数理解析研究所講究録 1211 巻 2001 年 44-53

44

Mild Solutions to the discrete Boltzmann
equation with linear and quadratic terms
for the initial data with locally finite entropy
Mitsuru YAMAZAKI
Institute of Mathematics, University of Tsukuba
Tsukuba-shi, Ibaraki 305-8571, Japan\文 q$\hslash^{4}$怯山嶋$\sqrt[-]{}\mathrm{a}\sim$

1. Introduction
We study the discrete velocity models of the Boltzmann equation in one
space dimension. These models describe the motion of particles in ararefied
gas. To observe the evolution of particles in athin infinite tube, we take into
account both collisions between particles and reflection over the inner wall oftube, which are represented by quadratic terms and alinear terms respectively.
The discrete velocity models consist in discretizing the velocity
andthen the velocity of particles in the models can be taken only in afinite set of. Let the variable be the direction of the axis of tube and thevariables and be transversal to the axis. The thinness of the tube enables us
to suppose that the behavior of the particles is homogeneous and uniform withrespect to the variables and . The distribution of the particles with velocity
is then represented by afunction $u:(x, t)$ . Denoting the -component of, we have the system of the hyperbolic partial differential equations
bywhich describes our models :$v\in \mathrm{R}^{3}$$\{C_{\dot{l}}\in \mathrm{R}^{3};i\in I\}$

$y$

$x$

$z$

$z$

$y$

$x$

$C_{\dot{1}}$$C_{i}$$c_{t}\in \mathrm{R}$$\frac{\partial u}{\partial}i$

(1.1)

$\{$

十果

$\frac{\partial u}{\partial}i$$u:|_{t=0}$$=Q_{:}(u)$$=u_{\dot{l}}^{0}(x)$,,$i\in I$,

represent the terms of binary collisions and the ones of
where
and
linear reflection respectively. These terms are in the form of:$Q_{:}(u)$$L_{:}(u)$(1.2)$Q_{:}(u)= \sum_{j,k,\ell\in I}(A^{k\ell}.u_{k}u_{\ell}-|jA^{\dot{l}j}uu_{j})k\ell:$(1.3)$L_{:}(u)= \sum_{k\in I}(\alpha_{\dot{l}}^{k}u_{k}-\alpha_{k}^{\dot{l}}u:)$

,

,

45
where the constants satisfy(1.4)$A_{k\ell}^{\dot{\iota}j}=A_{k\ell}^{j\dot{1}}$$=Ai_{k}^{j}\geq 0$,$A_{k\ell}^{\dot{l}\dot{1}}=0\alpha_{k}^{\dot{1}}$$\geq 0$.

In this talk, we prove the time global existence and the uniqueness of thesolutionsto the system (1.1) for initial data which are not necessary
bounded but with locally finite entropy, provided that the distinct velocity aesumption :$(u_{i})_{i\in I}$(dv)$i\neq j\Rightarrow c:\neq c_{j}$,and the weak microreversibility condition :
(pr)

for

$\sum_{k,\ell\in I}A_{k\ell}^{ij}=\sum_{k,\ell\in I}A_{\dot{\iota}j}^{k\ell}$

The condition(1.5)$(\mu r)$$\forall i,j\in I$.

is weaker than the usual microreversibility condition :$A_{ij}^{k\ell}=A_{k\ell}^{ij}$for$\forall i,j$

, ,
$k$

$\ell\in I$.

It is worthy to remark that, in the mesonic process $h_{\nu}+Parrow N+\psi+$ , theis satisfied ([3,4]).
condition (1.5) is violated but the condition
For the bounded data, we [7] obtained more precise results without condition (dv), which show the existence of solution [resp. locally] bounded andglobal in time for data positive and [resp. locally] bounded. We $[7,8]$ havemoreover the explicit estimates of solutions for bounded data. For generalized
Broadwell models, we [11] derive amore precise concrete estimates for bounded
and summable data. Nevertheless, for merely summable data, it is necessary
suppose the condition (dv) in order to define asolution to the initial problem
in some sense which is weaker than the distribution sense, as we will see in
Proposition 2.1.
In the case that the right hand side includes only the quadratic terms,
Toscani [6] showed the global existence of solutions for the data bounded,summable with weight $(1+|x|)^{\alpha}(\alpha>0)$ and with ‘globally’ finite entropy.
$(\mu r)$

2. Bounded data
To consider the solutions to the initial value problem (1.1), we introduce a
and aHk\’echet space $B_{\ell oc}(\mathrm{R}\cross[0, T])(T<\infty$ fixedBanach space, the former being introduced by Toscani [6].
$B(\mathrm{R}\cross[0, T])$$)$

[resp.
] the Banach
Definition.– We denote by[resp. R\’echet] space of classes of measurable functions $u=(u_{i}(x, t))_{i\in I}$ definedonsuch that the following norm [resp. semi-norm] is finite :$B(\mathrm{R}\cross[0, T])$

$B_{\ell oc}(\mathrm{R}\cross[0,T])$$\mathrm{R}\mathrm{x}[0, T]$(2.1)$||u||_{B}= \sum_{i\in I}\int_{\mathrm{R}}\mathrm{e}\mathrm{s}\mathrm{s}\sup_{0t\in[,T]}|u_{i}^{\#}(x,t)|dx$, $u_{i}^{\#}(x, t)=^{f}u_{i}(x+c_{i}t, t)de$ ,46[$resp.||u||_{B,K}= \sum_{\dot{l}\in I}\int_{K}\mathrm{e}\mathrm{s}\mathrm{s}\sup_{0t\in[,T]}|u!.(x,t)|dx,\forall K\omega mpad$

subset of R].

Proposition 2.1.– We assume the condition (dv). For[resp.
, we have,.
fresp.
$=\int_{0}^{t}(Q_{i}(u)+L_{i}(u))(x-\mathrm{q}.(t-s),s)ds$
We denote (Ku):(x, ). Then we obtain$u\in B(\mathrm{R}\mathrm{x}[0, T])$

$Q_{:}(u)$$u\in B_{\ell o\mathrm{c}}(\mathrm{R}\mathrm{x}[0,T])]$$L_{i}(u)\in L^{1}(\mathrm{R}\mathrm{x}[0,T])$$L_{\ell o\mathrm{c}}^{1}(\mathrm{R}\mathrm{x} [0,T])]$

$t$

,

(2.2)

$||Ku||_{B}\leq C^{st}(||u||\epsilon^{2}+T||u||_{B})$(2.3)$||Ku-Kv||_{B}\leq C^{st}||u-v||_{B}(||u||_{B}+||v||_{B}+T)$

Proof. It is crucial to suppose that the condition (dv) is verified.
We give aproof only for the case that $u\in B(\mathrm{R}\mathrm{x}[0,T])$ . Another casecan be proved similary. We take $u\in B(\mathrm{R}\mathrm{x}[0,T])$ and denote. Then, using
andfor
any $t\in[0,T]$ , we have
$U_{\dot{l}}(x)=$$U_{\dot{l}}\in L^{1}(\mathrm{R})$$\mathrm{e}\mathrm{s}\mathrm{s}\sup_{t\in[0,T]}|u_{\dot{1}}^{\#}(x,t)|$(2.4)$|u_{\dot{l}}^{\#}(x,t)|\leq \mathrm{U}\mathrm{i}(\mathrm{x})$$\int_{\mathrm{R}}\int_{0}^{T}Q:(u)dtdx\leq C^{st}\sum_{k\neq\ell}||U_{k}||_{L^{1}}||U\ell||_{L^{1}}\leq C^{st}||u||_{B}^{2}<\infty$(2.5)$\int_{\mathrm{R}}\int_{0}^{T}L:(u)dtdx\leq C^{st}T||u||_{B}<\infty$,.

Therefore we obtain(2.6)$\sum_{\dot{l}}\int_{\mathrm{R}}\mathrm{e}\mathrm{s}\mathrm{s}\sup_{0t\in[,T]}|(Ku)_{\dot{l}}^{\#}(x,t)|dx\leq C^{st}(||u||\epsilon^{2}+T||u||_{B})$

Similarly we have
(2.7)$\sum\int_{\mathrm{R}}\mathrm{e}\mathrm{s}\mathrm{s}\sup_{0t\in[,T]}|(Ku)_{\dot{1}}^{\#}(x,t)-(Kv)_{\dot{l}}^{\#}(x,t)|$$\leq.C^{st}||u-v||_{B}(||u||_{B}+||v||_{B}+T)|$.

$\iota$In order to prove the global existence for data with locally finite entropy, wedefine weak solution, so called mild solutions :Deflnition.– Let beandamild solution of the initial value problem (1.1) if$u_{\dot{l}}^{0}\in L_{\ell o\mathrm{c}}^{1}(\mathrm{R})$(2.8)$u\in B_{\ell o\mathrm{c}}(\mathrm{R}\mathrm{x}[0,T])$. We say that$u_{\dot{1}}^{\#}(x,t)=u_{\dot{l}}^{0}(x)+ \int_{0}^{t}\{Q!.(u)(x, s)+L_{\dot{l}}^{\#}(u)(x, s)\}ds$

$u$

is

.

, there is an equivalence between the
Remark :For bounded functions
notion of mild solutions and one of solutions in the distribution sense$u=(u_{\dot{l}})$47
We obtain, in aclassical way, the local results for bounded data :Proposition 2.2.– We suppose that the data are bounded.
(local existence) There exists aunique bounded solution at least up to the, we havetime $T_{0}=C/(1+||u^{0}||_{L}\infty)$ Furthermore, for$C$ depend only
where a constanton the system.
$ii)$ (uniqueness) If we have two bounded solutions on the same interval $[0, T]$ for
the same initial data, then they coincide on this interval. We can then dehne
as supremum of such that the solution exists and is
the existence timebounded up to the time .
$iii)$ (positivity) If the data are positive, the solution is also positive
uP to thetime$iv)$ (conservation of the mass) If the data are positive and summable, we have$i)$$t\leq T_{0}$$||u(\cdot,t)||_{L\infty}\leq$$C||u^{0}||_{L}\infty$$T$$T^{*}$$T$$T^{*}$

(2.9)

for

$\int_{\mathrm{R}}\sum_{i\in I}u_{i}(x,t)dx=\int_{\mathrm{R}}\sum_{i\in I}u_{i}^{0}(x)dx$$\forall t\in[0,T^{*}[$,.

(finite velocity propagation) If $two$ dataandcoincide in the interval$[a, b]$ , then the solutions
and coincide in the triangle or the trapezoid$u^{0}$$v)$$u$$v^{0}$

$v$

.
where
$(\mu
r)$
, then,(entropy with controlled increase) If we assume the condition
supported in $[-R, R]$ and with its finite entropy i.e.
for positive data
, we have $H(t)\leq H(\mathrm{O})+C_{*}t$ for $t\in[0,T_{1}$ ,depends only on the system and $H(t)= \sum_{i\in I}\int_{\mathrm{R}}u_{i}\log u_{i}(x, t)dx$ .
where$\{(x,t) :t\in[0, T_{1} [, T_{1}\leq T^{*}, a+\gamma t\leq x\leq b-\gamma t\}$$\gamma=\max_{i\in I}|c_{i}|$$v\mathrm{i})$$u^{0}$$\sum_{i\in I}\int_{\mathrm{R}}u_{i}^{0}\log u_{i}^{0}(x)dx<\infty$$T^{*}$$[$$T_{1}\leq$$C_{*}$

Proof. The classical iteration method enables us to show ) ). For details, werefer to [8].
On the increase of the entropy $H(t)$ , noting that the support of the solutions, we see thatwith respect to is contained in $[-R’, R’]$ with $R’=$$H(t)=\sum_{i}\int_{\mathrm{R}}u_{i}\logu_{i}(x,t)dx$
is well-defined. It follows from thethe quantity
system that$\mathrm{i}$$- \mathrm{v}$$u(\cdot, t)$(2.10)$7\mathrm{T}1$$x$$\sum_{i}(\frac{\partial}{\partial t}+c_{i}\frac{\partial}{\partial x})u_{i}\log u_{i}\leq-\sum_{ik}\alpha_{i}^{k}u_{k}\log\frac{u_{k}}{u_{i}}$.

Applying the Jensen’s inequality to the convex function $x\log x$ , we have$\int_{\mathrm{R}}u_{k}\log\frac{u_{k}}{u_{i}}dx$$= \int_{\mathrm{R}}\frac{u_{k}}{u_{i}}\log\frac{u_{k}}{u_{i}}\cdot u_{i}dx$(2.11)$\geq\int_{\mathrm{R}}u_{k}dx\cdot\log\int_{\mathrm{R}}u_{k}dx$

Integrating the inequality (2.10) on$H(t)-H(0)$$\mathrm{R}$

in

$x$

.

then between 0and$t$in , we have$t$$\leq-\sum_{ik}\alpha_{\dot{l}}^{k}\int_{0}^{t}d\tau\int_{\mathrm{R}}u_{k}dx\cdot\log\int_{\mathrm{R}}u_{k}dx$(2.12)$\leq t\underline{\sum_{ik}\alpha_{i}^{k}}$

$e$

48
because we have

4.

$x\log x\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$

3. Local existence
We show the time local existence for data with small mass :Theorem 3.1.– We assume the condition (dv). Then there exists a $\delta>0$, there exists aunique solution insuch that, for, and we have
. Fbrthermore, the mapping $G(t)$ :iscontinuous from the ball with radius 6of
into the ball with radius 25 of.
for$||u^{0}||_{L^{1}}\leq\delta$$u$$||u||_{B}\leq 2\delta$$B(\mathrm{R}\mathrm{x}[0, \delta])$$u^{0}\mapsto \mathrm{u}(\mathrm{x}, t)$$L^{1}$$L^{1}$$t\leq\delta$(x-c%t).
Proof. We write the system in the form $u-Ku=f$ , where
We put $u_{\nu+1}=Ku_{\nu}+f$, $u_{0}=0$ . By virtue of Proposition 2.1, for asufficiently
small 6, we haveand$f_{\dot{1}}$$=u_{\dot{l}}^{0}$$u_{\nu}\in B(\mathrm{R}\mathrm{x}[0, \delta])$$||u_{\nu}||_{B}\leq 2\delta$

(3.1)

,

$||Ku_{\nu}||_{B} \leq\frac{1}{2}||u_{\nu}||_{B}$

,

$\{$

$||Ku_{\nu+1}-Ku_{\nu}||_{\mathcal{B}} \leq\frac{1}{2}||u_{\nu+1}-u_{\nu}||_{\mathcal{B}}$.

converges to asolution
The fixed point theorem permits us to conclude that.
and we have
We suppose that is asolution for data
and we put $g:=v_{\dot{l}}^{0}(x-c_{i}t)$
Then we have $u-v=Ku-Kv+f-g$ . By virtue of the inequality (3.1), we
have $||u-v||_{\mathrm{B}} \leq\frac{1}{2}||u-v||_{B}+||f-g||_{B}$ therefore $||u-v||_{B}\leq 2||u_{0}-v_{0}||_{L^{1}}$ . It
implies the continuity of the mapping $G(t)$ .
$u_{\nu}$$u\in B(\mathrm{R}\mathrm{x}[0, \delta])$$||u||_{g}\leq 2\delta$$v^{0}$$v$$\bullet$, there is
Corollary 3.2.– We suppose the condition (dv) andbrought by Theorem 3.1.
a) (finite velocity propagation) The value $u(x, t)$ depends only on { $u^{0}(y)$ : $y\in$$[x-\gamma t, x+\gamma t]\}$. In particular, if the data are supported
whereis included in $[a-\gamma t, b+\gamma t]$ .
in $[a, b]$ , then the support of$b)$ (conservation of positivity) If the data are positive, then the solution is alSo
positive up to the time 6.
$c)$ If
is bounded, then is bounded up to the time 6and we$2(1+||u^{0}||_{L\infty})$ for.
$||u^{0}||_{L^{1}}\leq\delta$$\delta$$\gamma=\max:\in I|\mathrm{q}.|$$u(\cdot,t)$$u^{0}$$have||u(\cdot,t)||_{L^{\infty}}\leq$$u$$t\in[0, \delta]$

is determined by $\{u^{0}(y) :y\in[x-\gamma t,x+\gamma t]\}$ , then$u_{\nu+1}=Ku_{\nu}+f$ does also. As the solution,is alimit of the sequence$u(x,t)$ depends only on $\{u^{0}(y) : y\in[x-\gamma t,x+\gamma t]\}$ .
b) We approximate the data by the $u_{n}^{0}= \inf(u^{0},n)$ . Then the solution
which
sequence
corresponds to the data
and theexists inconverges toby the continuity of $G(t)$ . By virtue of Proposition 2.2, theare positive, then is positive

Proof. a) If$u_{\nu}(x,t)$$u$$u_{\nu}$$u_{n}$$u_{n}^{0}$$u(\cdot, t)$$u_{n}$$u$$B(\mathrm{R}\mathrm{x}[0, \delta])$$u_{n}(\cdot, t)$

49

.

,
6 the supremum of T such thatis
Let [0, T[ be with Tbounded up to the time T. Then, by Proposition 2.2, we have T $>0$ . For$\sup$. By the system, we havee $>0$ , we put Af.
$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT})$$\sup_{\mathrm{i}},$$\sup|u_{\ovalbox{\tt\small REJECT}}(\mathrm{z},$$\ovalbox{\tt\small REJECT}$$t\ovalbox{\tt\small REJECT}$$|\ovalbox{\tt\small REJECT}_{\mathrm{i}(\mathrm{L}\ovalbox{\tt\small REJECT}\cdot\ovalbox{\tt\small REJECT}}$$<\mathrm{o}\mathrm{o}$$\#\mathrm{E}[0,T-\mathrm{g}_{\ovalbox{\tt\small REJECT}}]\ovalbox{\tt\small REJECT}_{\mathrm{t}}x$$\leq||u^{0}||_{L}\infty+C^{st}M_{\epsilon}||u||_{B}+C^{st}||u||_{B}+M(T-\epsilon)$$\sup_{x}|u_{i}^{\#}(x, t)|$(3.3)$\leq C^{st}\delta M_{\epsilon}+C^{\epsilon t}\delta+||u^{0}||_{L}\infty$’for$t\in[0,T-\epsilon]$,

where constants depend only on the system. For $\delta<(2C^{st})^{-1}$ , we obtain(3.3)$M_{\epsilon}= \sup_{0t\in[,T-\epsilon]}\sup_{i,x}|u_{i}(x, t)|\leq 2(1+||u^{0}||_{L^{\infty}})$The right hand side being independent of , we have$\epsilon$(3.4)$M= \sup_{t\in[0,T]}\sup_{\dot{l},x}|u_{i}(x,t)|\leq 2(1+||u^{0}||_{L}\infty)$

be the existence time
This bound depends only on the initial data. Letof bounded solution which is associated by Proposition 2.2. If we had $T<\delta$, we would obtain a, taking as initial data the $u_{i}(x,T-\epsilon’)$ with
contradiction.
$T^{*}$$\epsilon’<T^{*}$$\mathrm{I}$be positive data in
Corollary 3.3.– We assume the condition (dv). Let.
for any
and areal number such that
witha) Then there exists aunique solution in
and $\gamma=\max_{i\in I}|c_{i}|$ Furthermore the value $u(x, t)$ depends only on { $u^{0}(y)$ :$u^{0}$$L^{1}$$\sum_{i\in I}\int_{a}^{a+h}u_{i}^{0}(x)dx\leq\delta$

$h$

$u$

$y\in[x-\gamma t, x+\gamma t]\}$$B(\mathrm{R}\cross[0, \theta])$$a\in \mathrm{R}$$\theta=\min\{\delta, h/\gamma\}$.

and we put $H(t)= \sum_{i\in I}\int_{\mathrm{R}}u_{i}\log u_{i}(x,t)dx$ . If
Assume the condition, wethe data are supported in $[-R, R]$ and verify
vvhich depends only on thehave, for $t\in[0,$ , $H(t)\leq H(0)+C_{*}t$ withsystem.
$b)$$(\mu r)$$\sum_{i\in I}\int_{\mathrm{R}}u_{i}^{0}\log u_{i}^{0}(x)dx<\infty$$C_{*}$$\ ]$Proof, a) If we restrict the initial data in $[a, a+h]$ , extending them by 0outside,

there exists asolution by Theorem 3.1. The restrictions of these solutions in
can be sticked
small triangles with base $[a, a+h]$ and with height
velocity
propagation.
together by virtue of the finiteb) For the calculus on the increase of $H(t)$ , we approximate the data by$\inf(u^{0}, n)$ . Then we have
which correspond to data
exist up to the time 0
Then the solutions
Furthermore, they are bounded and positive up to the time . Moreover,the solutions have their support in included in $[-R’, R’]$ with $R’=R+\gamma\theta$
Therefore the quantity $H_{n}(t)= \sum_{i}\int_{\mathrm{R}}u_{n,i}\log u_{n,:}(x, t)dx$ is well-defined. By
virtue of Proposition 2.2, we have $H_{n}(t)\leq H_{n}(0)+C_{*}t\leq H(0)+C_{*}t$ .
converge to the solution
Theorem 3.1. enables us to conclude that the. By extracting asub-sequence if necessary, theinfor eachconverges toalmost everywhere. By the fact that $u_{n,i}\log u_{n,i}(\cdot,t)$$\min\{\delta, h/\gamma\}$$u_{n}^{0}=$$\sum_{i}\int_{\mathrm{R}}u_{n,i}^{0}\log u_{n,i}^{0}(x)dx\leq\sum_{i}\int_{\mathrm{R}}u_{i}^{0}\log u_{i}^{0}(x)dx<\infty$$u_{n}^{0}$$u_{n}$$\theta$$x$$u_{n}(\cdot, t)$$u(\cdot, t)$$u_{n}(\cdot,t)$$L^{1}$$t\in[0, \theta]$$u(\cdot,t)$

50
are estimated from below

and that they are supported in afixed compact set, the Fatou’s lemma implies that
$H(t)$$\mathrm{b}\mathrm{y}-1/e$$= \sum_{\dot{l}}\int_{\mathrm{R}}u:\log u_{i}(x, t)dx$(3.5)$\leq\lim\inf\sum_{\dot{|}}\int_{\mathrm{R}}u_{\mathrm{n}},:\log u_{n},:(x,t)dx$. 1$= \lim\inf H_{n}(t)\leq H(\mathrm{O})+C_{*}t$

4. Global existence
In this section, we show the time global existence for the initial data withlocally finite entropy.
Proposition 4.1.– We assume the condition (dv).
a) We suppose that there exist two solutions and incorresponding to the summable and positive initial data which coincide in ainterval $[a, b]$ . Then the solutions coincide in the triangle or the trapezoid$u$$\{(x,t) ; t\in[0,T],a+\gamma t\leq x\leq b-\gamma t\}$

$B(\mathrm{R}\mathrm{x}[0,T])$$v$.

Let the initial data be supported in $[-R, R]$ , summable and positive. We
suppose that there exists asolution in. Then the support ofis included in $[-R-Ct, R+Ct]$ .
$b)$$B(\mathrm{R}\mathrm{x}[0, T])$$u$$u(\cdot,t)$

Proof. a) Let

be the infimum of such that $u(\cdot,t)\neq v(\cdot,t)$ . We have then
, there exists a such thatAs and are in$i\inI$:( , to)dx. Taking $h=\delta/(2q)$ , we have, forfor any$t$$t_{0}$$u(\cdot,t_{0})=v(\cdot,t_{0})\in L^{1}$$\int_{\{x:\mathrm{u}(x,t_{0})\geq q\}}:u$any$a\in \mathrm{R}$

(4.1)

$u$

$L^{1}$$v$$q$$<\delta/2$$x$,$\int_{a}^{a+h}u:(x,t_{0})dx$$\leq\frac{\delta}{2}+hq\leq\delta$, i$\in I$.

Using Corollary 3.3, are coincide in small triangles with base $[a,a+h]\mathrm{x}\{t=$and with height 0and we are led to acontradiction.
b) It is sufficient to apply the previous result toand 0as two initial data. 1
$u$

$v$

$t_{0}\}$$u^{0}$

lemma 4.2.– We assume the conditions (vd) and. Let $u(x, t)$ be apositive solution defined inwith its support in $[-R, R]$ . We supposethat $\sum_{:\in I}\int_{\mathrm{R}}u:\log u:(x, t)dx$ are estimated $kom$ above for any $t\in[0, T]$ by aconstant $C$ which does not depend on . Then, for any $\delta>0$ , there exists a, which depends only on $R$ , $C$ and , such that $\sum_{:\in I}\int_{a}^{a+h}u:(x,t)dx\leq\delta$ for
any a
and any t $\in[0,$ T].
$(/\mathrm{x}\mathrm{r})$$\mathrm{R}\mathrm{x}[0,T]$

$t$

$h$

$\delta$$\in \mathrm{R}$Proof. If not, for any $h>0$ , there would exist a
and a $t_{*}\in[0,T]$ such$\sum_{:}\int_{a}^{a.+h}.u:(x,t_{*})dx>\delta$. We use the argument owing to Toscani [6] andthat
Tartar-Craiidall [5]. We put, for $m\geq 1$ ,$a_{*}\in \mathrm{R}$(4.2)$B_{1,:}=${x $\in[a_{*},a_{*}+h]$:$u:(x,t_{*})\geq e^{m}\}$

51

(4.3)

$B_{2,i}=[a_{*},a_{*}+h]\backslash B_{1,i}$.

Then we would have$\int_{a_{*}}^{a_{*}+h}u_{i}(x, t_{*})dx$(4.4)$\leq\leq\frac{\frac{1}{m1}}{m}\int_{a_{*}}^{a.+h}.\cdot u_{\dot{1}}1\mathrm{o}\mathrm{g}^{+}u_{i}(y\int B_{1},u_{\dot{l}}1\mathrm{o}\mathrm{g}^{+}u_{i}(y,t_{*},)dy+he^{m}t_{*})dy+he^{m}$

On the other hand, we would have
$C$

(4.5)

$\geq\sum_{i}\int_{\mathrm{R}}u:\log u_{i}(y,t_{*})dy$$\geq\sum_{i}\int_{-R}^{R}(u_{i}\log^{+}u_{i}(y,t_{*})-1)dy$

where $p=\# I$ .

$= \sum_{i}\int_{-R}^{R}u_{i}\log^{+}u_{i}(y,t_{*})dy-2pR$

Therefore we would obtain$\delta$(4.6)$< \int_{\leq\frac{\sum_{1^{i}}}{m}(c}a_{*}u_{i}(y,t_{*})dy\leq\frac{1}{m}\sum_{i}\int_{a_{*}}^{a_{*}+h}a_{*}+hu_{i}\log^{+}u_{i}(y,t_{*})dy+phe^{m}+2pR)+phe^{m}$Choosing a $m$ such thatwould have

, then a such that, which is acontradiction.
$h$$\frac{1}{m}(C+2pR)<\frac{\delta}{4}$$\delta<\frac{\delta}{4}+\frac{\delta}{4}=\frac{\delta}{2}$$phe^{m}< \frac{\delta}{4}$, we$\mathrm{I}$Corollary 4.3.– We assume the conditions (vd) and (pr). We suppose thatthe initial data are supported in $[-R, R]$ , positive and with finite entropy and. Then we havethat there exists asolution in$B(\mathrm{R}\cross[0, T])$

(4.7)

$H(t)\leq H(0)+C_{*}t$for$\forall t\in[0, T]$

depends only on the system and where
where the constant$R$on the system, on and on .
$C_{*}$$J$

depends only$T$in is contained in $[-R’, R’]$ with $R’=R+\gamma T$ .
Proof. The support ofLet
be the infimum of such that the estimates does not hold at the time
, we have $H(t_{0}-\epsilon)\leq H(0)+C_{*}(t_{0}-\epsilon)\leq H(\mathrm{O})+C_{*}T$. Taking asmall
is positive and supported in $[-R’, R’]$ , theOwing to the fact that
previous lemma shows that there exists a independent of such that we have,. By virtue of Corollary 3.3, we,for anyhave$u(\cdot,t)$

$x$

$t$

$t$

$t_{0}$$\epsilon>0$$u(\cdot,t_{0}-\epsilon)$$h$$a\in \mathrm{R}$(4.8)$\epsilon$$\sum_{i}\int_{a}^{a+h}u:(x, t_{0}-\epsilon)dx\leq\delta$$H(t_{0}-\epsilon+\theta)$$\leq H(t_{0}-\epsilon)+C_{*}\theta$$\leq H(0)+C_{*}(t_{0}-\epsilon)+C_{*}\theta$.

52
The estimate is then verified uP to the timeof , which is acontradiction.
We state our main result :$t_{0}-\epsilon+\theta$

with$\theta>0$

independent$\epsilon$$\mathrm{I}$

Theorem 4.4.– We assume the conditions (dv) andFor the initial
data positive and with locally finite entropy, there exists aunique mild solution
to (1.1) defined on.
$(\mu r)$$\mathrm{R}\mathrm{x}[0,$$\infty[$

Proof. By virtue of Proposition 4.1, we are led to the case that the initial
data are supported in $[-R, R]$ and with finite entropy. Let
be the existence
time for these data. Suppose thatis finite. By Corollary 4.3, the entropy isbounded for $t<T^{*}$ : $H(t)\leq H_{T}<\infty$ . Owing to Lemma 4.2, the solution
being supported in $[-R-CT^{*},R+CT^{*}]$ , there exists a $h>0$ suchthat $\sum_{:}\int_{a}^{a+h}u:(x,t)dx\leq\delta$ for any
and $t<T^{*}$ . For $t<T^{*}$ , byvirtue of Corollary 3.3, applied to the initial data
, there exists a $\theta>0$
independent of such that the solution can be extended in. It issufficient to choose $t>T^{*}-\theta$ for reaching to acontradiction. 1$T^{*}$$T^{*}$$u(\cdot, t)$$a\in \mathrm{R}$$u:(\cdot,t)$$t$$\mathrm{R}\mathrm{x}[0,t+\theta]$

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