On the Cauchy problem for systems of conservation laws

On the Cauchy Problem for Systems of Conservation Laws

Alb ertoBressan

URL: http://www.emath.fr/proc/Vol.3/

Resume

This paper contains an informal discussion of recent results concerning the well posedness of the Cauchy problem for a nonlinear system of conservation laws in one space dimension. The system is assumed to be strictly hyperbolic, with each characteristic eld either linearly degenerate or genuinely nonlinear.

1 The semigroup approach

This paper contains an informal discussion of recent results concerning the well posedness of the Cauchy problem for a nonlinear n n system of conservation laws in one space

dimension:

u

t+

F(u)

x= 0

(1)

u(0x) = u(x): (2)

The system is assumed to be strictly hyperbolic, with each characteristic eld either linearly degenerate or genuinely nonlinear 23, 28]. In this framework, a basic existence problem can be formulated as follows.

(EP1)

Show that there exists some (nontrivial) domainD

L

1 such that, for every initial

data u2 D, the Cauchy problem (1)-(2) has at least one global, entropy-admissible, weak

solutionu: 017!D.

We recall that, for smooth initial data u, the local existence of a solution is well known

27]. Due to genuine nonlinearity, however, this solution may lose its regularity. Indeed, its gradientu

xmay become innite within nite time 21]. For this reason, in general, solutions

can be constructed globally in time only within a space of discontinuous functions. A natural choice is the space

BV

of integrable functions u: IR7! IR

n with bounded variation. The

existence problem

(EP1)

was solved in the fundamental paper of Glimm20], on a domainD

consisting of functions with suitably small total variation. For future reference, we recall the construction of the positively invariant domainD. Consider a piecewise constant functionu,

say with jumps at the pointsx 1

<<x

N. For

= 1:::N, call 1

:::

n

the strengths

of the waves in the standard self-similar solution of the Riemann problem with datau(x

;), u(x

+). The total strength of waves in

uand the potential for future wave interactions are

dened respectively as

V(u) :

= N X

=1 n

X

i=1 j

i

j Q(u) :

=X A

i

j

(3)

S.I.S.S.A.,ViaBeirut4,Trieste34014Italy.

where the second sum ranges over all couples of approaching waves. One then denes

D :

=cl n

u2

L

1(

IR IR n)

uis piecewise constant, V(u) + 0

Q(u)< 0

o

(4)

for suitable constants 0

0

>0, where cldenotes closure.

Concerning uniqueness, we remark that, for smooth solutions, (1) is equivalent to the quasilinear system

u

t+ A(u)u

x= 0

: (5)

Here A(u) =D F(u) is the Jacobian matrix ofF atu. Ifuv are both solutions of (1), their

dierence w=u;v satises

w

t+ A

;

u(tx)

w

x+

D A(tx)w

v

x(

tx) = 0 (6)

with

D A(tx) :

=Z 1

0 D A

;

u(tx) + (1; )v(tx)

d :

Observe that (6) is a linear homogeneous hyperbolic system forw, with coecients depending

ontx. If the functionsuvare Lipschitz continuous, then these coecients have some degree

of regularity, which allows one to derive a priori estimates on the norm ofw. In particular,

fromw(0) = 0 we deducew(t) = 0 for allt>0, hence u v, proving uniqueness.

If the solutionsuvare discontinuous, however, the above approach is no longer valid, and

a proof of uniqueness becomes considerably more dicult. Progress in this direction has been achieved recently by a new approach, based on the preliminary construction of a semigroup of solutions of (1). Instead of tackling the uniqueness question directly, it is convenient to consider rst a more comprehensive existence problem.

(EP2)

Show that there exists a (nontrivial) domain D

L

1 and a continuous ow S : D017!Dwith the properties:

(i) S 0

u=u S s

S

t u=S

s+t u,

(ii) kS t

u;S

s

v kL ;

jt;sj+ku;v k

, (iii) every trajectoryt7!S

t

uis a weak, entropy-admissible solution of the partial dierential

equation (1).

Observe that (i) is the semigroup property, while (ii) states that the ow is globally Lipschitz continuous, both with respect to time and to the initial data.

The main dierence between the problems

(EP1)

and

(EP2)

is that, in the latter, we wish to construct solutions simultaneously for all u2 D, continuously depending on

the initial data. Establishing the existence of a semigroupSsatisfying (i){(iii) is thus a

more dicult task than proving the existence of one single solution. This was achieved in 9] for 22 systems, and in 11] for generalnnsystems. More precisely, we have

Theorem 1.

Let the system (1) be strictly hyperbolic in a neighborhood of the ori-gin, and assume that each characteristic eld is either linearly degenerate or genui-nely nonlinear. Then there exist a domain D as in (4) and a continuous semigroup S :D017!Dsatisfying the above conditions (i){(iii).

(iv) If u2Dis piecewise constant, then for allt>0 suciently small the functionu(t) = S

t

ucoincides with the solution of (1)-(2) obtained by piecing together the standard

self-similar solutions of the Riemann problems determined by the jumps in u.

One of the main results in 7] states that, for a given domainDof the form (4), there can

exists at most one semigroupS:D017!Dsatisfying (i), (ii) and (iv). In the positive

case, the condition (iii) is also satised. We then call S a Standard Riemann Semigroup

generated by the system (1).

Having established the existence and uniqueness of the semigroup, we are in a much better position to study the uniqueness of the solution to a particular Cauchy problem, and its dependence on the initial data. Indeed, the analysis relies on the following simple lemma.

Lemma 1.

LetS :D017!Dbe a continuous ow satisfying the properties (i)-(ii) in

(EP2)

. For every Lipschitz continuous mapw: 0T]7!Done then has the estimate

w(T);S T

w(0)

L 1

L Z

T

0 (

liminf

h!0+

w(t+h);S h

w(t)

L 1

h

)

dt: (7)

A proof will be given in Section 6.

Observe that the only relevant assumption of the lemma is the Lipschitz continuity of the Semigroup. No reference is here made to the particular equation (1). Our main application, however, will be in connection with the Cauchy problem (1)-(2). Let w = w(t) be any

approximate solution, Lipschitz continuous as a function of time, taking the correct initial conditionw(0) = u. If (iii) holds, the exact solution is then provided by the functiont7!S

t u.

At timeT, the

L

1distance between the approximate and the exact solution is estimated by

(7). Observe that the integrand in (7) determines the instantaneous error rate. The estimate thus states that these local errors are magnied at most by a factor ofL, and summed up

over the interval 0T]. The formula (7) has two main applications.

1.

Letu=u(tx) be a weak solution of (1), Lipschitz continuous as a function of time with

values in

L

1. If one can show that

liminf

h!0+

w(t+h);S h

w(t)

L 1

h

= 0 for a.e. t>0 (8)

then one can conclude thatw(t) =S t

w(0) for allt. Since we already know that the semigroup S is unique, this yields a new method for proving uniqueness of solutions to a given Cauchy

problem. This approach was pursued in 7], establishing the uniqueness of limit solutions obtained by the Glimm scheme 20, 24] or by wave-front tracking approximations 2, 5, 19, 26]. Using similar ideas, a more general uniqueness result for entropy-weak solutions was proved in 13], provided that the total variation ofualong segments in the t-xplane can be

suitably controlled.

2.

Let w=w(tx) be an approximate solution of (1) constructed by a wave-front tracking

method. For everyT >0, the distance betweenw(T) and the exact solutionS T

w(0) can then

be estimated by computing the integrand in (7). More precisely, assume that at a generic timetthe functionwis piecewise constant, with jumps located at pointsx

1(

t)<<x N(

t),

travelling with speed _x ,

= 1:::N. Call! =

!

(

tx) the self-similar solution of the

Riemann problem with dataw(tx

;),w(tx

+). Letting ^

speeds and choosingh 0

>0 suitably small, we can write

lim

h!0+

w(t+h);S h

w(t)

L 1

h

= N X

=1

1

h Z

^

h0

; ^

h0

w ;

t+h 0

x

( t) +y

;!

( h

0 y)

dy : (9)

In other words, the instantaneous error rate in (9) is measured by the dierences between the travelling fronts ofwand the exact self-similar solutions of the corresponding Riemann

problems. This provides an explicitly computable error bound for the wave-front tracking method 7]. By a more rened analysis, error estimates for the Glimm scheme were obtained in 15].

2 Examples of contractive metrics

By the remarks in the previous section, in order to establish the well posedness of the Cauchy problem (1)-(2), a key step is the construction of a Lipschitz semigroup of solutions. Toward this goal, one needs to construct a sequence of approximationsu

= u

(

tx), carefully

controlling:

(1)

The total variation Tot.Var.; u

( t)

of each approximate solution.

(2)

The distance

u

( t);v

( t)

L

1 between any two approximate solutions.

For small

BV

initial data, the Glimmscheme 20, 24] and the wave-front tracking algorithms 2, 5, 19, 26] provide control of the total variation, in terms of a wave interaction potential. Unfortunately, none of the algorithms found in the standard literature yields estimates on the dependence on the initial data. In particular, all convergence proofs are based on a compactness argument, which does not provide informations on the uniqueness of the limit. New constructive algorithms must therefore be devised.

A useful technique for estimating the distance between solutions is the introduction of a new distance d

, which is equivalent to the old distance and nonexpansive w.r.t. the ow

generated by (1). In connection with hyperbolic systems, this approach was rst introduced in 4], then extended in 9, 11] to more general systems. To familiarize the reader with this method, we collect here a few simple examples of dynamical systems

d

dt

u(t) = ;

u(t)

(10)

whose ow (ut)7!S t

uis non-expansive for a suitable distanced .

A well known class of evolution problems generating a contractive semigroup are those of the form

02 du

dt

+B(u) (11)

whereBis ahyperaccretiveoperator on a Banach spaceE(see Chapter 3 in 18] for details).

Scalar conservation laws, even in several space dimensions, t into this framework 17, 22]. In this case, the ow is contractive w.r.t. the standard distance d(uv) = ku;v k. Below,

we consider situations where the ow is contractive not for d, but only for some equivalent

distance d

Example 1.

On the domainD :

= ]01, consider the dierential equation

_

x=cx: (12)

Of course, the corresponding ow isS t

x=e ct

x. The Euclidean distancejS t

u;S

t

v jbetween

two solutions thus increases in time. However, one easily checks that this ow is non-expansive w.r.t. the distance

d

( xy)

:

= log(

x=y)

: (13)

Example 2.

Letbe a smooth scalar function satisfyingC ;1

(x)Cfor some constant C>1. Then the evolution equation

u

t+ (x)u

x= 0 :

generates a linear semigroup on the space

L

1, which is non-expansive for the distance

d

( uv)

:

=Z

u(x);v(x)

(x)

dx:

By the assumptions on, the aboved

is uniformly equivalent to the standard

L

1distance.

In other cases, the distance is dened in terms of a Riemann-type metric, taking the point of view of dierential geometry. More precisely, letDbe a closed domain in a Banach space E. The construction of a metricd

on

Dwill involve:

A dense subset D 0

D.

At each pointu2D

0, a space T

uof tangent vectors, equipped with a weighted norm k

v

k

u.

A familyRP of suitably regular paths : ab]7!D 0.

We assume that the familyRP is closed under concatenation of paths. Moreover, for

each 2RP, we require that the dierential

v

=D =d( )=d is a well dened tangent

vector inT

( ), for almost every

2ab]. The weighted length of a regular path can thus

be dened as

kk

:

=Z b

a

D ( )

( )

d : (14)

Let nowuw2D

0. Call

uw the family of all paths

2RP joininguwithw. By setting

d

( uw)

:

= inf kk

2 uw

(15)

one denes a distance on the dense subset D 0

D. By continuity,d

can then be extended

to the whole domainD.

At this stage, the relevant question is whether the metricd

is non-expansive in connection

with the ow of (10). Clearly, this is the case provided that, for every2RP, the weighted

length of the path S t

: 7! S t

;

( )

is a non-increasing function of time. To check this condition, for each solution u of (10) we consider the corresponding linearized evolution

equation for tangent vectors

d

dt

v

(t) =D ;

u(t)

v

(t): (16)

(A1)

For every solutionu=u(t) of (10) taking values insideD

0, and every solution

v

=

v

( t)

of the variational equation (16), the weighted norm of the tangent vector

t7!

v

( t)

u(t) (17)

is a non-increasing function of time.

Condition

(A1)

is \almost sucient" for the contractivity of the ow of (10). A formal proof goes as follows. Let uw 2D

0 and

" >0 be given. Call u(t) :

=S t

u, w(t) :

=S t

w the

trajectories of (10) with initial data uw, respectively. Choose a regular path: ab]7!D 0

joining uwith w, such that

kk

d

(

uw) +": (18)

Dene

u

(t) :

=S t

;

( )

v

(t) :

= @ @

S

t ;

( )

:

Each u

is thus a solution of (10), while

v

solves the corresponding linearized variational

equation (16). Using

(A1)

we now compute

d

;

u(t) w(t)

kS

t k

= R b

a

v

(

t)

u

(t) d

R

b

a

v

(0)

u

(0) d

d

(

uw) +":

(19)

Since " > 0 was arbitrary, this achieves the proof. In the previous argument, we tacitly

assumed that the regularity of every path is preserved by the ow:

(A2)

For every 2 RP and every t > 0, the pathS t

still belongs to the family RP of

regular paths.

In many cases involving nite dimensional evolution equations, the above assumption is trivially satised. In connection with the system of conservation laws (1), however,

(A2)

fails, due to the possible loss of regularity in piecewise Lipschitz solutions. This technical diculty will be addressed in Section 4.

We conclude this section with two more examples of ows which are contractive w.r.t. a Riemann-type distance.

Example 3.

For the evolution _x = cx considered in Example 1, introduce the Riemann

metric

k

v

k x

:

= 1

x j

v

j:

Along any solutionx=x(t) of (12), we now have

_

v

=c

v

d

dt k

v

(t)k

x(t)= 0 :

It is thus clear that the corresponding distance

d

( xy) =

Z

y

x

1

s ds

xy>0

is non-expansive for the ow of (12). Actually, we have just carried out an alternative construction of the same distance d

Example 4.

On the space E =

L

1

;

01]IR

L

1

;

01]IR

, consider the discontinuous evolution equation 8 < : u t(

x) =

meas y v(y)<u(x)

v

t(

y) =

meas x u(x)>v(y)

(20)

where are smooth functions satisfying

(s)>1 (s)<;1

d ds < d ds

< 8s201]

for some constant > 0. The corresponding ow is then contractive w.r.t. the weighted

distance d on

E, dened in terms of the Riemannian metric (

w

z

) (uv ) : = R 1 0

w

( x) exp

;meas y v(y)<u(x) dx +R 1 0

z

( y) exp

;meas x u(x)<v(y)

dx:

It is interesting to observe that, ifuvsolve (20), then the two functions

U(t) :

= meas x u(tx)>

V(t) :

= meas x v(ty)<

provide a solution to the linearly degenerate, strictly hyperbolic system

U

t+ (V)U

= 0

V

t+ (U)V

= 0 :

Another example of a Riemann-type metric onIR

n, contractive for the ow of a

discon-tinuous dierential equation, can be found in 8].

3 A contractive metric for hyperbolic systems

In this section we outline the consruction of a distance, equivalent to the

L

1metric, which

is contractive w.r.t. the ow generated by a system of conservation laws.

Consider the domainDof functions with small total variation, dened as in (4). LetD PL

be the dense subset of Piecewise Lipschitz functions.

We recall below the denition of generalized dierential of a path : ab]7!

L

1,

intro-duced in 14]. For anyu2

L

1, on the family

uof all continuous paths : 0

0] 7!

L

1such

that (0) =u, consider the equivalence relation

0 i lim

!0+

1

( ); 0( )

L 1 = 0

; 0 2 u : (21)

Now assume that u is piecewise Lipschitz, say with jumps at the points x 1

< < x

N.

The space of generalized tangent vectors at u is then dened as T u

:

=

L

1 IR

N. To each

(v )2T u, with

= ( 1

:::

N), we associate the path

(v u) 2

u dened by

(v u)( ) :

=u+ v+ X

<0

u(x +

) ;u(x

; ) x + x ] ; X >0

u(x +

) ;u(x

; ) x x + ] :(22)

More generally, we say that a path2

ugenerates the generalized tangent vector(

v )2T u,

if is equivalent to

(v u), under the relation (21).

In other words, for small values of , the function u

:

=( ) can be obtained fromuby

adding v and by shifting the positions of the jumps fromx to

x

+

. As

procedure yields a rst order approximation tou

, with an error

o( ) in the

L

1 norm, with

o( )= ! 0 as ! 0. In connection with the above dierential structure, one can dene

a kind of continuous dierentiability property for maps : 7! u

2

L

1, with piecewise

Lipschitz values. Following 14], we say that a map :]ab7!

L

1 is aregular path if there

exists an integerN such that:

(i) Every functionu

:

=( ) is piecewise Lipschitz continuous with jumps at pointsx

1 <

<x

N continuously depending on . Outside the jumps, each u

is continuous with a

Lipschitz constant Lindependent of . All functionsu

coincide outside some interval

;MM].

(ii) The map 7!u

xis continuous with values in

L

1.

(iii) There exists a continuous map 7!(v ) 2

L

1 IR

N such that for every

lim "!0+ 1 "

( +"); (v u )( ") L 1 = 0

: (23)

More generally, we say that a continuous map : ab]7!

L

1 is a piecewise regular path if

there exist points a= 0

<

1

<<

=

b such that the restriction of to each open

subinterval ] j;1

j is a regular path.

Now consider any u2D

PL, say with jumps at x

1

<<x

N. For every

= 1:::N, i= 1:::n, let

i

be the strength of the

i-th wave in the Riemann problem atx

Given a

generalized tangent vector (v )2T u=

L

1

IR

N, we dene its weighted norm as

( v )

u : = N X =1 n X i=1 j i jj jW u i ( x ) + n X i=1 Z 1 ;1 v i( x) W u i (

x)dx (24)

Here v i( x) : =l i ;

u(x)

v(x) is the i-th component ofv, whileQis the interaction potential

dened at (3). The weight functions W u

i are dened by

W u

i ( x)

:

= 1 + 1

R u

i( x) +

1

2

Q(u) (25)

R u i( x) : = h P ji R 1 x + P ji R x ;1 i u j x( y) dy+ P k i x >x +P k i x <x j k j + i if

x=x and

i

>0,

0 otherwise, (26)

for suitably large constants 1

2. Intuitively, R

u

i(

x) can be regarded as the total strength

of all waves inuwhich approach an innitesimali-shock located atx. Finally, let: 7!u

be a piecewise regular path dened on ab], and let (v

) be its generalized tangent vector

at u

. In analogy with (14), we dene the weighted lengthof as kk : =Z b a ( v ) u

d : (27)

The formula (15) now denes a weighted distance d on

D

PL. Due to the choice of the

weights in (25), it is not dicult to check thatd

is uniformly equivalent to the standard

L

1

4 Construction of the semigroup

In the previous section we dened a weighted distance d

on the domain

D in (13),

uniformly equivalent to the

L

1 distance. At this stage, we are still a long way from proving

that d

is contractive w.r.t. a semigroup generated by the system (1). Indeed, the very

existence of the semigroup still needs to be established. The main clue that d

may be contractive for the ow generated by the conservation

laws is the fact that

(A1)

holds. More precisely, let u=u(tx) be a weak solution of (1),

with u(t) 2 D

PL for every

t. Assume that all shocks in u are entropy-admissible and

interect at most two at a time. Then, in analogy with (16), the generalized tangent vectors

;

v(t) (t)

2T

u(t)satisfy a linearized evolution equation, say

d

dt

(v ) = " ;

u(t)

(v ): (28)

The precise form of " and the restarting conditions at times of shock interactions were derived in 14].

In this setting, for a suitable choice of the constants 0

iin (4) and (25), the main result

in 6] states that the map

t7!

;

v(t)(t)

u(t)

(29) is a non-increasing function of time, also in the presence of shock interactions.

An immediate consequence is the following. Let : 7! u

be a regular path of initial

data. Assume that, for each 2ab], the solutionu

of the corresponding Cauchy problem

remains regular (i.e., piecewise Lipschitz). Then the weighted length of the path S t

: 7! u

(

t) is a non-increasing function of time.

Exploiting the above results, we now try to construct a semigroup generated by (1). A naive attempt goes as follows. Start with a dense family of suciently regular solutions, say piecewise Lipschitz. For any two solutionsuw, and">0, consider a regular path

0joining u(0) withw(0), such that

k

0 k

d

;

u(0) w(0)

+":

Lett7!u (

t) be the solution of (1) with initial datau (0) =

0( ). For any

>0, consider

the path :

7!u (

). We then have

d

;

u() w()

k

k

k

0 k

d

;

u(0) w(0)

: (30)

Since " is arbitrary, this shows that the ow of regular solutions is contractive w.r.t. the

weighted distance d

. This ow can thus be extended by continuity to a unique semigroup S dened on the whole domainD.

This approach was carried out in 4], in connection with systems whose characteristic elds are linearly degenerate. In the general case, a major technical diculty arises. Indeed, piecewise Lipschitz initial data may lose their regularity in two ways:

The number of jumps becomes innite in nite time, due to repeated shock interactions.

The Lipschitz constant becomes innite, due to genuine nonlinearity.

If any one of the solutionsu

loses its regularity at some time

t>0, then for all >t

the length of

can no longer be computed by (14), and the whole estimate breaks down.

1.

The system of conservation laws (1) is approximated by an auxiliary evolution equation, not necessarily in conservation form. In particular, for the new system, shock and rarefaction curves will locally coincide.

2.

Given a family of solutionsu

, at the rst time

1when one or more of these functions loses

its regularity, a restarting procedure is performed. The path

1 : 7!u

(

1) is thus replaced

by a new regular path

1+, close to the old one and with almost the same length. The

corresponding solutions will remain regular up to a time 2

>

1when a second restarting is

performed, etc:::In a nite number of steps, a path of approximate solutions is constructed

on any given interval 0T]. The basic estimate (30) can still be recovered.

The rst approximation technique was introduced in 9]. Alone, it suces to handle the case of 22 systems. The restarting procedure was rst used in the paper 11], to which we

refer for all details.

Remarks.

WhenB is a hyperaccretive operator 18], the contractive semigroup generated

by (11) can be obtained as limit of the approximating ows

du

dt

+B (

u) = 0:

For each >0, the Lipschitz continuous operatorB

is here dened as

B

:

= ;1(

I;R

)

R

:

= (I+B) ;1

:

On the other hand, for semigroups which are contractive w.r.t. a Riemann-type distance

d

, no general approximation procedure is known.

An alternative attempt to construct the semigroup generated by a system of conservation laws is to extend the formula (27). Namely, one can consider more general paths: 7!u

,

with each u

not necessarily piecewise Lipschitz. A rst step in this direction was taken in

12] by introducing a notion ofshift dierential, for paths taking values in the wider class of

BV

functions.

5 Solutions with unbounded variation

Theorem 1 states the well posedness of the Cauchy problem (1)-(2) within a set of func-tionsu2

L

1 with small total variation. In 1, 10], semigroups of solutions were constructed

on domainsDcontaining also functions with large variation.

A natural question is whether these ows can be extended to

L

1functions, possibly with

unbounded variation. This was accomplished in 3, 16] for special classes of 22 systems.

The following example, however, shows that such a continuous extension is not possible for general nnsystems.

Example 5.

Consider the 33 strictly hyperbolic, linearly degenerate system

u

t+ u

x = 0 v

t ;v

x = 0 w

t+ ;

1;uv

6 w

x = 0

(31)

with initial data

where is any smooth function and

'(x) :

=

8

<

:

1 if 2;(2n+1)

<jxj<2 ;2n,

;1 if 2 ;(2n)

<jxj<2 ;2n;1.

(33)

Clearly we have

u(tx) ='(x+ 1;t) v(tx) ='(x;1 +t): (34)

To obtain the componentw, calls7!y(stx) the characteristic line through the point (tx),

i.e. the solution to the Cauchy problem _

y(s) =f ;

s y(s)

:

= 1;u(sy)v(sy)

6 (35)

y(t) =x (36)

withuvgiven at (34). Forst201 all these Cauchy problems have a unique solution and

the valuesy(stx) are well dened. Together with (34), the function

w ( )(

tx) :

= @y(0tx) @x

;

y(0tx)

(37) provides the unique solution to the Cauchy problem (31)-(32), on the strip 01IR.

Observe that the two functions

y(s) = s;1

3 y(s) = 0

are both solutions of (35) with initial data y(1) = 0. For any">0, applying the divergence

theorem on the domain #"

:

=n

(tx) t20 1;"] (t;1)=3x0 o

we obtain

Z

0

;"=3

w(1;"x)dx= Z

0

;1=3

(x)dx :

= 0

: (38)

If in (38) we have 0

6

= 0, then ast!1, the solutionw(t) has no limit in

L

1

loc. Its weak limit

is a measure having a mass

0 concentrated at the origin. We now show that the solution can

be prolonged to the strip 02]IRin innitely many dierent ways. Indeed, the components uvare always given by (34). To obtainwfor t212], we rst choose any smooth positive

function

0such that

0(

x) =(x) if x2=;1=3 0]

Z

0

;1=3

0(

x)dx= 0

: (39)

Callingw

0

the corresponding solution of (1) dened fort<1, we can now dene

w 0

(tx) :

= w

(

tx) if t<1, w

0

(2;t ;x) if 1<t2.

(40) Using the relations (39), since all characteristics starting at time t = 0 within the interval

;1=3 0] meet at the point (tx) = (10), one checks that the functionw 0

in (40) provides a solution to the Cauchy problem (31)-(32). Clearly,w

0

(2x) = 0(

;x). Dierent choices

of

6 Proof of Lemma 1

As a preliminary, observe that the integrand

(t) :

= liminf

h!0+

w(t+h);S h

w(t)

h

in (7) is measurable. Indeed, for every h>0, the functionf h(

t) :

=

w(t+h);S h

w(t)

is

continuous. By continuity of the mapsh7!f h(

t), we have

(t) = lim "!0+ inf h2Q\]0"] f h( t)

where the inmum is taken only over rational values of h. Hence is Borel measurable.

Next, let" >0 be given. To x the ideas, let M >L be a Lipschitz constant for w, so

that

w(t);w(s)

Mjt;sj ts20T]: (41)

By Lusin's theorem, there exists a compact setJ 0T] and a continuous functionsuch

that

meas(J)>T ;" Z

T

0

(s);(s)

ds<" (x) =(x) for all x2J: (42)

Fort20T], dene

%"( t) : =L Z t 0

"+(s)

ds+ 2MLmeas ;

0t]nJ

(43)

:

= supn

t20T]

S

T;t

w(t);S T w(0) % "( t) o :

We claim that =T. Indeed, assume the contrary. By continuity, at t= we have

S

T;

w();S T

w(0)

= % "(

): (44)

If 2=J, sinceJ is closed we can chooseh>0 such that +h]\J =. By (44) and (41)

this implies

S

T;;h

w( +h);S T w(0) S T;

w();S T w(0) + L

w(+h);S h

w()

%

"(

) +L2Mh

%

"( +h):

In the case 2J, by (42) and the continuity of, there existsh>0 such that

w( +h);S h

w()

h

() + "

2 min s2 +h]

(s) +":

By (44), this yields

S

T;;h

w( +h);S T w(0) S T;

w();S T w(0) + L

w(+h);S h

w()

%

"( ) +L

R

+h

(s) +"

ds

%

"( +h)

again in contradiction with the maximality of . Therefore, =T. Recalling (42) and the

denition of %", we deduce

w(T);S T w(0) % "( T) = R T 0

(t)dt+L"T+ 2MLmeas ;

0T]nJ

L R

T

0

(t)dt+L("+"T) + 2ML":

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