Uniqueness of Renormalized Solutions of Degenerate Elliptic–Parabolic Problems

journal of differential equations 156, 93121 (1999) Article ID jdeq.1998.3597, available online at http:www.idealibrary.com on

Uniqueness of Renormalized Solutions of Degenerate EllipticParabolic Problems Jose Carrillo* Departamento de Matematica Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain

and Petra Wittbold Departement de Matematique, Universite Louis Pasteur, 7, rue Rene Descartes, 67084 Strasbourg Cedex, France Received March 12, 1998

We consider a general class of degenerate ellipticparabolic problems associated with the equation b(v) t =div a(v, Dv)+f. Using Kruzhkov's method of doubling variables both in space and time we prove uniqueness and a comparison principle in L 1 for renormalized solutions.  1999 Academic Press

1. INTRODUCTION Let 0 be a bounded domain in R N with Lipschitz boundary if N2, T>0. Consider the quasi-linear ellipticparabolic problem

(EP)(v 0 , f )

{

b(v) t =div a(v, Dv)+f

on

Q= ]0, T [_0

v=0 b(v)(0, } )=b(v 0 )

on on

7= ]0, T [_0, 0

where (H1) b: R Ä R is a continuous, non-decreasing function, satisfying the normalization condition b(0)=0; (H2) a: R_R N Ä R N is a continuous vector field satisfying, for some 1k |v| k.

if

v0;

(ii)

for all h # C 1c (R), ` # D([0, T [_0),

(iii)

| `|

v

h(r) db(r)+` fh(v)=

t

Q

v0

|

a(v, Dv) } D(h(v) `)

(3)

Q

and, moreover,

|

Q & [n |v| n+1]

(a(v, Dv)&a(v, 0)) } Dv Ä 0

as

n Ä .

(4)

UNIQUENESS OF RENORMALIZED SOLUTIONS

95

Remark 1.2. (i) Note that in (3) and (4) each term is well-defined. Indeed, the first member of (3) is well-defined as | vv0 h(r) db(r)| &h& _ |b(v)&b(v 0 )| and b(v) # L 1(Q), b(v 0 ) # L 1(0). The term on the right-hand side of (3) has to be understood as

|

a(v, DT k(v)) } D(h(T k(v)) `)

(5)

Q & [ |v| 0 such that supp h/[&k, k]. Indeed, if supp h/[&k, k], then h(v)=h(T k(v)) and h(v)=0 a.e. on [|v| k]. Since T k(v) # L p(0, T; W 1,0 p(0)), it is the same for h(v) `, and D(h(v) `)=0 a.e. on [ |v| k]; due to the growth condition (1), a(v, DT k(v)) / [ |v| 0 sufficiently small, for some continuous function C = : R 2 Ä R +. Let us now state our main result. As usual, sign + denotes the multivalued function defined by sign +(r)=0 if r0 and we denote by sign + 0 its single-valued section which takes the value 0 in r=0. Theorem 2.3. Assume that (H1)(H2) and the additional condition (8) hold. For i=1, 2, let v 0i : 0 Ä R be measurable with b(v 0i ) # L 1(0),

98

CARRILLO AND WITTBOLD

fi # L 1(Q). Let v i be a renormalized solution of (EP)(v 0i , f i ), i=1, 2. Then there exists } # sign +(v 1 &v 2 ) such that, for a.e. 00 and almost every t>0, (b(v(t))&b(v(t&'))) h(v(t))

|

v(t)

h(r) db(r),

(32)

h(r) db(r)

(33)

v(t&')

(b(v(t))&b(v(t&'))) h(v(t&'))

|

v(t)

v(t&')

almost everywhere in 0 where, for t0, for any '>0, where, this time, for t0, the function ` (t)=1'  t&' `(s) ds belongs to W 1, (Q) & L p(0, T; W 1,0 p(0)), ` '(T)=0, hence ` ' is admissible in (27). According to (34), using similar rearrangements as above, for ' sufficiently small, we find

|

T

( G, ` ' ) dt

0

=

|

(` ' ) t (b(v(t))&b(v 0 )) Q

|

Q

1 (`(t)&`(t&'))(b(v(t))&b(v 0 )) '

T

=

0

1 & ' 

'

| |

`(t&')(b(v 0n )&b(v 0 ))

0

0

1 !(t&') Q '

|

1 & ' =

1 (b(v(t&'))&b(v(t))) `(t&') 0 '

| |

|

v(t&')

h(r) db(r)

v(t)

0

| | &'

!(t) h(v 0n )(b(v 0n )&b(v 0 )) 0

1 (!(t)&!(t&')) Q '

|

1 + ' 1 & '

0

| | &'

|

!(t) 0

v0n

|

v(t)

h(r) db(r) v0

h(r) db(r)

v0

0

| | &'

!(t) h(v 0n )(b(v 0n )&b(v 0 )). 0

Note that 1 '

0

| | &'

0

!(t)

|

v0n v0

h(r) db(r)=

|

!(0) 0

1 + '

|

v0n

h(r) db(r)

v0

0

| | &'

0

(!(t)&!(0))

|

v0n v0

h(r) db(r),

UNIQUENESS OF RENORMALIZED SOLUTIONS

119

and the last integral converges to 0 as ' Ä 0. In the same way we have 1 '

0

| | &'

!(t) h(v 0n )(b(v 0n )&b(v 0 ))

0

=

|

!(0) h(v 0n )(b(v 0n )&b(v 0 )) 0

+

1 '

0

| | &'

(!(t)&!(0)) h(v 0n )(b(v 0n )&b(v 0 )),

0

where again the last integral converges to 0 as ' Ä 0. Combining the last p three estimates, using the fact that ` ' Ä h(v) ! a.e. and in L p(0, T; W 1, 0 (0)) and remains uniformly bounded as ' Ä 0, we obtain

|

T

( G, h(v) !) dt 0

|

!t

Q

|

|

v

h(r) db(r)+ v0

|

!(0)

0

|

v0

n

h(r) db(r)

v0

!(0) h(v 0n )(b(v 0n )&b(v 0 )).

&

0

v

As n is arbitrary, b(v 0n ) Ä b(v 0 ) in L 1(0) and | v0n h(r) db(r)| sup |h|_ 0 |b(v 0n )&b(v 0 )|, passing to the limit with n Ä  in the preceding inequality yields (28) in the case h is non-increasing. It follows that (28) is satisfied for any h=h 1 _h 2 with h 1 , h 2 monotone; indeed, it is sufficient to apply the preceding result for monotone functions two times: first with v, b and G for h1 , then with v, B h1(r)= r0 h 1(r) db(r) and Gh 1(v) for h 2 . As any h # W 1, (R) may be approximated by convex combinations of these ``product'' functions, the assertion of the lemma follows. K Proof of Lemma 1.4. This result is actually a simple consequence of Lemma 4.2. Indeed, according to the assumptions, v satisfies both inequalities in (27) with G=&b(v) t and, therefore, by the result of Lemma 4.2, we obtain an equality in (28) in the case of non-negative h # W 1, (R), ! # D([0, T[ _R N ), hence by approximation for all non-negative ! # W 1, (Q) with p !(T )=0 and h(v)! # L p(0, T; W 1, 0 (0)). Due to the linearity in h and ` the assertion of Lemma 1.4 follows. K It remains to give the Proof of Proposition 1.3. (i) Let v be a weak solution of (EP). In particular, b(v) t =div a(v, Dv)+f in D$(Q), and thus b(v) t # L p$(0, T; W &1, p$(0))

120

CARRILLO AND WITTBOLD

+L 1(Q). Moreover, b(v)(0, } )=b(v 0 ) in D$(0). Then, for h # C 1c (R), ! # D([0, T[_0), by Lemma 1.4, T

| |

!t

0

0

|

v

|

T

h(r) db(r)=& v0

=

( b(v) t (t), !(t) h(v(t))) dt 0

|| a(v, Dv) } D(h(v) !)& fh(v) !,

where ( } , } ) denotes the duality pairing between W &1, p$(0)+L 1(0) and p  W 1, 0 (0) & L (0). Next, for n # N, let T n+1, n =T n+1 &T n . Again by Lemma 1.4, we obtain

| | 0

v(t)

T n+1, n(r) db(r)= v0

= =

|

T

( b(v) t (t), T n+1, n(v(t))) dt 0

||

Q & [n< |v|