigskip \noindent{\bf 3. Moment Bounds} \bigskip \numsec= 3 \numfor= 1 \vskip.2cm In this section we establish {\it a--priori} moment bounds for solutions of (2.1). In estimating the evolution of the moments, we shall use one set of methods to treat the effects of collisions, and another set to treat everything else. Thus it is natural to rewrite (2.1) as $${\partial\over\partial t}f(v,t) = {\cal L}f(v,t) + {1\over \epsilon} Q(F)(v,t)\Eq(3.1)$$ where ${\cal L}f = -E\cdot\nabla f + Lf$. We shall use the standard ``bracket notation'' notation for averages: $\langle \phi \rangle_t$ denotes $\int_{\IR^3}\phi(v)f(v,t){\rm d}^3v$ for any positive or integrable function $\phi$. Throughout this paper, $K$ will denote a computable constant that depends at most on the electric field $E$, the paramters $a$, $b$, $c$ and $\nu$ specified in \equ(2.3), and where indicated, also on the fourth moment of the initial distribution: $\langle |v|^4 \rangle_0$. The constant will, however, change from line to line. In these terms, the main result of this section is: \bigskip \noindent{\bf Theorem 3.1}\ {\it Let $f$ denote a solution to (2.1). Then for $Q=Q_B$ or $Q=Q_{BGK}$, we have $$\langle |v|^4 \rangle_t \le \langle |v|^4 \rangle_0 + K\ .$$} \bigskip Note that by Jensen's inequality, this imediately controls all lower order moments as well. The first step, however, is to directly conrol the second moments. \bigskip \noindent{\bf Lemma 3.2}\ {\it Let $f$ denote a solution to \equ(2.1). Then for both choices of the collision kernel $Q$, we have $$\langle |v|^2 \rangle_t \le \langle |v|^2 \rangle_0 + K\Eq(3.2)$$} \bigskip \noindent{\bf Proof:} Since $\langle |v|^2 \rangle_t$ is a collision invariant, and $\int v^2 L_2f{\rm d}^3v=0$, $$\eqalign{&{{\rm d}\over {\rm d}t}\langle |v|^2 \rangle_t = \int_{\IR^3}|v|^2{\cal L}f(v,t)\d\cr &+2E\cdot\langle v \rangle_t -2 \int_{\IR^3}D(v)M(v)v\cdot\nabla\biggl({f(v,t)\over M(v)}\biggr) {\rm d}^3v =\cr &2E\cdot\langle v \rangle_t + 6\langle D \rangle_t +2(v\cdot\nabla D)\langle (v\cdot\nabla D) \rangle_t -2\langle D|v|^2 \rangle_t\cr}$$ Now observe that $$(v\cdot\nabla D) \le 0\Eq(3.3)$$ and that $$c \le D \le (a+c)\Eq(3.4)$$ for all $v$. Finally, by Jensen's inequality, $$|\langle v \rangle_t| \le \bigl(\langle |v|^2 \rangle_t\bigr)^{1/2}\Eq(3.5)$$ These facts, combined with the previous calculation, yield the estimate $${{\rm d}\over {\rm d}t}\langle |v|^2 \rangle_t \le -2c\langle |v|^2 \rangle_t +6(a+c) +2|E|\bigl(\langle |v|^2 \rangle_t \bigr)^{1/2}$$ Straightforward estimation now leads to $${{\rm d}\over {\rm d}t}\langle |v|^2 \rangle_t \le -c\langle |v|^2 \rangle_t + K\ .$$ Then \equ(3.2) in turn follows from the fact that any solution of the differential inequality $\dot x(t) \le -cx(t) + K$ satisfies $x(t) \le x(0) + K/c$. \ $\square$ \bigskip We next parley these bounds into bounds on the fourth moments; i.e., $\vf$. Since $\vf$ is not a collision invariant, these depend on the particular collision kernel under consideration. \bigskip \noindent{\bf Lemma 3.3}\ {\it Let $Q=Q_{BGK}$. Then for any density $f$, $$\int_{\IR^3}|v|^4Q(f)\d \le (160/3)\bigl(\vt\bigr)^2 - \vf$$} \bigskip \noindent{\bf Proof:} By an easy calculation, $$\int_{\IR^3}|v -\langle v \rangle_t|^4M_f(v)\d =$$ $$ (5/3)\biggl(\int_{\IR^3}|v -\langle v \rangle_t|^2M_f(v)\d\biggr)^2 = (5/3)\langle|v -\langle v\rangle_t|^2 \rangle^2 _t$$ Next, note that $|v| \le |v-\langle v \rangle_t| + |\langle v \rangle_t|$, and thus, $|v|^4 \le 16(|v-\langle v \rangle_t|^4 + |\langle v \rangle_t|)^4$. Combining this and Jensen's inequality as in \equ(3.5) with the above, we have $$\int_{\IR^3}|v|^4M_f(v)\d \le (160/3)\bigl(\vt\bigr)^2\ .$$ The result now follows directly from the form of $Q(f)$.\quad $\square$ \bigskip % \noindent{\bf Lemma 3.4}\ {\it Let $Q=Q_{B}$. Then there are positive constants $c_1$ and $c_2$ such that for any density $f$, $$\int_{\IR^3}|v|^4Q(f)\d \le c_1\bigl(\vt\bigr)^2 - c_2\vf$$} \bigskip \noindent{\bf Proof:} The result follows once we prove that $$\int d^3 v |v|^4 f\circ f(v) \le (\ell - c_2)\vf + c_1\bigl(\vt\bigr)^2.$$ To check this, let $\pi_\o$ and $\pi^\perp_\o$ denote the projection in the direction $\o$ and the complementary projection respectively. Then we can write $v$ as $$v= \pi_\o v_* + \pi^\perp_o v'_*,$$ and $$|v|^2= |\pi_\o v_*|^2 + |\pi^\perp_o v'_*|^2.$$ Hence $$|v|^4 = |\pi_\o v_*|^4 + |\pi^\perp_o v'_*|^4 + 2|\pi_\o v_*|^2 |\pi^\perp_o v'_*|^2.$$ Averaging on $S_2$ we get $$\eqalign { &\int_{s_2} d\/\o B(|v-v_*|,\o)[|\pi_\o v_*|^4 +|\pi^\perp_o v'_*|^4 ] = |v|^4 \pi\int_0^\pi d\/\theta |\sin\/\theta |\/h(\theta) [\cos^4 \theta +\sin^4 \theta]\cr &= |v|^4\Big( \ell - 2 \pi\int_0^\pi d\/\theta |\sin\/\theta |\/h(\theta) \cos^2 \theta \sin^2 \theta\Big)\equiv |v|^4( \ell -c_2 ). } $$ \ $\square$ \bigskip \noindent{\bf Proof of Theorem 3.1:} Calculating as before, we have: $$\eqalign{&{{\rm d}\over {\rm d}t}\langle |v|^4 \rangle_t = \int_{\IR^3}|v|^4{\cal L}f(v,t)\d + {1\over \e} \int_{\IR^3}|v|^4Q(f)(v,t)\d=\cr &4E\cdot\langle |v|^2v \rangle_t -4\int_{\IR^3}D(v)M(v)|v|^2v\cdot\nabla\biggl({f(v,t)\over M(v)}\biggr) {\rm d}^3v + {1\over \e}\int_{\IR^3}|v|^4Q(f)\d=\cr &4E\cdot\langle |v|^2v \rangle_t + 12\langle |v|^2D \rangle_t +4(|v|^2v\cdot\nabla D)\langle (v\cdot\nabla D) \rangle_t -2\langle D|v|^4 \rangle_t + {1\over\epsilon}\int_{\IR^3}|v|^4Q(f)\d\cr\ .}$$ We now estimate using \equ(3.3) and \equ(3.4) just as in the proof of Lemma 3.2, together with Lemmas 3.3 and 3.4, to control the collision term. (Note that once again $L_2$ makes no contribution.) The result is $$\eqalign{&{{\rm d}\over {\rm d}t}\langle |v|^4 \rangle_t \le\cr &4|E|\bigl(\vf\bigr)^{3/4} + 12(a+c)\vt - 4c\vf +{1\over \epsilon} \bigl(c_1\bigl(\vt\bigr)^2 - c_2\vf\bigr)\ .\cr}$$ Next, by Lemma 3.1 together with Jensen's inequality, we know that $12(a+c)\vt$ is bounded above by a universal constant plus $\bigl(\langle |v|^4\rangle_0\bigr)^{1/2}$. Thus, if we introduce the scaled time parameter $$\tau = (1/\epsilon)t\ ,$$ and define $x(\tau) := \langle |v|^4\rangle_\tau$, we have that $x(\tau)$ satisfies a differential inequality of the form $$\dot x(\tau) \le -x(\tau) + K$$ from which the result follows.\quad$\square$ %\vfill\break \bigskip \noindent{\bf 4. Interpolation inequalities} \bigskip \numsec= 4 \numfor= 1 \bigskip The lemmas in this section are several interpolation inequalities related to the familiar Gagliardo--Nirenberg inequalites, but with some special features adapted to our applications. The inequality of Lemma 4.2 is the most novel and interesting of these. \bigskip \noindent{\bf Lemma 4.1}\ {\it Let $f\in L^1(\IR^3)$. Then there is a universal constant $C$ such that if $f$ has a square integrable distributional Laplacean, then $f$ has a square integrable gradient, and $$\|\nabla f\|_2 \le C\|f\|_1^{2/7}\|\Delta f\|_2^{5/7}\Eq(4.1)$$ Similarly, there is a universal constant $C$ such that if $\Delta^2f$ is square integrable, then $\Delta f$ is also square integrable, and $$\|\Delta f\|_2 \le C\|f\|_1^{4/11}\|\Delta^2 f\|_2^{7/11}\Eq(4.2)$$} \bigskip \noindent{\bf Proof:}\ Taking Fourier transforms, we have $$\eqalign{&\|\nabla f\|_2^2 = \int_{\IR^3}|p|^2|\hat f(p)|^2{\rm d}^3p =\cr &\int_{|p|\le R}|p|^2|\hat f(p)|^2{\rm d}^3p + \int_{|p|\ge R}|p|^2|\hat f(p)|^2{\rm d}^3p\le \cr &CR^5\|f\|_1^2 + R^{-2}\int_{|p|\ge R}|p|^2|\hat f(p)|^4{\rm d}^3p \le\cr &CR^5\|f\|^2_1 + R^{-2}\|\Delta f\|_2^2\cr}$$ where the computable constant $C$ changes from line to line. Optimizing over $R$ now yields \equ(4.1). The proof of \equ(4.2) is done in the same way. \quad $\square$ \bigskip The next inequality is similar in effect to an ``uncertainty principle''. We shall use it to obtain uniform lower bounds on the variance of our density $f$. \bigskip \noindent{\bf Lemma 4.2}\ {\it Let $f\in L^1(\IR^3)$, and suppose that $f$ has a square integrable distributional gradient. Then there is a universal constant $C$ such that $$\int_{\IR^3}f(v)\d \le C\|\nabla f\|_2^{4/9}\biggl(\int_{\IR^3} \biggl|v - \int_{\IR^3}vf(v)\d\biggr|^2f(v)\d\biggr)^{5/9}\Eq(4.3)$$} \bigskip \noindent{\bf Proof:}\ The right side of \equ(4.3) is decreased when we replace $f$ by its spherically symmetric decreasing rearrangement, while the left side is unchanged. We may therefore assume without loss of generality that $f$ is spherically symmetric and radialy decreasing. Now fix $R>0$, and define $g(v) := f(v)$ for $|v|\le R$, and $g(v) = 0$ otherwise. Define $h$ by $f= g + h$. Now clearly, $$\int_{\IR^3}h(v)\d \le R^{-2}\int_{|v|>R}|v|^2f(v)\d\ .$$ Also, $$\eqalign{&\int_{\IR^3}g(v)\d =\cr &(4\pi/3)R^3f(R) + (4\pi/3)R^3\biggl((4\pi/3)^{-1}R^{-3}\int_{|v|0$. Similarly, suppose that $\|\Delta f(\cdot,0)\|_2$ is finite. Then there is a constant $K$ such that $$\|\Delta f(\cdot,t)\|_2 \le K\bigl( 1 + \|\Delta f(\cdot,0)\|_2\bigr)\Eq(5.2)$$ for all $t>0$. If $Q=Q_B$ the same results hold provided $\|f(\cdot,0) - M_{f(\cdot,0)}\|_1$ and $\epsilon$ are both sufficiently small. } \bigskip {\bf Remark:} The smallness condition on $\|f(\cdot,0) - M_{f(\cdot,0)}\|_1$ poses no problem here since we are avoiding an initial layer by assuming that $f(\cdot,0) = M_{f(\cdot,0)}$. However, it seems likely that it would be strightforward to include an inital layer analysis, and to dispense with this condition -- even the present proof does not require $\|f(\cdot,0) - M_{f(\cdot,0)}\|_1$ to be particularly small. \noindent{\bf Proof:} Once again, we write (2.1) in the form (3.1). Then differentiating, and integrating by parts, we have $${{\rm d}\over {\rm d}t}\|\nabla f(\cdot,t)\|^2 = -2\int_{\IR^3} \Delta f(v,t) \biggl({\partial f(v,t)\over \partial t}\biggr)\d =$$ $$ -2\int_{\IR^3}\Delta f(v,t){\cal L}f(v,t)\d -2\int_{\IR^3}\Delta f(v,t)Q(f)(v,t)\d\ .$$ As with the moment bounds, we begin by estimating the individual contributions to the ${\cal L}$ term. \bigskip \noindent{\bf Lemma 5.2}\ {\it There is a universal constant $K$ so that for all solutions $f$ of (2.1), $$ -2\int_{\IR^3}\Delta f(v,t){\cal L}f(v,t)\d \le K$$ for all $t \ge 0$} \bigskip First, $2E\cdot\int_{\IR^3}\Delta f\nabla f\d = 0$. Next, $$-2 \int_{\IR^3}\Delta f\bigl(\nabla D\cdot \nabla f)\d = \int_{\IR^3}|\nabla f|^2(\Delta D|\d \le (a+c)\|\nabla f\|_2^2$$ Now, repeatedly integrating by parts: $$\eqalign{&-2\int_{\IR^3}\Delta f\bigl(\nabla D\cdot v\bigr)f\d =\cr &2\int_{\IR^3}\nabla f\bigl(v(\Delta D)f +3 \nabla D f + (v\cdot \nabla D) \nabla f\bigr)\d \le \cr &-3\int_{\IR^3}f^2(\Delta D)\d -\int_{\IR^3}f^2\bigl(v\cdot \nabla(\Delta D) \bigr)\d\cr}$$ where we have once again used \equ(3.3). This gives us a bound of the form $$-2\int_{\IR^3}\Delta f\bigl(\nabla D\cdot v\bigr)f\d \le K\|f\|_2^2\ .$$ We now use a standard interpolation inequality, the Nash inequality: $$\|f\|_2 \le C\|\nabla f\|_2^{3/5}\|f\|_1^{2/5}\ .$$ This allows us to eliminate $\|f\|_2$ in favor of $\|\nabla f\|_2$, the quantity of interest, and $\|f\|_1$, the conserved quantity. $$-2\int_{\IR^3}\Delta f\bigl(\nabla D\cdot v\bigr)f\d \le K \|\nabla f\|_2^{6/5}\|f\|_1^{4/5}\ .$$ Next: $$-2\int_{\IR^3}\Delta f\bigl(Dv\cdot \nabla f\bigr)\d = \int_{\IR^3}|\nabla f|^2\nabla\cdot(Dv)\d \le K\|\nabla f\|_2^2$$ Apart from the collision term and the favorable dissipation term, the only other term to be bounded is $-2\int_{\IR^3}\Delta fDf$. However, integrating by parts, we obtain terms identical to terms that we have already bounded. The dissipation term is bounded using Lemma 4.1 as follows: $$\eqalign{-&2\int_{\IR^3}D(\Delta f)^2\d \le -2c\|\Delta f\|_2^2\le\cr &K\|\nabla f\|_2^{14/5}\|f\|_1^{-4/5} = K\|\nabla f\|_2^{14/5}\cr}$$ since $\|f\|_1 = 1$ for all times $t$. Thus, with $x(\tau) := \|f(\cdot,\tau)\|_2^2$, we have established $$-2\int_{\IR^3}\Delta f(v,t){\cal L}f(v,t)\d \le K(-x(t)^{7/5} + K\bigl(x(t)^{3/5} + x(t)^{3/10} + x(t)\bigr)\bigr) \le K$$ since the largest power if $x(t)$ has a negative coefficient. \ $\square$ \bigskip To conclude the proof of \equ(5.1), we need smoothness bounds for the collision kernel; in particular, we need an estimate on the smoothness of the gain term in the collision kernel. For the BGK case, this is given by the following: \bigskip \noindent{\bf Lemma 5.3:}\ {\it For any positive integer $n$ there is a constant $K$ depending only on the second moment of $f$, such that $$ 2\int_{\IR^3}(-\Delta)^{n} f(v,t)Q(f)(v,t)\d \le K - \|(-\Delta)^{n/2} f\|_2^2\Eq(5.3)$$} \bigskip \noindent {\bf Proof:} We only check the case $n=1$; the rest are similar. We have $$\eqalign{-&2\int_{\IR^3}\Delta fQ_{BGK}(f)\d = \cr -&2\int_{\IR^3}(\Delta M_f)f\d - 2\|\nabla f\|_2^2\ . }\Eq(5.3.3)$$ Now, since $$-2\int_{\IR^3}(\Delta M_f)f\d\le C T_f^{-1}\|\n f\|_2,$$ we can use Lemma 4.2 to get the bound $$ T_f^{-1}\le C \|\n f\|_2^{4/5}\Eq(5.3.5)$$ and hence $$-2\int_{\IR^3}(\Delta M_f)f\d\le C T_f^{-1}\|\n f\|_2\le C\|\n f\|_2^{9/5}\le K+ \|\n f\|_2^2.$$ Combining this with \equ(5.3.3) we get the Lemma 5.3 for $n=1$\ $\square$ \bigskip The analog of Lemma 5.3 for $Q_B$ is slightly more complicated. \bigskip \noindent{\bf Lemma 5.4:}\ {\it For any positive integer $n$ there are constants $K$ and $\delta$ depending only on the {\it a-priori} bound on the second moments of $f$, such that $$ 2\int_{\IR^3}(-\Delta)^{n} f(v,t)Q_B(f)(v,t)\d \le K - (\ell/2) \|{(-\Delta)^{n/2}} f\|_2^2\Eq(5.4)$$ whenever $$\|f(\cdot,t) - M_{f(\cdot,t)}(\cdot)\|_1 \le \delta\ .$$ where $\ell$ is the constant in \equ(2.3.9).} \bigskip \noindent {\bf Proof:} It is sufficient to note that $$\eqalign{&2\int_{\IR^3}(-\Delta)^{n} fQ_B(f)\d = \cr &2\ell\int_{\IR^3}f(-\Delta)^n f\circ f\d - 2\ell\|(-\Delta)^{n/2}f\|_2^2\le\cr & 2\ell\bigl(\|(-\Delta)^{n/2}f\|_2\|(-\Delta)^{n/2}f\circ f\|_2 - \|(-\Delta)^{n/2}f\|_2^2\cr}$$ What we need now is control over $\|(-\Delta)^{n/2}f\circ f\|_2$. This is provided by an inequality from [\rcite{CGT}], where it is shown that for any $\gamma>0$ there are constant $K$ and $\delta >0$ such that $$\|(-\Delta)^{n/2}f\circ f\|_2^2 \le K + \gamma \|(-\Delta)^{n/2} f\|_2^2.\Eq(5.6)$$ whenver $$\|f - M_f\|_1 \le \delta\ .$$ This inequality is proved in [\rcite{CGT}] assuming that $f$ has zero mean and unit variance, and under these conditions, the constants $K$ and $\delta$ are universal. Scaling the inequality, we have it holding with constants $K$ and $\delta$ depending only on the second moments of $f$. (The inequality is applied in [\rcite{CGT}] to get stong exponential convergence estimates for the spatially homogeneous Boltlzmann equation with physically realistic constants in the bounds.) With this inequality, we need only take $\gamma = 1/2$. \ $\square$ \bigskip \noindent{\bf Proof of Theorem 5.1:} We begin with the proof of \equ(5.1). To put all of the lemmas together, let $\tau := (1/\epsilon)t$ as before, and put $x(\tau) := \|f(\cdot,\tau)\|_2^2$. Combining Lemmas 5.2 and 5.3, we get in the BGK case that $$\dot x(\tau) \le \epsilon K + K - x(\tau)$$ and any solution of this differential inequality satisfies $x(\tau) \le x(0) + (1+ \epsilon)K$ for all $\tau$. This establishes \equ(5.1) in the BGK case. The proof of \equ(5.2) in the BGK case is entirely analogous. To handle the Maxwellian collision kernel case, first define $$\bar t= \inf \{t \hbox{ such that } \gamma \|f(\cdot,t)-M_{f(\cdot,t)}\|_1^2\ \ge\ \delta/2 \} \Eq(5.7)$$ where $\delta$ is the universal constant from Lemma 5.4. We take our initial condition small enough that $\bar t > 0$. Then, for all $t \le \bar t$, we have the following differential ineqaulity by combining Lemmas 5.2 and 5.4: $$\dot x(\tau) \le \epsilon K + K - \ell x(\tau)$$ and any solution of this differential inequality satisfies $x(\tau) \le x(0) + (1+ \epsilon)K/\ell$ for all $\tau \le (1/\epsilon) \bar t$. It only remains to show that actually $\bar t = +\infty$. We shall do this in the next section using an entropy inequality. The entropy inequlity requires the {\it a-priori} smoothness bounds from Theorem 5.1, and shows that a long as they hold; i.e., as long as $t \le \bar t$, we have an upper bound of the form $$\|f(\cdot,t)-M_{f(\cdot,t)}\|_1 \le \delta_{\rm entropy}(\epsilon)\ . $$ All we have to do now is to take $\epsilon$ so small that $$\delta_{\rm entropy}(\epsilon) \le \delta/4$$ and then it is clear from the definition of $\bar t$ that $\bar t = \infty$. Thus, borrowing the entropy bound from the next section, \equ(5.1) is established for $Q_B$ for all $\epsilon$ sufficiently small, and all initial data sufficently close to a Maxwellian. Again, \equ(5.2) for $Q_B$ is handled in an entirely analogous way. \ $\square$ \bigskip \noindent{\bf 6. Entropy bounds} \bigskip \numsec= 6 \numfor= 1 \bigskip Let $h(\rho_1|\rho_2)$ denote the relative entropy of two probability densities $\rho_1$ and $\rho_2$ on $\IR^3$: $$h(\rho_1|\rho_2) = \int_{\IR^3} \rho_1 \biggl({\rho_1\over \rho_2} \biggr)\ln \biggl({\rho_1\over \rho_2}\biggr)\d\ .$$ Here we are primarily interested in bounds on $h(f|M_f)$. It will be convenient however, to first obtain bounds on $h(f|M)$, and to then relate the two relative entropies. We do this in the next two lemmas. \bigskip \noindent{\bf Lemma 6.1} {\it Let $f$ be any solution of \equ(2.1) with $$\langle |v|^4\rangle_0 \le C \qquad{\rm and}\qquad \|\nabla f\|_2 < C$$ for all $t\ge 0$. Then there is a constant $K$ depending only on $C$ such that $${{\rm d}\over {\rm d}t}h(f(\cdot,t)|M) \le K + {1\over \epsilon} \int_{\IR^3}f(\cdot,t)Q(\cdot, t))\d\ .\Eq(6.1)$$} \bigskip \noindent{\bf Proof:} Differentiating, we have $$\eqalign{&{{\rm d}\over {\rm d}t}h(f(\cdot,t)|M) =\cr &\int_{\IR^3}{\partial f\over \partial t}\bigl(\ln f - \ln M\bigr)\d = \cr &\int_{\IR^3}\bigl(-E\cdot\nabla f + Lf + \epsilon^{-1}Q(f)\bigr) \bigl(\ln f - \ln M\bigr)\d\cr}$$ Now, integration by parts reveals that $$\int_{\IR^3}Lf\bigl(\ln f - \ln M\bigr)\d \le 0$$ and since $\ln M$ is linear in $|v|^2$, $$-\int_{\IR^3}\ln M Q(f) = 0\ .$$ Finally, $$\eqalign{&\int_{\IR^3}\bigl(E\cdot\nabla f\bigr)\bigl(\ln f - \ln M\bigr)\d =\cr &\int_{\IR^3}\bigl(E\cdot\nabla f\bigr)\ln f\d - \int_{\IR^3}\bigl(E\cdot\nabla f\bigr)\ln M\d =\cr &E\cdot\langle v\rangle_t\cr}$$ Using Jensen's inequality to bound $|\langle v\rangle_t|$ in terms of the uniformly bounded $\vt$, we have the assreted result.\quad $\square$ \bigskip \noindent{\bf Lemma 6.2} {\it Let $f$ be any solution of (2.1) with $$\langle |v|^4\rangle_0 \le C \qquad{\rm and}\qquad \|\nabla f\|_2 < C$$ for all $t \le T$, some $T>0$. Then there is a constant $K$ depending only on $C$ so that such that $${{\rm d}\over {\rm d}t}h(f(\cdot,t)|M_{f(\cdot,t)}) \le K + {1\over \epsilon} \int_{\IR^3}f(\cdot,t)Q(\cdot, t))\d\ .\Eq(6.2)$$ for all $t \le T$.} \bigskip \noindent{\bf Proof:} By the definitions, we have $$\eqalign{& h(f(\cdot,t)|M_{f(\cdot,t)}) - h(f(\cdot,t)|M) =\cr &\langle \ln M\rangle_t - \langle \ln M_{f(\cdot,t)}\rangle_t = \cr &(3/2)\ln\langle |v-\langle v\rangle_t|^2\rangle_t + (3/2) - (1/2)\vt\cr}$$ Thus, $$\eqalign{&{{\rm d}\over {\rm d}t}h(f(\cdot,t)|M_{f(\cdot,t)}) =\cr &{{\rm d}\over {\rm d}t}h(f(\cdot,t)|M) +\cr &(3/2){{\rm d}\over {\rm d}t}\ln \langle |v-\langle v\rangle_t|^2 \rangle_t - (1/2){{\rm d}\over {\rm d}t}\vf\cr}$$ Computing further with the logarithmic derivative term, $$\eqalign{&{{\rm d}\over {\rm d}t}\ln\langle |v-\langle v\rangle_t|^2 \rangle_t =\cr &\bigl(\langle |v-\langle v\rangle_t|^2\rangle_t\bigr)^{-1} {{\rm d}\over {\rm d}t}\bigl(\langle |v|^2\rangle_t -\langle v\rangle_t^2\bigr)\cr}$$ To control this term, we need an upper bound on $\bigl(\langle |v-\langle v\rangle_t|^2\rangle_t\bigr)^{-1}$ But since $\langle |v-\langle v\rangle_t|^2\rangle_t\ge C\|\nabla f\|_2^{-4/5} \ge K$ by Lemma 4.2, we have on application of Lemma 5.1 that $${{\rm d}\over {\rm d}t}\ln\langle |v-\langle v\rangle_t|^2\rangle_t \le K{{\rm d}\over {\rm d}t}\bigl(\langle |v|^2\rangle_t -\langle v\rangle_t^2\bigr)$$ The lemmas of Section 3 provide uniform bounds on this last term. and therefore provide uniform bounds on the derivative of $h(f(\cdot,t)|M_{f(\cdot,t)}) - h(f(\cdot,t)|M)$. The result now follows from the previous Lemma.\quad$\square$ \bigskip \noindent{\bf Lemma 6.3}\ {\it Let $Q=Q_{BGK}$. For any density $f$ with finite second moments, $$\int_{\IR^3}\ln fQ(f) \le -h(f|M_f)\Eq(6.3)$$} \bigskip \noindent{\bf Proof:} Let $S(f) = -\int_{\IR^3}f\ln f\d$ denote the entropy of $f$. For any density $f$ and for any $\tau\in [0,1]$, put $f(\tau) =(1-\tau)f +\tau M_f$. We have $${{\rm d}\over {\rm d}\tau}f(\tau)=M_f-f= Q(f)$$ and $f(0) = f$. Then $$\int_{\IR^3}\ln f Q(f)\d = -{{\rm d}\over {\rm d}\tau }S(f(\tau)).$$ However, the entropy functional is concave, so that $$S(f(\tau)) \ge (1-\tau)S(f) + \tau S(M_f)$$ and hence $${S(f(\tau))-S(f(0))\over \tau} \ge S(M_f) - S(f).$$ Since $M_f$ and $f$ share the same hydrodynamic moments, $S(M_f)- S(f) = h(f|M_f)$\quad$\square$ \bigskip \noindent{\bf Lemma 6.4} {\it Let $Q=Q_{BGK}$ and $f$ be any solution of \equ(2.1) with $$\langle |v|^4\rangle_0 \le C \qquad{\rm and}\qquad \|\nabla f\|_2 < C$$ for all $t$ Then there is a constant $K$ depending only on $C$ so that $${{\rm d}\over {\rm d}t}h(f(\cdot,t)|M_{f(\cdot,t)}) \le K - {1\over \epsilon} h(f(\cdot,t)|M_{f(\cdot,t)})\ .\Eq(6.4)$$} \bigskip \noindent{\bf Proof:} This follows immediately upon combining the last three lemmas.\quad$\square$ In the case of the Boltzmann collision kernel, Lemma 6.3 is replaced by the following proposition proved in [\rcite{CC2}]: \bigskip \noindent{\bf Proposition 6.5} {\it For all $C>0$, there is a positive function $\Phi_C(r)$ strictly increasing in $r$, such that for all densities $f$ with $$\int_{\IR^3}|v|^4f(v)\d \le C\qquad{\rm and}\qquad \|\nabla f\|_2 \le C$$ $$\int_{\IR^3}\ln fQ_B(f) \le -\Phi_C[h(f|M_f)].\Eq(6.5)$$} Consequently, for $Q=Q_B$, \equ(6.4) is replaced by $${{\rm d}\over {\rm d}t}h(f(\cdot,t)|M_{f(\cdot,t)}) \le K - {1\over \epsilon}\Phi_C[ h(f(\cdot,t)|M_{f(\cdot,t)})]\ .\Eq(6.6)$$ Now if $f(\cdot,0) = M_{f(\cdot,0)}$, so that $h(f(\cdot,0)|M_{f(\cdot, 0)} = 0$, then it is evident from this differential inequality that $$\Phi_C[ h(f(\cdot,t)|M_{f(\cdot,t)})] \le \epsilon K$$ for all $t\ge 0$. This together with Kullback's inequality $$\|f(\cdot,t) -M_{f(\cdot,t)} \|_1^2 \le 2h(f(\cdot,t)|M_{f(\cdot,t)}) \Eq(6.8)$$ clearly implies that there is a function $\delta_{\rm entropy}(\epsilon)$, decreasing to zero with $\epsilon$ so that $$\|f(\cdot,t) -M_{f(\cdot,t)} \|_1 \le \delta_{\rm entropy}(\epsilon) $$ for all $t \le \bar t$ --the times for which we know that $f(\cdot, t)$ will satisfy the bounds in the hypothesis of Proposition 6.5. But as explained at the end of the proof of Theorem 5.1, this is enough to show that $\bar t = \infty$ for all sufficiently small $\epsilon$. Thus, the entropy bound provides the information needed to complete the proof of Theorem 5.1 as claimed, and moreover, \equ(5.7) hold globaly in time. \bigskip \noindent{\bf 7. Proof of Theorems 2.1 and 2.2} \bigskip \numsec= 7 \numfor= 1 \bigskip \noindent{\bf Proof of Theorem 2.1}\ Let us consider first the case $Q=Q_{BGK}$. Since Maxwellian initial data satisfies the hypotheses of Lemma 6.4, we have that \equ(6.4) holds for the solution $f(\cdot,t)$ of \equ(2.1) under consideration in Theorem 2.1. But then $${{\rm d}\over {\rm d}t}\bigl(e^{t/\epsilon}h(f(\cdot,t)|M_{f(\cdot,t)})\bigr) \le K e^{t/\epsilon}\ .$$ Since by hypothesis, $h(f(\cdot,0)|M_{f(\cdot,0)}) = 0$, we have $$h(f(\cdot,t)|M_{f(\cdot,t)}) \le \epsilon K$$ for all $t\ge 0$. This together with Kullback's inequality \equ(6.8) yields the first inequality asserted in Theorem 2.1, with $\delta_1(\e)=\e^{1/2}$. To obtain the second, note that since $f(\cdot,0)$ is Maxwellian, there is a bound on $\|\Delta f\|_2$ depending only on $\langle |v|^2\rangle_0$. Inequality \equ(5.2) of Lemma 5.1 now fives us a uniform bound on $\|\Delta f(\cdot,t)\|_2$. Combining this with the interpolation inequality \equ(4.1) finally yields the second inequality of Theroem 2.1 with $\delta_2(\e)=\e^{1/7}$ for $Q=Q_{BGK}$. The proof for $Q=Q_B$ is only slightly more involved, but in fact we have already given the proof of the first part of Theorem 2.1 in our ``back and forth'' proof of the smoothness bounds and entropy bounds for this case. As observed at the end of Section 6, $$\|f(\cdot,t) -M_{f(\cdot,t)} \|_1 \le \delta_{\rm entropy}(\epsilon) $$ for all $t\ge 0$, and $\delta_{\rm entropy}(\epsilon)$ does decrease to 0 with $\epsilon$ as required. The second part follows in an entriely similar way.\ $\square$ \bigskip \noindent{\bf Proof of Theorem 2.2}\ This will follow from Theorem 2.1, and it is now no longer necessary to separate the cases $Q = Q_{BGK}$ and $Q = Q_B$. As we have computed in the proof of Lemma 3.1, $$\eqalign{\phantom{.}&{{\rm d}\over {\rm d}t}\langle |v|^2 \rangle_t =\cr \phantom{.}&-2E\cdot\langle v \rangle_t + 6\langle D \rangle_t +2(v\cdot\nabla D)\langle (v\cdot\nabla D) \rangle_t -2\langle D|v|^2 \rangle_t\cr}$$ If we replace the density $f(\cdot,t)$ everywhere on the right by $M_{f(\cdot,t)}$, by definition we obtain the function $G(u(t),e(t))$ where $u(t)$ and $e(t)$ are the moments of $f(\cdot,t)$ figuring in Theorem 2.2. The error we make has to be estimated term by term. The least trivial of these terms concerns the contribution from $\langle D|v|^2 \rangle_t$, and is estimated as follows: $$|\langle D|v|^2 \rangle_t - \int_{\IR^3}D(v)|v|^2M_{f(\cdot,t)}(v)\d | \le$$ $$\int_{\IR^3}D(v)|v|^2|f(v,t) - M_{f(\cdot,t)}(v)|\d\ \le$$ $$\int_{|v|\le R}D(v)|v|^2|f(v,t) - M_{f(\cdot,t)}(v)|\d\ +$$ $$\int_{|v|\ge R}D(v)|v|^2|f(v,t) - M_{f(\cdot,t)}(v)|\d\ \le$$ $$C\biggl(R^2\|f(\cdot,t) - M_{f(\cdot,t)}\|_1 + R^{-2} \int_{\IR^3}|v|^4|f(v,t) - M_{f(\cdot,t)}(v)|\d\biggr)\le$$ $$C\bigl(R^2\|f(\cdot,t) - M_{f(\cdot,t)}\|_1 + R^{-2}K\bigl(\vt + 1\bigr)\bigr)$$ Optimizing over $R$ now yields the result $$|\langle D|v|^2 \rangle_t - \int_{\IR^3}D(v)|v|^2M_{f(\cdot,t)}(v) \d |\le K\|f(\cdot,t) - M_{f(\cdot,t)}\|_1^{1/2}$$ Theorem 2.1 yields a bound of the size $\delta_1(\e)^{1/2}$. The other error terms in the time derivatives of $\vt$ and $\langle v\rangle_t$ are bounded by a direct application of Theorem 2.1 (and hence yield errors of order $\delta_1(\e)$ instead of $\delta_1(\e)^{1/2}$). \quad$\square$ \bigskip \noindent{\bf 8. Stationary solutions} \bigskip \numsec= 8 \numfor= 1 \bigskip In this section we discuss stationary solutions of \equ(2.1), and prove Theorem 2.4. The stationary solutions of \equ(2.1) are the positive normalized solutions of $$\e{\cal L} f+ Q(f)=0. \Eq(8.1)$$ The existence of such solutions relies on a simple fixed point argument. Recalling the expression of $Q(f)$, we put $$J(f)=\cases{ M_f \quad \hbox{\rm in the BGK case },\cr f\circ f \quad\hbox{\rm in the Boltzmann case }.} $$ Equation \equ(8.1) can be rewritten as $$f=J(f)+ {\e\over \ell} {\cal L} f,\Eq(8.2)$$ where $\ell=1$ in the BGK case. The explicit form of the linear operator ${\cal L}$ implies that, for $\e$ sufficiently small, the operator $${\cal G} = (1-{\e\over \ell} {\cal L})^{-1},\Eq(8.3)$$ is positivity and normalization preserving on $L_1(\IR^3)$. (That is, it is a Markovian operator). Moreover, it is clear from the smoothing properties of of ${\cal G}$ that it is compact on $L_1(\IR^3)$. Also, $J$ is a positivity and normalization preserving map of $L_1(\IR^3)$ into itself. Then since we can rewrite (8.2) as $$f={\cal G}J(f).\Eq(8.4)$$ we see that the solutions of \equ(8.1) that we seek are the fixed points of the map $f\mapsto{\cal G}J(f)$. The properties of this map listed above prove the existence of fixed points (see [\rcite{Kra}]). Let us denote by $f_*$ one of them. It is then easy to check that it solves \equ(8.1) pointwise. Next we note that the same arguments used in previous sections imply that, if $f$ solves\equ(8.1) then $$\|f_*-M_{f_*}\|\le\delta_{\rm entropy}(\epsilon).$$ This concludes the first part of Theorem 2.4. Everything done so far would apply in the Boltzmann case as well as the BGK case. What we don't know at this point is: Are there fixed points in {\it each} of the stable neighborhoods, and if so is there {\it exactly one} in each stable neighborhood. These questions can be positively answered in the BGK case in a simplre way. It seems likely that one could also provide a positive answer for at least the first of them for the Boltzmann kernel, but we have not done more than skecth a lengthly argument, and so will confine ourselve to the BGK case. In fact, since ${\cal G}=I+\e{\cal L}{\cal G}$ with $I$ the identity map and $J(f)=M_f$, we can write \equ(8.4) as $$f=M_\e+\e{\cal L} {\cal G} M_\e.\Eq(8.5)$$ Note that the right hand side of \equ(8.5) depends only on $u$ and $e$, the first and second moments of $f$. If we multiply \equ(8.5) by $v$ or by $v^2$ and integrate, we get $$\eqalign{ F_\e(u,e)=0,\cr G_\e(u,e)=0, }\Eq(8.6)$$ because $f$ and $M_f$ have the same first two moments. The functions $F_e$ and $G_e$ are quite complicated, but for $\e=0$ they reduce to the functions $F$ and $G$ in the right hand side of \equ(2.5.3). Then, by Proposition 2.3 we know there are solutions $(u_*,e_*)$ to \equ(8.6) for $\e=0$. Moreover the differential of the map $(u,e)\to (F_\e(u,e),G(u,e))$ has eigenvalues with non vanishing real part, in $\e=0$ and $(u,e)=(u_*,e_*)$, when $E$ is in the appropriate range. Therefore, by the implicit function theorem, for $\e$ sufficiently small, we have a unique solution $(u_\e, e_\e)$ in a neighborough of $(u_*,e_*)$ to \equ(8.6). Let $M_\e$ be the Maxwellian with moments $(u_\e, e_\e)$. Then it is easy to check that $$f=M_\e+{\cal L}{\cal G}M_\e$$ is solution to \equ(8.1). This concludes Theorem 2.4. \bigskip \noindent{\bf 9. Long time behavior.} \bigskip \numsec= 9 \numfor= 1 \bigskip It is natural to ask whether the stationary solution are the asymptotic limits as $t\to+\infty$ of the evolution starting in appropriate neighborhoods of the fixed point Maxwellian. To this we have only a partial answer even in the BGK case: {\bf Proposition 9.1} {\it Choose a stable fixed point $(u^*,e^*)$ of \equ(2.5.3) and let $f_*$ be a stationary solution of \equ(2.1) in the neighborhood of $M_{(u^*,e^*)}$. Assume that the solution $f_t$ of the time dependent problem starting near $M_{(u^*,e^*)}$ has moments $u(t)$ and $e(t)$ converging to $u_*$ and $e^*$ respectively. Then $$\lim_{t\to+\infty} ||f-f_*||_2=0.$$ } Unfortunately we do not have enough control on the time behavior of the solution to check the convergence of the moments. We expect however such convergence and this can be proven for a modified model where we consider, instead of a diffusion coefficient $D$ depending on the velocity, one depending only on the average $e$. A straightforward calculation then shows that one gets closed equations for the first two moments and the long time asymptotics is easily obtained. In this case all our results still apply and the conditions of Proposition 2.5 are fulfilled. First we prove Proposition 2.5. Let $f_*$ be a fixed point and $f$ the distribution at time $t$. Calculating as in previous sections, we have:$$\eqalign { &{1\over 2}{{\rm d}\over {\rm d}t}\|f-f_*\|_2^2= \int_{\IR^3}\d (f(v)-f_*(v))[{\cal L}f(v) +{1\over \e}Q(f)(v)]\cr &= \int_{\IR^3}\d (f(v)-f_*(v))[{\cal L}(f(v)-f_*(v)) +{1\over \e}(Q(f)(v)-Q(f_*)(v))]\cr &=\int_{\IR^3}\d (f(v)-f_*(v)){\cal L}(f(v)-f_*(v))\cr &+{1\over \e}\int_{\IR^3}\d (f(v)-f_*(v))(M_f(v)-M_{f_*}(v))-{1\over\e}\|f-f_*\|_2^2. }$$ It is easy to chech that $$\int_{\IR^3}\d (f(v)-f_*(v)) L_2(f(v)-f_*(v))\le 0,$$ $$\int_{\IR^3}\d(f(v)-f_*(v))E\cdot\n_v(f(v)-f_*(v))=0.$$ On the other hand $$\eqalign { &\int_{\IR^3}\d (f(v)-f_*(v)) L_1(f(v)-f_*(v))\cr &=-\int_{\IR^3}\d \n_v (f(v)-f_*(v)) D(v)M(v)\n_v\left({f(v)- f_*(v)\over M}\right)\cr &=-\int_{\IR^3}\d |\n_v (f(v)-f_*(v))|^2 D(v) - \int_{\IR^3}\d v\cdot \n_v (f(v)-f_*(v)) D(v)(f(v)-f_*(v))\cr &=-\int_{\IR^3}\d |\n_v (f(v)-f_*(v))|^2 D(v) - {1\over 2}\int_{\IR^3}\d v\cdot \n_v (f(v)-f_*(v))^2 D(v)\cr &=-\int_{\IR^3}\d |\n_v (f(v)-f_*(v))|^2 D(v) + {1\over 2}\int_{\IR^3}\d (f(v)-f_*(v))^2\n_v\cdot(v D(v))\cr &=-\int_{\IR^3}\d |\n_v (f(v)-f_*(v))|^2 D(v) + {1\over 2}\int_{\IR^3}\d (f(v)-f_*(v))^2v\cdot \n_v D(v))\cr &+{3\over 2}\int_{\IR^3}\d (f(v)-f_*(v))^2 D(v)\cr &\le {3\over 2}\int_{\IR^3}\d (f(v)-f_*(v))^2D(v)\cr } $$ The last ineqality is consequence of the \equ(3.3). Since $$\int_{\IR^3}\d (f(v)-f_*(v))(M_f(v)-M_{f_*}(v))\le {1\over 2} \|f-f_*\|^2_2 +{1\over 2} \|M_f -M_{f_*}\|^2_2,$$ we conclude that $${{\rm d}\over {\rm d}t}\|f-f_*\|_2^2\le {1\over \e} \|M_f -M_{f_*}\|^2_2 -[{1\over \e}-3(a+c)]\|f-f_*\|_2^2.$$ With $x(t)=\|f-f_*\|_2^2$ and $a= \|M_f-M_{f_*}\|^2_2$ we have, for $\e$ sufficiently small: $$\dot x + c x\le a$$ and hence $$x(t)\le x_0{\rm e}^{-{t/e}}+\int_0^td\/s {\rm e}^{-{(t-s)/e}a(s)}.$$ This implies $x(t)\to 0$ as $t\to \infty$, provided that $a(t)\to 0$ as $t\to \infty$. This concludes Proposition 2.5. The convergence of $M_f$ to $M_{f_*}$ is not easy to get for \equ(2.1). A simple answer is obtained if one replaces the operator $L_1$ given by \equ(2.2) with $$L_1f(v)=D(e)\nabla\cdot\biggl(M(v)\nabla\/f\biggl({f(v)\over M(v)} \biggr) \biggr),\Eq(9.2)$$ This model is much simpler than the one already considered, but still has a non trivial behavior on the hydrodynamical sale. In particular, it is easy to see that, if we write the equations for $u$ and $e$, we get a closed system in those variables. Namely, \equ(2.5.3),\equ(2.5.5) and \equ(2.5.7) are replaced by $${{\rm d}\over {\rm d}t}\left(\matrix{u(t)\cr e(t)\cr}\right) = \left(\matrix{F(u(t), e(t))\cr G(u(t), e(t))\cr}\right),\Eq(9.5.3)$$ the functions $F$ and $G$ being given explicitly by $$ F(u,e) = E - u[\nu +D(e)], \Eq(9.5.5) $$ % $$ G(u,e) = Eu +3 D(e)-2eD(e). \Eq(9.5.7) $$ Equation \equ(9.5.3) is exact for this model independently of $\e$ and for suitable choices of the functions $D(e)$ has several critical points for $E$ in an appropriate range. Moreover the asymptotic behavior for large times is easy to establish. All our results apply to this model without substantial changes. 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