In this note we are examining again the mode1 proposed by S. Paveri-Fontana in(SJ and studied in various papers, in particu1ar [1) and (2].

- 2-

l. Introduction.

In this note we are examining again the mode1 proposed by S. Paveri-Fontana in(SJ and studied in various papers, in particu1ar [1) and (2].

The prob1em of evolution, connected with sl./cha mode1 is

u(x,v,w;O) = u.(x,v,w)

(1 )

( - - + d v dt

d d w-v

dX)U + dV ( T u) = ^F(u) xeR;t>O; v,we(v

1 ,v

2 ) - V (O<v

1 <v

2 <+=ì xeR;v,weV

u(x,v,w;t) = O -

t>O;xeR;v,w~V

where, i f f = f(x,v,w),

v ₂

Jlf - _J f(x,v,w' ) dw' v

v ₂

J / - I (v'-v)f(x,v' ,w) dv' v

q constant i n [O, 1J

J f =

3 I ^v (v-v' )f(x,v' ,w) dv' . v'

The meani ng of the symbo l s can be found

n [SJ ' [lJ and [2]. In [2], the prob1em (l) is studied

and bounded functions

when u be10ngs to the space of the uniform1y continuous X =U.C.B.(R) 3 and the existence and uniqueness of the 10ca1 (in time) stria solution is proved. Noted that u = u(x,v,w;t) ^lS a car density and that

+<o V 2

I ^dx I ^dv

-00

Vl

f V2 u(x,v,w;t)dt

vI

glVes the tota1 number of cars on the motorwòy at the time t, the most natural space to study the prob1em (l) is L 1

(R 3

). In [lJ , mollifying the non-linear part

of the equation, i.e. F, we obtainedthe existence and uniqueness of the global

- 3 -

strictsolution. Mo11ifyng, in our case, means rep1acing ^F with

where (4)

and

(K f

e )(x,v,w) _ J+a>

X

k (x' - x) f (x' ,v, w) dx '

e

(5 )

k eL (O,

^+00); k (y) ^> O; k (y) = O

E E - E if y , ^(D,e); J k~y)dy - 1 .

o

The aim of this work is to study the origina1 prob1em, i .e.(l) , ln L1 and to find the connexion between the solution u(t) of (1) and the solution

u (t) of the mo11ified prob1em.

e

Precisely we prove that if ^U

^€ L ^ln L

then (1) has a unique loca ^l "mild"

solution, i.e. the integra1 version of (1) has a unique local solution. If [D,t] is the existence time interva1 of such solution u(t), we ha ve

1im Ilu (t) - u(tHI = O

e:-+o+ e:

uniformlyrespectto t inCO,t). 11·11 lstheusualnormin L 1 .

We sha11 use ^the we11-known resu1ts of 1inear semigroup theory for which we refer to [4J ^Chapter 9. For the results on the non 1inear evo1ution equations

(in particular for semi-1inear ones) we refer to [3],[6] ^{and [8J.}

2. THE A8STRACT PROBLEM.

Denote x = (f =f(x,v,w); feL (R xV)} and ^{1 2-} X _o =(f;feX, f(x,v,x) = O a.e. if vé V} X

is a c10sed subspace of X and we use it to get the third relation

ln ^(1).

Defi ne

A f = v f - w-v f ⁺ ~ f

l x T v T

(6 )

f w-v ^f

x ⁺ T ^v