数理解析研究所講究録 1425 巻 2005 年 98-112

98

Global Solutions of the Boltzmann Equation
with the External Force
横浜国立大学大学院工学研究院 鵜飼 正二 (Seiji Ukai)
Faculty of Engineering, Yokohama National University

Tong Yang
Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong
Huijiang Zhao
Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences andSchool of Political Science and Economics, Waseda University

1

Introduction and Main Theorem

The Boltzmann equation in the field of a potential force is written as(1.1)$\frac{\partial f}{\partial t}+\xi\cdot\nabla_{x}f-\nabla_{x}\Phi(x)\cdot\nabla_{\xi}f=Q(f, f)$.

Here $f=f(t, x,\xi)$ is the unknown scalar function which stands for the distributional
density of gas particles at time in the phase space of positionand velocity, while is a given potential function and is the collision operator. We restrict
ourself to the hard sphere model for .
We also assume that (!} is -independent, i.e. (I) $=\Phi(x)$ . Then, the local Maxwellian
given by$x\in \mathbb{R}^{3}$$t$$\xi\in \mathbb{R}^{3}$$\Phi$$Q$$Q$$\mathrm{t}$(1.2)$\overline{M}(x, \xi)=\frac{\overline{\rho}}{(2\pi R\overline{\theta})^{3/2}}\exp\{-\frac{1}{R\overline{\theta}}(\Phi(x)+\frac{|\xi|^{2}}{2})\}\equiv \mathrm{M}_{[\hat{\rho}\langle x),0,\overline{\theta}]}(\xi)$,

where $R>0$ is the gas constant andare some constants, is a stationary solution
to (1-1) which describes, physically, the distributional density of a gas in an equilibrium
state with the mass density$\overline{\rho},\overline{\theta}>0$

$)$

(1.3)

$\hat{\rho}(x)=\overline{\rho}\exp\{-\frac{\Phi(x)}{R\overline{\theta}}\}$,

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zero bulk velocity, and absolute temperature 0.

Our aim is to show that this local Maxwellian is asymptotically stable. More precisely,with the initial
we consider the Cauchy problem of (1.1) for $t\geq 0$ andcondition$(x, \xi)\in \mathbb{R}^{3}\mathrm{x}$(1.4)$f(0, x, \xi)=f_{0}(x, \xi)$For a constant$\theta_{-}>0$,$(x, \xi)\in \mathbb{R}^{3}\rangle\langle \mathbb{R}^{3}$$\mathbb{R}^{3}$

.

, let
$M_{-}( \xi)=\frac{1}{(2\pi R\theta_{-})^{3/2}}\exp(-\frac{|\xi|^{2}}{2R\theta_{-}})$,

be the Maxwellian specified by 0,. We will prove theth $ere$ exist
and for any integer
Main Theorem. For any positive constantspotential
thesuch that if
a small positive constant and a positive constantsatisfy
4 and the initial data of the form
$N\geq 4_{\gamma}$$\overline{\rho},\overline{\theta}$$\theta_{-}\in(\overline{\theta}/2,\overline{\theta})$$\epsilon$$f_{0}=\overline{M}+M_{-}^{1/2}g_{0}$

,

(1.5)

$|| \Phi||_{L_{x}^{2}}+\sum_{1\leq|\alpha|\leq N+1}\mathrm{I}\partial_{x}^{\alpha}\Phi||_{L_{x}^{3}}+\sum_{|\alpha|+|\beta|\leq N}||\partial_{x}^{\alpha}\partial_{\xi}^{\beta}g\mathrm{o}||_{L_{x,\xi}^{2}}\leq\in$then, the Cauchy problem (1.1), (1.4) has a unique classical solution in the large in timeof the form
$f=\overline{M}+M_{-}^{1/2}g$

with

$g$

,

satisfying,$\partial_{t}^{\gamma}\partial_{x}^{\alpha}\partial_{\xi}^{\beta}g\in C(\overline{\mathbb{R}}_{+}\mathrm{i}L_{x,\xi}^{2})\cap L^{\infty}(\mathbb{R}_{+} ; L_{x,\xi}^{2})$for any$\gamma+|\alpha|+|\beta|\leq N$

, and

.

$(1+|\xi|)^{1/2}\partial_{\ell}^{\gamma}\partial_{x}^{\alpha}\partial_{\xi}^{\beta}g\in L_{\mathrm{t},x,\xi}^{2}(\mathbb{R}_{+}\mathrm{x} \mathbb{R}^{3}\mathrm{x} \mathbb{R}^{3})$for any$\gamma+|\alpha|>0,\gamma+|\alpha|+|\beta|\leq N$. Moreover, the asymptotic property$(t\prec\infty)$$\sup_{x\in \mathbb{R}^{3}}||\partial_{t}^{\gamma}\partial_{x}^{\alpha}\partial_{\xi}^{\beta}g(t, x, \cdot)||_{L_{\xi}^{2}}arrow 0$

holds for any$\gamma$$+|\alpha|+|\beta|\leq N-4$,.

solution under the additional
Remark 1. A similar result was announced in [1] on thethe compressible Navierassumption that the support of is compact. On the otherandStokes equations with the potential force was solved in [7] on the same global existencethe
Recently,
asymptotic property under the same assumption on as in our theorem.
ininitial value problem of the Vlasov-Poisson-Boltzmann system has been solved globally$L_{\xi}^{\infty}$$\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{d}_{t}$$\Phi$$\Phi$

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time in the torus in [3], and in the whole space in [9]. See also [4] for the Vlasov-MaxwellBoltzmann system.
, but this does
Remark 2. The assumption (1.5) requires, among others, thatnot contradict to the fact that the potential is unique only up to an additive constant,because this constant can be absorbed into the constant .
$\Phi\in L_{x}^{2}$$\Phi$$\overline{\rho}$

Our proof relies on the energy method based on the macro-micro (fluid dynamickinetic) decomposition of the Boltzmann equation which was developed recently in [6].
The energy estimates for the macroscopic (fluid) component of are obtained withthe -theorem for the lower order derivatives and by the usual integration by parts on
the differential equation for higher order derivatives. Both estimates contain Sobolev
norms of the microscopic (kinetic) component of . It should be noted that if thesenorms are dropped, our estimates coincide with those derived in [7] for the compressible
Navier-Stokes equation.
The norms of the microscopic component can be estimated by virtue of the microscopic-theorem of a new type, i.e., the negative definiteness of the linearized collision operator
with different weight functions on the space of microscopic components, and again by
the integration by parts on the differential equations for derivatives of the microscopic$f$$\mathrm{H}$$f$$\mathrm{H}$component.
This technique has been developed in [6] for the force-free case, where the energy
estimates can be closed only with $(t, x)$ derivatives of . In our case, however, 4 derivatives
should be also included. Recently, in [5], anotherenergy method has been proposed
for the Boltzmann equation. Although the technique is quite different from [6], it applies
also to our case, to deduce the same result.
The global existence is concluded by combining the local existence and the energy estimates. Our local solutions should be, therefore, in consistence with our energy estimates,that is, they should besolutions with respect to 4 as well as ?, $x)$ . Such solutions
can be constructed by using theenergyestimate of derived in [2] and time local
estimates.
The present paper is organised as follows. In the next section, it is shown that themacro-micro decomposition of [6] works also in the presence of the external force. Actually,
we will derive a system of fluid-type equations governing the macroscopic component
whose main parts consist of the compressible Navier-Stokes equations, and the equation
governing the microscopic component. These equations will be used in \S 3 to establish
an a priori estimate for
Sobolev norm of the solution. The norms of lower orderderivatives of the macroscopic component are estimated by the celebrated H-theoremof the Boltzmann equation and those of higher order derivatives by applying theenergy technique developed for the compressible Navier-Stokes equations in [7]. The-energy method is also shown to work for norms of the microscopic component thanks
to a microscopic version of the -theorem. Since the computation is delicate and lengthy,
only an outline is presented. The detaill will appear in [8]. The a priori estimate in \S 3
$f$

$L^{2}$$L^{2}$$($$L^{2}$$Q$$L^{2}$$L^{2}$$L^{2_{-}}$$L^{2}$$\mathrm{H}$101

is combined with the local existence theorem established in \S 4, to conclude the global
existence theorem under the smallness conditions both on the initial data and potentialfunction.theorem. -local solutions are constructed by the contraction mapping principle
based on local energy estimates. It is noted that the smallness conditions on the initial
data and potential function are necessary even for the local result.
$L^{2}$

Ma

2-Micro Decomposition$\mathrm{c}\mathrm{r}\mathrm{o}$The decomposition of [6] has two significances: It made possible to develop a new theoryof the Boltzmann equation, that is, the -theory based on the energy method familiarin the theory of partial differential equations, and it provided a new method to reveal themathematical and physical relation between the Boltzmann equation and the compressible
Navier-Stokes equations.
Originally, it was developed for the force-free case, but it works also for our case.
Thus, we decompose the solution $f(t, x, \xi)$ of (1.1) into the macroscopic (fluid) compoand the microscopic
nent specified by the local Maxwellian:(kinetic) component$L^{2}$$\mathrm{M}=\mathrm{M}(t, x, \xi)=\mathrm{M}_{[\rho,u,\theta]}(\xi)$$\mathrm{G}=\mathrm{G}(t, x,\xi)$(2.1)$f(t, x, \xi)=\mathrm{M}(tx, \xi\})+\mathrm{G}(t, x, \xi)$

.

Here,

(2.2)

$\mathrm{M}\equiv \mathrm{M}_{[\rho,u,\theta]}(t, x, \xi)\equiv\frac{\rho(t,x)}{\sqrt{(2\pi R\theta(t,x))^{3}}}\exp(-\frac{|\xi-u(t,x)|^{2}}{2R\theta(t,x)})$;,, bulk velocity $u(t, x)=$is the local Maxwellian with the macroscopic density, and temperature 0 associated with this particular by$(u_{1}, u_{2}, u_{3})$(2.3)$\mathrm{E}$$\rho(t, x)$$f$’$\rho(t,x)\equiv,\int_{\mathrm{E}=\frac{3}{2}R\theta}\mathrm{R}^{3}f\rho(t,x)u_{i}(t,x)\equiv[\rho(\mathrm{E}+\frac{1}{2}|u|^{2})](\int_{t,x)\equiv}^{(t,x_{)}\xi)d\xi}\mathrm{R}^{3}’ x\psi(\xi)f(t,,\xi).d\xi \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{i}=1,2,3\int^{i}\mathrm{R}^{3}\psi_{4}(\xi)f(t,x,\xi)d\xi,$being the inner energy density. And(2.4)$\psi_{0}(\xi)\equiv 1$,$\psi_{i}(\xi)\equiv\xi_{\mathrm{i}}$$\psi_{\alpha}(\xi)$, $\alpha=0$ , 1,

for $i=1,2,3$ ,$\cdots$, 4, are the collision invariants,$\psi_{4}(\xi)\equiv\frac{1}{2}|\xi|^{2}$satisfying$\oint_{\mathrm{R}^{3}}\psi_{\alpha}(\xi)Q(h, g)d\xi=0$, for $\alpha=0,1$ , 2, 3, 4.

,

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With this local Maxwellian, we now define the inner product$\langle h, g\rangle_{\mathrm{M}}\equiv\oint_{\mathrm{R}^{3}}\frac{1}{\mathrm{M}}h(\xi)g(\xi)d\xi$.

Then, the functions$\chi_{0}(\xi;\rho, u, \theta)$$\chi_{l}(\xi).\rho$

(2.5)

, ,
$u$

$\theta)$$\equiv\frac{1}{\sqrt{\rho}}\mathrm{M}_{\mathrm{J}}$$\equiv\frac{\xi_{i}-u_{i}}{\sqrt{\mathrm{f}1\theta\rho}}\mathrm{M}$for $i=1,2,3$ ,

$\{$

X4

$(\xi;\rho, u, \theta)$$\langle\chi_{i}, \chi j\rangle_{\mathrm{M}}$$\equiv\frac{1}{\sqrt{6\rho}}(\frac{|\xi-u|^{2}}{R\theta}-3)\mathrm{M}$$=\delta_{ij}$,, for $i,j=0,1,2,3,4$ ,form an orthonormal basis of the space of macroscopic (fluid) components of the solution,
so that(2.6),$\mathrm{P}_{0}h\equiv\sum_{j=0}^{4}\langle h, \chi_{\mathit{1}}\rangle_{\mathrm{M}}\chi_{j}$$\mathrm{P}_{1}h\equiv h-\mathrm{P}_{0}h$,define orthogonal self-adjoint projections w.r.t. the inner product { ,.
is called themacroscopic projection andthe microscopic projection, respectively. A function
is called microscopic or kinetic if it has no fluid components, that is,$\cdot$$\cdot\rangle_{\mathrm{M}}$$\mathrm{P}_{0}$$h(\xi)$$\mathrm{P}_{1}$(2.7)$\oint_{\mathrm{R}^{3}}h(\xi)\psi_{\alpha}(\xi)d\xi=0$, for cx $=0$ , 1,

$2_{?}3,4$.

It is clear that such a function is in the range of thethat the decomposition (2.1) satisfies

microscopic projection

(2.8)

,

$\mathrm{P}_{0}f=\mathrm{M}$,$\mathrm{P}_{1}f=\mathrm{G}$$\mathrm{P}_{1}$. Notice

Plug (2.1) into (1.1) to deduce(2.9)$(\mathrm{M}+\mathrm{G})_{t}+\xi\cdot$$\nabla_{x}(\mathrm{M}+\mathrm{G})-\nabla_{x}\Phi\cdot$ $\nabla_{\xi}(\mathrm{M}+\mathrm{G})=2Q(\mathrm{G}, \mathrm{M})+Q(\mathrm{G}, \mathrm{G})$By applying Po to (2.9), we have

(2.10)$\mathrm{M}_{t}+\mathrm{P}_{0}(\xi\cdot\nabla_{x}\mathrm{M})+\mathrm{P}_{0}(\xi\cdot$.

$\nabla_{x}\mathrm{G})-\nabla_{x}\Phi\cdot\nabla_{\xi}\mathrm{M}=0$

where we have used the fact

(2.11).

$\mathrm{P}_{0}f_{t}=(\mathrm{P}_{0}f)_{t}=\mathrm{M}_{t}$

.

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, but does not if
andNotice that (2.11) holds only for the particular specifying
independent of .
is defined withAs usual, the system of five conservation laws follow by taking the inner product of:(2.10) and the collision invariants$\mathrm{M}$

$f$

$f$

$(\rho, u, \theta)$$\mathrm{P}_{0}$$\mathrm{P}_{0}$$\psi_{\alpha}(\xi)$,(2. i2)$\ovalbox{\tt\small REJECT}$$\rho_{t}+\mathrm{d}\mathrm{i}_{\mathrm{V}_{x_{3}}}=0(\rho u_{i})_{t}+\sum_{j=1}^{\rho u}(\rho u_{i}u_{j}’)_{x_{j}}+p_{x_{i}}-\hat{p}_{x_{i}}(\rho-\hat{\rho})\Phi_{x_{i}}[\rho(\frac{1}{2}|u|^{2}+\mathrm{E})]_{t}+\sum_{=j1}^{3}(u_{j}(p(\frac{1}{2}|u|^{2}+\mathrm{E})+p))_{x_{j}}+\rho u\nabla_{x}\Phi=-\int_{=-}^{+}\mathrm{R}^{\mathrm{s}_{I_{\mathrm{R}^{3}}^{\psi_{4}(\xi\nabla_{x}\mathrm{G})d\xi}}}\psi_{i}(\xi\cdot\nabla_{x}\mathrm{G})d\xi,i.=1,2,.3$

Here, is the pressure for the monatomic gases:, in (1.2),
fact that for$p= \frac{2}{3}\rho \mathrm{E}=R\rho\theta$$p$$\hat{\rho}(x)=\overline{\rho}e^{-\Phi(x)/(R\overline{\theta})}$$\hat{p}(x)=R\hat{\rho}(x)\overline{\theta}$,$\hat{p}_{x_{n}}+\hat{\rho}\Phi_{x_{i}}=0$

On the other hand, the microscopic equation for
and using again (2.11):(2.13), and we have used the$\overline{\theta}$$\mathrm{G}_{t}+\mathrm{P}_{1}(\xi\cdot\nabla_{x}\mathrm{G}+\xi\cdot\nabla_{x}\mathrm{M})-\nabla_{x}\Phi$where$L_{\mathrm{M}}g=L_{[\rho,u,\theta]}g\equiv Q$(.

$\mathrm{M}+g$$\mathrm{G}$.

is obtained by applying$\nabla_{\xi}\mathrm{G}=L_{\mathrm{M}}\mathrm{G}+Q(\mathrm{G}, \mathrm{G})$,$\mathrm{M}+g$$\mathrm{P}_{1}$to (2.9),) $-Q(g, g)$ .

is self-adjoint and negative definite on the space of microscopic. Therefore, it has a bounded inverse and,components w.r.t. the inner product(2.13) can be rewritten as

It is classical that

$L_{\mathrm{M}}$$<\cdot$$\mathrm{G}$(2.14)$\cdot>_{\mathrm{M}}$$=L_{\mathrm{M}}^{-1}(\mathrm{P}_{1}(\xi\cdot\nabla_{x}\mathrm{M}))$$+L_{\mathrm{M}}^{-1}(\mathrm{G}_{t}+\mathrm{P}_{1}(\xi\cdot\nabla_{x}\mathrm{G})-\nabla_{x}\Phi\cdot$,$:=L_{\mathrm{M}}^{-1}(\mathrm{P}_{1}(\xi\cdot\nabla_{x}\mathrm{M}))+\Theta$$\nabla_{\xi}\mathrm{G}-Q(\mathrm{G}, \mathrm{G}))$

104

Set

$\{\begin{array}{l}A_{\mathrm{j}}(\xi)=\frac{1}{2}(\xi|^{2}-5)\xi_{j},j=\mathrm{l}_{2}2,3B_{ij}(\xi)=\xi_{i}\xi_{j}-\frac{1}{3}\delta_{\mathrm{i}j}|\xi|^{2},i,j=1,23)\mu(\theta)=\kappa(\theta)=-R\theta\oint_{-R^{2}\theta}\mathrm{R}^{3}B_{ij}(\frac{\xi}{\sqrt{R\theta}})L_{\mathrm{M}_{[1,u,\theta]}}^{-1}(\int_{\mathrm{R}^{3}}A_{l}(\frac{\xi}{\sqrt{R\theta}})L_{\mathrm{M}_{\lfloor 1,u,\theta]}}^{-1}\end{array}$$B_{ij}( \frac{\xi}{\sqrt{R\theta}})\mathrm{M}_{[1,u,\theta]})d\xi>0$$(A_{l}( \frac{\xi}{\sqrt{R\theta}})\mathrm{M}_{[1,u,\theta])d\xi>0}$,$i\neq j$,.

After straightforward but lengthy computations, we have$- \int_{\mathrm{R}^{3}}\psi_{i}\xi$$\{$$- \oint_{\mathrm{R}^{3}}\psi_{4}\xi$.
.

$\nabla_{x}L_{\mathrm{M}}^{-1}(\mathrm{P}_{1}(\xi\cdot\nabla_{x}\mathrm{M}))d\xi=\sum_{j=1}^{3}[\mu(\theta)(u_{ix_{\mathrm{j}}}+u_{jx_{i}}-\frac{2}{3}\delta_{ij}\mathrm{d}\mathrm{i}\mathrm{v}_{x}u)]_{x_{\mathrm{J}}}$,$i=1,2,3$ ,$\nabla_{x}L_{\mathrm{M}}^{-1}(\mathrm{P}_{1}(\xi\cdot\nabla_{x}\mathrm{M}))d\xi=\sum_{j=1}^{3}(\kappa(\theta)\theta_{x_{j}})_{x_{i}}$$+ \sum_{i,j=1}^{3}\{\mu(\theta)u_{i}(u_{ix_{\mathrm{j}}}+u_{\mathrm{i}x_{i}}-\frac{2}{3}\delta_{ij}\mathrm{d}\mathrm{i}\mathrm{v}_{x}u)\}_{x_{j}}$.

Plugging these relations and (2.14) into (2.12) yield now another representation of theequation (11) which contains a system of fluid-type equations:$\mathrm{s}$$\rho_{t}+\mathrm{d}\mathrm{i}\mathrm{v}_{x}(pu)=0$,$( \rho\Re)_{t}+\sum_{j=1}^{3}(\rho u_{i}u_{j})_{x_{\acute{\mathrm{J}}}}+(p-\hat{p})_{x_{i}}+(\rho-\hat{p})\Phi_{x_{i}}$$= \sum_{j=1}^{3}[\mu(\theta)(u_{ix_{\mathrm{j}}}+u_{jx_{i}}-\frac{2}{3}\delta_{ij}\mathrm{d}\mathrm{i}\mathrm{v}_{x}u)]_{x_{j}}$(2.15)$- \int_{\mathrm{R}^{3}}\psi_{i}(\xi\cdot\nabla_{x}\Theta)d\xi$,$\mathrm{i}=1,2,3$,$[p( \frac{1}{2}|u|^{2}+\mathrm{E})]_{\mathrm{t}}+\sum_{j=1}^{3}(u_{j}(\rho(\frac{1}{2}|u|^{2}+\mathrm{E})+p))_{x_{j}}$$+\rho u\cdot$$\nabla_{x}\Phi=\sum_{i,j=1}^{3}\{\mu(\theta)u_{i}(u_{ix_{j}}+u_{jx_{i}}-\frac{2}{3}\delta_{ij}\mathrm{d}\mathrm{i}\mathrm{v}_{x}u)\}_{x_{\mathrm{j}}}$$+ \sum_{j=1}^{3}(\kappa(\theta)\theta_{x_{j}})_{x_{\mathrm{j}}}-\oint_{\mathrm{R}^{3}}\psi_{4}(\xi\cdot\nabla_{x}\Theta)d\xi$.

Notice that if one drops all the terms containing , then (2.15) reduces to the system of thecommpressible Navier-Stokes equations with the external force; which has been solved in$\Theta$

105

Sobolev space by using the energy method.
[7] on the existece of global solutions in an
This suggests the possibility of developing the -theory of the Boltzmann equation. Thisis true, indeed, as seen below.
$L^{2}$$L^{2}$

3

Energy Estimates

The construction of global$L^{2}$

solutions relies on an a priori estimate for the norm

(3.1)
$N(t)^{2}= \sup_{0\leq\tau\leq t}(_{|\gamma|\leq 4}\sum||\partial_{t_{)}x}^{\gamma}(\rho-\hat{\rho})u,$$\theta-\overline{\theta})(\tau)||_{L_{x}^{2}}^{2}+\sum_{|\gamma|+|\beta|\leq 4}||\partial_{t,x}^{\gamma}\partial_{\xi}^{\beta}\mathrm{G}(\tau)||_{L_{\mathrm{b}\alpha_{-},oe,\xi}^{2}})$lr$+ \sum_{1\leq|\gamma|\leq 4}\int_{0}^{t}||\partial_{t,x}^{\gamma}(\rho-\hat{\rho}, u\} \theta-\overline{\theta})(\tau)||_{L_{x}^{2}}^{2}$

The lower order norms ofTheorem(3.2)$(\rho, u_{1}\theta)$.

$+ \sum_{|\gamma\}+|\beta|\leq 4}I_{0}^{t}||\nu_{\mathrm{M}}^{1/2}\partial_{t,x}^{\gamma}\partial_{\xi}^{\beta}\mathrm{G}(\tau)||_{L_{\mathrm{M}oe,\xi}^{2}}^{2}-,d\tau$involved here are controlled by the celebrated H-$\int_{\mathrm{R}^{3}}Q(f, f)\ln fd\xi\leq 0$,

and the higher order norms by the -method developed in [7], while the estimate forare derived by using the microscopic -Theorem of a new type, namely, the negativewith different weight functions,definiteness of$L^{2}$$\mathrm{H}$$\mathrm{G}$$L_{\mathrm{M}}$(3.3)$- \int_{\mathrm{R}^{3}}\frac{\mathrm{G}L_{\mathrm{M}}\mathrm{G}}{\tilde{\mathrm{M}}}d\xi\geq\overline{\sigma}\int_{\mathrm{R}^{3}}\frac{\nu_{\mathrm{M}}(\xi)\mathrm{G}^{2}}{\tilde{\mathrm{M}}}d\xi$,

are rather close to each other, with a constantwhich holds if two Maxwellians, see [6] for a detail.
depending only on
estimate of the nonlinear operator established in [2]: $\exists C>0$We also need thesuch that$\mathrm{M},\tilde{\mathrm{M}}$$(u,\overline{u}, \theta,\tilde{\theta})$$\overline{\sigma}>0$$Q$$L^{2}$$\leq$$C \{\int_{\mathrm{R}^{3}}\frac{\nu_{\mathrm{M}}(\xi)f^{2}}{\mathrm{M}}d\xi$.

$\int_{\mathrm{R}^{3}}\frac{g^{2}}{\mathrm{M}}d\xi+\int_{\mathrm{R}^{3}}\frac{f^{2}}{\mathrm{M}}d\xi$.

$\int_{\mathrm{R}^{3}}\frac{\nu_{\mathrm{M}}(\xi)g^{\gamma}\sim}{\mathrm{M}}d\xi\}$,

is any Maxwellian such that the above integrals are well defined.
whereis the collision frequency associated with the classicalIn the above, the function,
decomposition of the operator$\overline{\mathrm{M}}$$\nu_{\mathrm{M}}(\xi)$$L_{\mathrm{M}}$(3.5)$L_{\mathrm{M}}=-\mathit{1}\mathrm{M}(\xi)\mathrm{x}$$+K_{\mathrm{M}_{\mathrm{J}}}$

106

where
explicit expressions ofis a symmetrictheintegralandkernels of$K_{\mathrm{M}}(\cdot)=-K_{1\mathrm{M}}(\cdot)+K_{2\mathrm{M}}(\cdot)$$\nu_{\mathrm{M}}(\xi)$$\nu_{\mathrm{M}}$$L^{2}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}$integral operator. Theare$K_{i\mathrm{M}}(\cdot)$,$= \frac{2\rho}{\sqrt{2\pi R\theta}}\{(\frac{R\theta}{|\xi-u|}+|\xi-u|)\int_{0}^{|\xi-u|}\exp(-\frac{y^{2}}{2R\theta})dy+R\theta\exp(-\frac{|\xi-u|^{9}\sim}{2R\theta})\}$$\{$$k_{1\mathrm{M}}(\xi, \xi_{*})$$= \frac{\pi\rho}{\sqrt{(2\pi R\theta)^{3}}}|\xi-\xi_{*}|\exp(-\frac{|\xi-u|^{2}}{4R\theta}-\frac{|\xi_{*}-u|^{2}}{4R\theta}))$$k_{2\mathrm{M}}(\xi, \xi_{*})$$= \frac{2}{}\sqrt{2\pi R\theta}E-|\xi-\xi_{*}|^{-1}\exp(-\frac{|\xi-\xi_{*}|^{2}}{8R\theta}-\frac{(|\xi|^{2}-|\xi_{*}|^{2}]_{-}^{2}}{8R\theta|\xi-\xi_{*}|^{2}})$.

In the below it is crucial thatas$\nu_{\mathrm{M}}(\xi)\sim(1+|\xi|)$

3.1$|\xi|arrow\infty$.

Lower Order Estimate

Introduce the entropy(3.6)$\mathrm{M}\ln$ $\mathrm{M}d\xi$$- \frac{3}{2}\rho S\equiv\oint_{\mathrm{R}^{3}}$.

By a direct computation, we see that
$S=- \frac{2}{3}\ln\rho+\ln(2\pi R\theta)+1$

A convex entropy-entropy flux pair(1.2) is then given by(3.7)$\eta$,$p=R\rho\theta=k\rho^{\frac{5}{3}}e^{S}$$(\eta, q)$$\overline{\mathrm{M}}=\mathrm{M}[\hat{\rho}(x),0,\theta]$of$= \frac{3}{2}\{\rho\theta-\overline{\theta}\rho S+p[(\overline{S}-\frac{5}{3})\overline{\theta}+\frac{|u|^{2}}{2}]+\frac{2}{3}\hat{\rho}\overline{\theta}\}7$

$\{$

,

$\mathrm{j}=1,2,3$

which satisfies, for some constant $C>1$ and withget

$k=1/(2\pi e)$ .

around the sationary solution$q_{j}=u_{j}\eta+u_{j}(p\theta-\hat{p}\overline{\theta})$

(3.8)

,

,

$\hat{E}$$= \frac{3}{2}R\overline{\theta}$,$C^{-1}(|p-\hat{\rho}|^{2}+|\rho u|^{2}+|\rho E-\hat{\rho}\hat{E}|^{2})\leq\eta\leq C(|\rho-\hat{\rho}|^{2}+|\rho u|^{2}+|pE-\hat{\rho}\hat{E}|^{2})$On the other hand, multiplying (2.10) by$\log \mathrm{M}$( In M)$(3.9)- \frac{3}{2}(\rho S)_{t}-\frac{3}{2}\mathrm{d}\mathrm{i}\mathrm{v}_{x}(\rho uS)+\nabla_{x}(\int_{\mathrm{R}^{3}}$

$\xi$

,

and integrating w.r.t, 4 by parts, we$\mathrm{G}d\xi)=\int_{\mathrm{R}^{3}}\frac{\mathrm{G}\mathrm{P}_{1}(\xi\cdot\nabla_{x}\mathrm{M})}{\mathrm{M}}d\xi$

,

107

which can be rewritten in terms of the pair$\eta_{t}+\mathrm{d}\mathrm{i}\mathrm{v}_{x}q=$$(\eta, q)$as$- \nabla_{x}(\int_{\mathrm{R}^{3}}(\xi \mathrm{G}\ln \mathrm{M}+\frac{3}{2}\psi_{4}\xi \mathrm{G})d\xi)-\frac{3}{2}\rho u\cdot\nabla_{x}\Phi$(3.10)$+ \int_{\mathrm{R}^{3}}\frac{\mathrm{P}_{1}(\xi\cdot\nabla_{x}\mathrm{M}\}\mathrm{G}}{\mathrm{M}}d\xi$.

(saywith some ffied
In the below, we assume that (1.5) holds for) . Plug (2.13) into(saywith some fixedand similarly that$(2.12)_{1}$, it holds that
(3.10) and integrate w.r.t. , . Owing to the mass conservation law$\epsilon$$t$$\int_{\mathrm{R}^{3}}$pu .

$\delta_{1}=1$$\delta_{1}>0$$N(t)\leq\delta_{1}$$\epsilon_{1}=1)$$\epsilon_{1}>0$$\in(0, \epsilon_{1}]$$x$$\nabla_{x}\Phi dx=-\int_{\mathrm{R}^{3}}\nabla_{x}$.

$( \rho u)\Phi dx=\frac{d}{dt}\int_{\mathrm{R}^{3}}(\rho-\hat{\rho})\Phi dx$,

and as a consequence, we get$) \Phi)(t)dx+\int_{0}^{t}\int_{\mathrm{R}^{3}}|\nabla_{x}(u, \theta)|^{2}dxd\tau\leq \mathit{0}(1)\epsilon^{2}$$\int_{\mathrm{R}^{3}}(\eta+\frac{3}{2}(\rho-\hat{\rho}$(3.11)$+O(1) \oint_{0}^{t}\int_{\mathrm{R}^{3}}\int_{\mathrm{R}^{3}}\frac{\mathrm{I}/_{\mathrm{M}}\langle\xi\}}{\mathrm{M}}($$|\mathrm{G}_{t}|^{2}+|\nabla_{x}\mathrm{G}|^{2}+\epsilon|\nabla_{\xi}\mathrm{G}|^{2}+N(t)|\mathrm{G}|^{2})(t, x,\xi)d\xi dxd\tau$Here, $N(t)$ comes by estimating$Q(\mathrm{G}, \mathrm{G})$.

by (3.4). Notice also that$\oint_{\mathrm{R}^{3}}|\rho-\hat{\rho}|\Phi dx\leq\in||\rho-\hat{\rho}||$,.
so that if 6 is small, this is absorbed into the entropy term
takingthe inner product of (2.13)
The microscopic component can be estimated by
and using (3.3), to result inand$\int\eta dx$$\mathrm{G}$$\mathrm{G}$$\int_{\mathrm{R}^{3}}\int_{\mathrm{R}^{3}}\frac{|\mathrm{G}|^{2}}{\mathrm{M}}d\xi dx+\int_{0}^{t}\oint_{\mathrm{R}^{3}}\int_{\mathrm{R}^{3}}\frac{\nu(\xi)|\mathrm{G}|^{2}}{\mathrm{M}}d\xi dxd\tau$(3.12)$\leq O(1)\epsilon^{2}+O(1)(\epsilon+N(t))\oint_{0}^{t}\int_{\mathrm{R}^{3}}\oint_{\mathrm{R}^{3}}\frac{\nu(\xi)\{\mathrm{G}|^{2}}{\mathrm{M}-}d\xi dxd\tau$$+O(1) \int_{0}^{t}\oint_{\mathrm{R}^{3}}(|\nabla_{x}(u, \theta)|^{2}+\int_{\mathrm{R}^{3}}\frac{\nu\{\xi)|\nabla_{x}\mathrm{G}|^{2}}{\mathrm{M}})$$dxd\tau$is needed to contorl the terms like
where the uncommon weight
gives,
replaced by
A similar calculation with the weight$\mathrm{M}_{-}$$\mathrm{M}$,$\partial_{t,x,\xi}^{\alpha}(\log \mathrm{M})$, $|\alpha|=1$ .

$\mathrm{M}_{-}$$\int_{\mathrm{R}^{3}}\int_{\mathrm{R}^{3}}\frac{|\mathrm{G}|^{2}}{\mathrm{M}_{-}}d\xi dx+\int_{0}^{t}\int_{\mathrm{R}^{3}}\int_{\mathrm{R}^{\mathrm{s}^{\mathrm{M}-}}}^{\underline{\nu(\xi\}|\underline{\mathrm{G}|^{2}}}}d\xi dxd\tau$(3.13)$\leq O(1)\epsilon^{2}+(\epsilon+N(t))\int_{0}^{t}\oint_{\mathrm{R}^{3}\acute{\mathrm{R}}^{3}}.\frac{\nu\{\xi)|\underline{\mathrm{G}|^{2}}}{\mathrm{M}-}d\xi dxd\tau$$+O(1) \int_{0}^{t}\int_{\mathrm{R}^{3}}(|\nabla_{x}(\rho-\hat{\rho},u, \theta)|^{2}+\int_{\mathrm{R}^{3}}\frac{\nu(\xi)|\nabla_{x}\mathrm{G}|^{2}}{\mathrm{M}_{-}})$

Note that an extra term forenergy estimates$\partial\rho$lxdr.

appears here. (3.11)-(3.13) give the complete lower order

108

3.2

Higer Order Estimates

The estimate forcan be deduced from (2.15) by proceeding just in thesame way as in [7], while the estimates forcan be derived from (2.13) applied byin a similar way as in (3.12) and (3.13), by knowing that the commutator$\partial_{t,x}^{\gamma}(\rho-\hat{\rho}, u, \theta-\overline{\theta})$$\partial_{t,x}^{\gamma}\partial_{\xi}^{\beta}\mathrm{G}$$\partial_{t,x}^{\gamma}\partial_{\xi}^{\beta}$$[\partial_{\xi}^{\beta}, L_{\mathrm{M}}]=\partial_{\xi}^{\beta}L_{\mathrm{M}}-L_{\mathrm{M}}\partial_{\xi}^{\beta}$

is bounded inand that the quantity
enjoyes a similar estimate as (3.4).
The computation is very much delicate and lengthy, however, and so, only the final
result is presented here. See [8] for the detail.
$\partial_{t,\mathrm{b}’}^{\beta}Q(f_{7}g)$$L_{\mathrm{M}_{t}\xi}^{2}$$\sum_{|\gamma|\leq 4}\oint_{\mathrm{R}^{3}}|\theta^{\gamma}(\rho-\overline{\rho},u, \theta-\overline{\theta})|^{2}dx+\sum_{|\gamma|+|\beta|\leq 4}\int_{\mathrm{R}^{3}}\oint_{\mathrm{R}^{3}}\frac{|\partial^{\gamma}\partial^{\beta}\mathrm{G}|^{2}}{\mathrm{M}}d\xi dx$(3.14)$+ \int_{0}^{t}\int_{\mathrm{R}^{3}}(\sum_{1\leq|\gamma|\leq 4}|\partial^{\gamma}(\rho-\overline{p}, u, \theta)|^{2}+\sum_{|\gamma|+|\beta|\leq 4}\int_{\mathrm{R}^{3}}\frac{\nu(\xi)|\partial^{\gamma}\partial^{\beta}\mathrm{G}|^{2}}{\mathrm{M}}d\xi)dxd\tau$$\leq$.

$o(1) \epsilon^{2}+O(1)(\epsilon+N(t))\sum_{|\gamma|+|\beta|\leq 4}\int_{0}^{t}\int_{\mathrm{R}^{3}}\int_{\mathrm{R}^{3}}\frac{\nu(\xi)|\partial^{\gamma}\partial^{\beta}\mathrm{G}|^{2}}{\mathrm{M}-}d\xi dxd\tau$

As in (3.13), the weightterm of the form$\mathrm{M}$can be replaced by M-, but again, at the cost of the extra$\sum_{1\leq|\gamma|\leq 4}\int_{0}^{t}\int_{\mathrm{R}^{3}}|\partial^{\gamma}(\rho-\hat{\rho}, u, \theta)|^{2}$dxdr.

on the right hand side.

3,3

A Priori Estimate

An appropriate linear combination of all the estimates obtained so far yields an inequalityof the form
$N(t)^{2}\leq C_{1}\epsilon^{2}+C_{2}(\epsilon+N(t))N(t)^{2}$,for some constants, $C_{2}>0$ independent ofexists a positive constant 62 such that for each
$C_{1}$$\epsilon$

and $N(t)$ . It is easy to see that there, the cubic equation$\epsilon\in(0_{2}\epsilon_{2})$$C_{2}x^{3}-(1-C_{2}\epsilon)x^{2}+C_{1}\epsilon^{2}=0$

has positive solutions. Denote the smallest one by , and set, to conclude that
holds for each$\mathit{5}_{2}$$\delta_{0}=\min(\delta_{1}.\delta_{2})$$N(t)\leq\delta_{0}$$\epsilon_{0}=\min(\epsilon_{1}, \epsilon_{2})$$\epsilon\in(0, \epsilon_{0})$

.

and

I09

Local Solution

4
Put$f=\overline{\mathrm{M}}+g$

and consider the Cauchy problem.

$g_{t}+\xi$(4.1)$\nabla_{x}g-\nabla_{x}\Phi$.

$\nabla_{\xi}g=L_{\overline{\mathrm{M}}}g+Q(g, g)$

,

$\{$

$g|_{t=0}=g_{0}(x, \xi)$.

In order to construct local solutions, we first analyse, for each given point, the backward -characteristic curve $(X(t),—(t))\equiv(X,---)(t;t_{0}, x_{0}, \xi_{0})$ of(4.1) passing through the point (tO, , ) which is given by$(t_{0}, x_{0}, \xi 0)\in$$\mathrm{R}^{+}><\mathrm{R}^{3}\mathrm{x}$$\mathrm{R}^{3}$$\mathrm{b}\mathrm{i}$$x_{0}$(4.2)$\xi_{0}$$\{\begin{array}{l}\frac{dX(t\}}{dt}=---(t)-_{d}\zeta \mathrm{E}[perp]_{t}t\rangle=-\nabla_{x}\Phi(X(t))(X(t),---(t))|_{t=t_{0}}=(x_{\mathrm{O}},\xi_{\mathrm{O}})\end{array}$Since is assumed suficiently smooth with bounded derivatives, a smooth bi-characterisitic
curve exists uniquely for all time .
Recall the decomposition$\Phi$$t$,$.L_{\overline{\mathrm{M}}}=-_{\mathrm{M}}(x,\xi)+K_{\overline{\mathrm{M}}}$$\nu_{0}(1+|\xi|)\leq _{\mathrm{M}}(x,\xi)\leq\iota/_{1}(1+|\xi|)$,$K_{\overline{\mathrm{M}}}$: bounded on,$L \frac{2}{\mathrm{M}},\xi$

and define the operator $U(t)$ by$U(t)g_{0}= \exp\{-\int_{0}^{t}_{M}(X(s, t, x, \xi),---(s, t, x, \xi))ds\}g_{0}(X(0, t, x, \xi),---(0.t, x, \xi))$.

Then, (4.1) can be rewritten formally in the form of the integral equation

(4.3)

.

$g(t)=U(t)g_{0}+ \oint_{0}^{t}U(t-s)\{K_{\overline{\mathrm{M}}}g(s)+Q(g(s), g(s)\}ds$$\equiv V(g)(t)$.

We shall show that the nonlinear map$V$

is a contraction map in the energy space$\frac{\partial^{\alpha}\partial^{\beta}g(t,x,\xi)}{\sqrt{\mathrm{M}-(\xi)}}\in \mathrm{B}\mathrm{C}_{t}([0, T),$

(4.4)

$L_{x,\xi}^{2}(\mathrm{R}^{3}\mathrm{x}\mathrm{R}^{3}))\}$$X_{T,a}=\{g(t, x, \xi)$$|||g|||\leq a$$|||g|||=$

(4.5)

,

$|\alpha|+|\beta|\leq 4$

,

110

provided that , $a>0$ are chosen sufficiently small
To this end, put $h=V(g)$ . First, assume that $g\in X_{T,a}$ . Then, is well defined (theintegral on the right hand side of (4.3) converges). If, in addition, 9 and go are assumed
is also suficiently smooth and solves the partial
sufficently smooth, then evidently,
differential equation$T$$\mathrm{h}$$h$$h_{t}+\xi$

(4.6)

.

$\nabla_{x}h-\nabla_{x}\Phi$.

$\nabla_{\xi}h+_{\mathrm{M}}(\xi)h=K_{\overline{\mathrm{M}}}g+Q(g, g)$

,

$\{$

$h|_{t=0}=g_{0}(x, \xi)$.

The extra smoothness assumptions on , go will be removed later, by appealing to thelimiting argument. See below the estimate (4.12).
and integrate w.r.t. , to deduce
Now, multiply (4.6) by
$g$

$x$

$h/\mathrm{M}_{-}$

$\xi$

,

$\frac{d}{dt}||h(t)||^{2}+2(1-C_{1}\epsilon)||\nu_{\frac{1/}{\mathrm{M}}h(t)||^{2}\leq C_{2}||g||||h||+||\mathcal{U}_{\frac{-1}{\mathrm{M}}Q(g(t),g(t))||||\nu_{\frac{1/}{\mathrm{M}}}^{2}h(t)||}}^{\mathrm{z}/2}$

where 6 is like in (1.5) for$\Phi$, or more precisely, it is assumed that

(4.7)
while

$\sum_{1\leq|\alpha|\leq 5}||\partial_{x}^{\alpha}\Phi||_{L_{x}^{3}}\leq\in$$C_{1}$,, $C_{2}>0$ are constants independent of , , and the norm is
$h$

$g$

.

$||h||^{2}= \oint_{\mathrm{R}}^{3}\int_{\mathrm{R}}^{3}\frac{h^{2}}{\mathrm{M}_{-}}dxd\xi$

Finally, we apply

to (4.6) and proceed as above. Fix$\partial_{t,x}^{\gamma}\partial_{\xi}^{\beta}$

so that $1-C_{1}\epsilon>0$ ,$\epsilon>0$

or,
(4.8)

$\epsilon<1/(2C_{1})$Then, by virtue of (3.4), we get for$|\gamma|+|\beta|\leq 4$.

,$\frac{d}{dt}||\partial_{t,x}^{\gamma}\partial_{\xi}^{\beta}h(t)||^{2}+||\nu_{\frac{1/}{\mathrm{M}}}^{2}\partial_{t,x}^{\gamma}\partial_{\xi}^{\beta}h(t)||^{2}$$(1+||\nu_{\frac{1/}{\mathrm{M}}}^{2}\partial_{t,x}^{\gamma}\partial_{\xi}^{\beta}g(t)||^{2}$$\leq C_{3}\sum_{|\gamma\}+|\beta|\leq 4}||\partial_{t,x}^{\gamma}\partial_{\xi}^{\beta}g||^{2}$Integrating this w.r.t.

(4.9)

$t$

).

yields$|||h|||=|||V(g)|||\leq|||g_{0}|||_{0}+C_{3}|||g|||(T+|||g|||)$,

where the norm $|||g|||$ is as in (4.5) andis defined also by (4.5) but with $T=0$ .
Consider the quadratic equation $C_{3}a^{2}-(1-C_{3}T)a+|||g_{0}|||_{0}=0$ . This has two positive
solutions if $1-C_{3}T>0$ and $D=(1-C_{3}T)^{2}-4C_{3}|||g_{0}|||_{0}>0$ , that is, if$|||g_{0}|||_{0}$(4.10)$T< \frac{1}{C_{3}}$,$|||g_{0}|||_{0} \leq\frac{(1-C_{3}T)^{2}}{4C_{3}}$

,

1]1

and the smaller one is given by$a= \frac{1}{2C_{3}}(1-C_{3}T-\sqrt{D})$

Under this choice of ,$T$

$g_{0}$.

, , (4.9) implis that$a$$|||g|||\leq$

a $\supset|||V(g)|||\leq a$ .

Further, let $g’\in X_{T,a}$ and put $h=V(g)-V(g’)$ . It is easy to see that if
assumed sufficiently smooth, solves$g’$is also$h$$h_{t}+\xi$(4.11).

$\nabla_{x}h-\nabla_{x}\Phi$.

$\nabla_{\xi}$

A $+_{\mathrm{M}}(\xi)h=K_{\overline{\mathrm{M}}}(g-g’)+Q(g+g’, g-g’)$ ,$\{$$h|_{t=0}=0$.

Quite similarly as in (4.9), we can deduce$|||h|||=|||V(g)-V(g’)|||\leq C_{3}|||g+g’|||||g-g’|||$ ,(4.12)

with the same constant$C_{3}$

as in $(4,9)$ . Since we are assuming)$g’\in X_{T,a}$, we get$\mu=2C_{3}a=1-\sqrt{D}<1$ .

$|||V(g)-V(g’)|||\leq\mu|||g-g’|||$ ,(4.13)$g$

We can now remove the extra smoothness assum ption on $g,g’7$ go required in the arguments given so far. Indeed, any $g\in X_{T,a}$ can be approximated by a sequenceand (4.13) givesin the norm of$\{g_{n}\}\subset C_{0}^{\infty}$$X_{T,a}$$|||V(g_{n})-V(g_{m})|||\leq\mu|||g_{n}-g_{m}|||$,

which shows that $V(g_{n})$ converges. The limit is of course $V(g)$ , and by the limiting argument, we see that the estimates (4.9), (4.12), and (4.13) hold without extra smoothness
assumption.
is a complete metric space with the metric induced by the norm
Sinceand since $\mu\in(0,1)$ , the above argument imph.es that the contraction mapping theorem. This fixed point is the desired local
has a unique fixed point inapplies andsolution.
Note that this local existence result requires the smallness conditions both on thepotential and initial data go, as seen by $(4,8)$ and (4.10).
$|||\cdot$$|||$$X_{T,a}$$V$$X_{T,a}$$\Phi$

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March 6, 2002

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