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Chapter 5 Stability and Exactness of Sequential and Weak Solutions for Polynomial Nonlinear PDEs
CHAPTER 5 S T A B I L I T Y AND EXACTNESS
OF SEQUENTIAL AND WEAK SOLUTIONS FOR POLYNOMIAL NONLINEAR POEs
0. I n t r o d u c t i o n The g e n e r a l i t y p r o p e r t i e s o f s e q u e n t i a l , i n p a r t i c u l a r weak o r d i s t r i b u t i o n s o l u t i o n f o r polynomial n o n l i n e a r PDEs b e i n g s e t t l e d i n t h e way s p e c i f i e d i n S e c t i o n 0, Chapter 3, t h e i n t e r p l a y between t h e s t a b i l i t y and exactness p r o p e r t i e s o f t h e mentioned s o l u t i o n s w i l l be d e a l t w i t h i n t h e p r e s e n t chapter. As s p e c i a l l y p o i n t e d o u t i n S e c t i o n 1, Chapter 1, one o f t h e b a s i c reasons f o r s t u d y i n g s e q u e n t i a l , weak o r d i s t r i b u t i o n s o l u t i o n s o f polynomial non= l i n e a r PDEs w i t h i n t h e framework o f q u o t i e n t algebras i s t h e n e c e s s i t y o f d e a l i n g w i t h t h e problem o f s t a b i l i t y o f these s o l u t i o n s . T h e r e f o r e , a p r i m a r y a t t e n t i o n w i t h i n t h e p r e s e n t c h a p t e r w i l l be g i v e n t o t h e s t a b i l i t y p r o p e r t i e s o f t h e s o l u t i o n s f o r n o n l i n e a r PDEs. We s h a l l o n l y deal w i t h t h e case o f chains o f q u o t i e n t a l g e b r a s ( 2 4 ) , Chap= t e r 3. The r e s u l t s p r e s e n t e d f o r t h a t case, can e a s i l y be r e f o r m u l a t e d f o r t h e chains o f q u o t i e n t a l g e b r a s ( 9 3 ) , Chapter 3, as i n d i c a t e d i n S e c t i o n 4. The r e s u l t s c o n c e r n i n g s t a b i 1 it y and exactness w i 11 be p r e s e n t e d on two l e v e l s . F i r s t , i n Sections 1-4, t h e problems w i l l be d e a l t w i t h on a gene= r a l l e v e l , i . e . r e s u l t s concerning t h e s t a b i l i t y and exactness o f any se= q u e n t i a l s o l u t i o n f o r any polynomial n o n l i n e a r PDE c o n s i d e r e d w i t h i n t h e chains o f q u o t i e n t a l g z a s ( 2 3 ) , Chapter 3, w i l l be presented. Then, i n S e c t i o n s 5 and 6, r e s u l t s concerning t h e s t a b i l i t y o f t h e weak s o l u t i o n s f o r t h e classes o f polynomial n o n l i n e a r PDEs s t u d i e d i n Chapter 4 w i l l be pre= sented.
1. B a s i c Facts and Remarks Concerning S t a b i l i t y Suppose g i v e n a C m - r e g u l a r i z a t i o n ( V , S ) . Then, t h e s t a b i l i t y p r o p e r t y o f s e q u e n t i a l weak o r d i s t r i b u t i o n s o l u t i o n s o f polynomial nonTinear PDEs, c o n s i d e r e d w i t h i n t h e framework o f t h e c h a i n s o f q u o t i e n t a l g e b r a s c o n t a i n = i n g t h e d i s t r i b u t i o n s (see ( 2 4 ) , Chapter 3)
(1)
A R ( V Y s ) = AR(V,S)/IR(V,S),
i s d i r e c t l y expressed b y t h e
(2)
IR(V,S), R
E
size o f
il E
8,
the ideals
R
t h a t i s , b e t t e r s t a b i l i t y means l a r g e r i d e a l s ( 2 ) , and i n e x t r e m i s , maximal s t a b i l i t y means maximal i d e a l s ( 2 ) .
163
164
E.E.
Rosinger
R e c a l l i n g now t h e way g i v e n i n (17-20), Chapter 3, t h e i d e a l s ( 2 ) were c o n s t r u c t e d , i t i s easy t o see t h a t these i d e a l s i n c r e a s e t h e i r s i z e , when= ever V o r S do so. However, g i v e n a C m - r e g u l a r i z a t i o n ( V , S ) , t h e s i z e o f S cannot be increased, i n view o f t h e f o l l o w i n g : Lemma 1 -~ If (V1,S1) and (V2,S2) a r e Cm-regularizations,
t h e n S1
C
S2 i m p l i e s S1 = S2.
Proof I t f o l l o w s e a s i l y from (21.2 , S e c t i o n 3, Chapter 3, which can a c t u a l y be w r i t t e n under t h e e q u i v a l e n t form m
S
=
vmQ S
0
Nevertheless, given a C m - r e g u l a r i z a t i o n ( V , S ) , t h e f o l l o w i n g changes i n S are possible. Lemma 2 I f ( V , S ) i s a C m - r e g u l a r i z a t i o n and S 1 i s a v e c t o r subspace i n Smsuch t h a t
then
(V,Sl)
i s also a
Cm-regularization.
Proof Obviously ( V , S l ) w i l l s a t i s f y (21.2) and (21.3) i n D e f i n i t i o n 1 , S e c t i o n 3, Chapter 3. Concerning t h e r e l a t i o n (21.1) i n t h e mentioned d e f i n i t i o n , we n o t i c e t h a t T ( V ) n SI c I ( V ) n = Z(V) n ( V
@ S)
(v
@ sl)
c Z(V)
ns
=
= Q
the l a s t i n c l u s i o n r e s u l t i n g from t h e f a c t t h a t V
c Z(V)
0
On t h e o t h e r hand, i n c r e a s i n g V i s a l s o n o t a t r i v i a l t a s k , i n view o f t h e r e l a t i o n (see ( 2 2 ) , Chapter 3 ) :
(3)
V C
#
V"
which i m p l i e s t h a t t h e maximum s i z e f o r V , g i v e n b y V", i s i n a c c e s s i b l e and we s h a l l have t o s e t t l e f o r V h a v i n g maximal s i z e s o n l y .
165
STABILITY AND EXACTNESS
Therefore, once again the necessit of dealing with various chains of quo= t i e n t algebras (24) or ( 9 3 ] d e d from the inexistence of canonical chains of the mentioned types, 2.
Maximal Stability
I n view of Lemma 1, we shall suppose given a fixed vector subspace S c Sm which s a t i s f i e s (see ( 2 1 . 2 ) and (21.3), Chapter 3) the conditions:
(4)
S" = V"
(5)
%"(n)
0
S
=
and study thg s t a b i l i t y of golutions as only spaces V c V which give C -regularizations such vector subspaces V , for instance V = p. Section 1, we shall be interested in maximal
depending on the vector sub= ( V , S ) . Obviously, there exist However, as mentioned in V.
Suppose now given a polynomial nonlinear PDE (6)
C ci(x) l ~ i Q h l < j < k i
and the corresponding PDO (7)
T(D) =
C
l < i < h
with the order (8)
m
=
max{ l p . . ] 1J
1
ci(x)
u(x)
DpiJ
=
f(x), x
E
il
.n
l < j < k i
1 < i < h, 1
Dpij
Q j Q
kil
We shall suppose that the PDE in ( 6 ) i s 0-smooth, that i s , c i , f (see Section 0 , Chapter 4 ) .
E
C"(n)
I t will be useful t o deal with the s t a b i l i t y of solutions for the PDE in ( 6 ) within the framework of the following rather general notion of solution.
Definition 1 A sequence of functions s E ( C m ( n ) )N i s caLled an S-sequential solution of the PDE i n (6), only i f there exists a C -regularization ( V , S ) , such that (9)
s E AR(V,S),
V R
E
W
-
u(f)
E
IR(V,S), 'd 1 E \
(10) w
=
T(D)s
The meaning of the above notion of solution results from:
-Proposition 1 N If s E ( C " ( 0 ) ) i s an 2-sequential solution o f the m-th order PDE in (6), then, for a certain C -regularization ( V , S ) , the following properties hold: 1) S
=
s
t l'l(V,S)
E A'(V,S),
Y 1E
fl
166
2)
E . E . Rosinger
S satisfies th plication i n A
i n t h e usual a l g e b r a i c sense, w i t h R (PDE U S )i nand(6), the p a r t i a l d e r i v a t i v e operators
Dp: A'(U,S)
Ak(VS)
+
,
p
ENn,
IpI
my with R , k g
m,
multi=
R-k> m
Proof I t f o l l o w s e a s i l y from ( 9 ) , ( 1 0 ) above and Theorems 2 and 3, S e c t i o n 3, Chapter 3.
n
The g e n e r a l i t y o f t h e above n o t i o n o f s o l u t i o n r e s u l t s from: Proposition 2 Ifa d i s t r i b u t i o n S E P ' ( R ) \ C m ( n ) i s a Cm-regular weak s o l u t i o n o f t h e PDE i n ( 6 ) , the! t h e r e e x i s t s a sequence o f f u n c t i o n s s E S and a v e c t o r sub= space S C S s a t i s f y i n g ( 4 ) and (5), such t h a t
(11)
s
(12)
s i s an S - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 )
=
*>
Proof I t f o l l o w s e a s i l y from (71-73), Chapter 4.
NOW, i n o r d e r t o p r e s e n t t h e r e s u l t on t h e maximal s t a b i l i t y o f S-sequential s o l u t i o n s o f t h e PDE i n ( 6 ) , i t i s convenient t o i n t r o d u c e t h e f o l l o w i n g d e f in i t i on. Definition 2
A sequence o f f u n c t i o n s s E ( C ~ ( R ) )i ~s c a l l e d a maximally s t a b l e S-sequen= t i a l s o l u t i o n o f t h e PDE i n ( 6 ) , o n l y i f t h e r e e x i s t s a c m - r e g u l a r i z a t i o n (V,s),w i t h V maximal, and s a t i s f y i n g ( 9 ) and ( 1 0 ) . Theorem 1 Any s - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 ) i s maximally s t a b l e . Proof Denotemby RV t h e s e t o f a l l t h e v e c t o r subspaces V C Vm f o r which (V,s) i s a C - r e g u f a r i z a t i o n . We s h a l l show t h a t R Vs i s c b a i n complete. Indeed, i n view o f (21.1), Chapter 3, a v e c t o r subspace U CmV belongs t o RVS , o n l y i f I( V ) n S = Q , where I( V ) i s t h e i d e a l i n ( C (f2))N g e n y a t e d b y V. But, i t i s easy t o see t h a t I( V ) i s t h e v e c t o r subspace i n ( C (f2))N generated b y V. (Cm(n))N. Therefore, RVS w i l l indeed be c h a i n complete. NOW, i n view o f ( 9 ) and ( l o ) , as w e l l as Z o r n ' s Lemma, t h e p r o o f i s com= pleted.
0
Remark 1 R e s u l t s on t h e s t r u c t u r e o f t h e maximal v e c t o r subspaces V w h i c h d e f i n e t h e m a x i m a l l y s t a b l e s e q u e n t i a l s o l u T E 6 - 3 n d whose e x i s t e n c e i s g r a n t e d by Theo= rem 1, w i l l be presented i n t h e n e x t chapter.
STABILITY AND EXACTNESS 3.
167
I n t e r p l a y between S t a b i l i t y and Exactness
As seen i n Sections 4 and 5, Chapter 1, t h e exactness p r o p e r t y o f sequen= t i a l , weak o r d i s t r i b u t i o n s o l u t i o n s f o r polynomial n o n l i n e a r PDEs c o n s i = dered w i t h i n t h e framework o f t h e c h a i n s o f q u o t i e n t a l g e b r a s ( 2 4 ) , Chapter size o f the ideals 3, o r ( 1 ) above, i s d i r e c t l y expressed by t h e -
(13)
ZR(V,S) , R
E
R
E
h'
and algebras (14)
A'(Y,S)
,R
t h a t i s , b e t t e r exactness means s m a l l e r i d e a l s ( 1 3 ) and l a r g e r a l g e b r a s ( 1 4 ) . I n e x t r e m i s , maximal exactness means minimal i d e a l s ( 1 3 ) and maximal a l g e = bras (14). NOW, i n view o f (17-20), Chapter 3, where t h e i d e a l s ( 1 3 ) and a l g e b r a s ( 1 4 ) were d e f i n e d , i t i s easy t o see t h a t t h e i d e a l s (13) decrease i n t h e i r s i z e t o g e t h e r w i t h V , w h i l e i n a c o n f l i c t i n manner, t h e a l g e b r a s ( 1 4 ) i n c r e a s e t h e i r size together w i t h V . a t concerns S , i t was seen i n Lemma 1, Sec= t i o n 1, t h a t n e i t h e r i n c r e a s e n o r decrease i s p o s s i b l e .
+
More p r e c i s e l y , suppose t h e PDO i n ( 7 ) i s considered t o a c t as f o l l o w s (see p c t . 2 and 4, Remark 3, S e c t i o n 3, Chapter 3): T(D) : AR(V,S)
+
Ak(V,S)
w i t h 2 , k E 4 , k < R , R-k > m. Then, maximal s t a b i l i t y f o r t h e s e q u e n t i a l , w&ak o r d i s t r i b u t i o n s o l u t i o n s o f t h e m)E i n (6)maximal ideals 7 ( V , S ) , t h a t i s , maximal V . On t h e p t h e r s i d e , maxima7 exactness f t h e same s o l u t i o n s m e a m m a 1 i d e a l s Z ( V , S ) , and maximal a l g e b r a s A ( V , S ) , t h a t i s , i n t h e same t i m e minimal and maximal V .
R
The way t h e above c o n f l i c t i s s e t t l e d r e q u i r e s i n each p a r t i c u l a r case o f s e q u e n t i a l , weak o r d i s t r i b u t i o n s o l u t i o n considered f o r t h e PDE i n ( 6 ) , a s p e c i f i c a n a l y s i s o f t h e s t a b i l i t y and exactness i n t e r e s t s i n v o l v e d . 4.
Remarks i n t h e F i n i t e l y Smooth Case k2 Suppose, i h e PDE i n ( 6 ) i n C -smooth, f o r a c e r t a i n g i v e n k2 E N, t h a t i s , c., f E C 2(Q). We s h a l l be i n t e r e s t e d i n t h e s t a b i l i t y o f s e q u e n t i a l s a l u t i o n o f t h e in-order PDE i n ( 6 ) , g i v e n b y sequences o f f u n c t i o n s s €(Ck1(Q))N, w i t h k l E N, kl > m. The framework w i l l be g i v e n t h i s t i m e by t h e chains o f q u o t i e n t algebras ( 9 3 ) , Chapter 3. More p r e c i s e l y , we s h a l l c o n s i d e r t h a t t h e PDO i n ( 7 ) a c t s as f o l l o w s (see Remark 5, S e c t i o n 5, Chapter 3): T(D) : A'( V,S) * Ak( V , S )
where ( V , S ) i s a C " - r e g u l a r i z a t i o n , k Q min {kl-m,k23.
w h i l e R, k
E
N, R-k > m y "kl,
I t i s easy t o see t h a t t h e remarks i n S e c t i o n 1, remain v a l i d , under t h e i r corresponding form, f o r C" - r e g u l a r i z a t i o n s and t h e c h a i n s o f q u o t i e n t a l = gebras ( 9 3 ) , Chapter 3, w h i c h t h e y generate.
168
E.E.
Rosinger
Therefore, given a v e c t o r subspace S c So which s a t i s f i e s t h e c o n d i t i o n s
so
(15)
v"
=
@S
Uco(n)
y
cs
kl N we c a l l a sequence o f f u n c t i o n s s E ( C (a)) an S - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 ) , o n l y i f t h e r e e x i s t s a C " - r e g u l a r i z a t i o n ( V , S ) such that
, tf
A'(v,s)
(16)
s
(17)
w = T(D)s-u(f)
E
R E N, R G kl E
k I (V,S), V k
E
a-k
N,
> m,k < min{kl-m,k2}
I n t h a t case, one o b v i o u s l y o b t a i n s t h a t S = s
+ I
R
( V , S ) E A ( V , S ) , U R E N, R
< kl
and S s a t i s f i e s t h e PDE i n ( 6 ) , i n t h e usual a l g e b r a i c sense, w i t h t h e m u l t i p l i c a t i o n i n Ak( V , S ) and t h e p a r t i a l d e r i v a t i v e o p e r a t o r s
DP : A ~ ( v , s ) f o r R, k
E
N, R-k
+
> my a
A ~ ( v , s ) , ~E Nn , I P I
Q
kl,
Q
my
k G min{kl-m,k21
k Moreover, ifa d i s t r i b u t i o n S E P ' ( R ) \ C " ( R ) i s a C ' - r e g u l a r weak s a l u t i o n o f t h e PDE i n ( 6 ) , then, t h e r e e x i s t s a sequence o f f u n c t i o n s s E S and and a v e c t o r subspace S c So s a t i s f y i n g (15), such t h a t S = < s, s i s an S - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 ) . kl N Now, c a l l i n g a sequence o f f u n c t i o n s s E ( C (n)) a maximally s t a b l e S - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 ) , o n l y i f t h e r e l a t i o n s (16) and ( 1 7 ) h o l d f o r a c e r t a i n C " - r e g u l a r i z a t i o n ( V , S ) , w i t h V maximal, one can e a s i l y o b t a i n t h e corresponding v e r s i o n o f Theorem 1. F i n a l l y , i t i s easy t o see t h a t t h e remarks i n S e c t i o n 3 can a l s o be adap= t e d t o t h e case o f f i n i t e smoothness. 5.
S t a b i l i t y o f Regular Weak S o l u t i o n s
Two general classes o f weak s o l u t i o n s f o r polynomial n o n l i n e a r PDEs were s t u d i e d i n Chapter 4 t i r s t , i n S e c t i o n 4, Chapter 4, t h e r e g u l a r weak s o l u t i o n s f o r a r b i t r a r y polynomial n o n l i n e a r PDEs were i n t r o d u c e d as an e x t e n s i o n o f t h e p i e c e w i s e smooth weak s o l u t i o n s o f s i m p l e polynomial n o n l i n e a r PDEs s t u d i e d i n S e c t i o n s 1-3, Chapter 4. Then, i n S e c t i o n 6, Chapter 4, t h e weak s o l u t i o n s w i t h d i s c o n t i n u i t i e s across hypersurfaces were s t u d i e d i n t h e case of r e s o l u b l e systems o f polynomial n o n l i n e a r PDEs, systems e x t e n d i n g t h e t y p e o f e q u a t i o n s o f magnetohydrodynamics and general r e l a t i v i t y , s t u d i e d i n S e c t i o n 5, Chapter 4.
.
The r e s u l t s on t h e mentioned weak s o l u t i o n s presented i n Chapter 4, con= cerned t h e r e s o l u t i o n o f t h e i r s i n g u l a r i t i e s and were o b t a i n e d b y construe= t i n g a r t i c u l a r i n s t a n c e s o f t h e chains o f q u o t i e n t algebras (24) o r (93), C h a p t b 3 T 5 T u c h a way t h a t t h e weak s o l u t i o n s s a t i s f i e d t h e polynomial n o n l i n e a r PDEs i n t h e usual a l g e b r a i c sense, w i t h t h e m u l t i p l i c a t i o n and p a r t i a l d e r i v a t i v e o p e r a t o r s i n t h e c h a i n s o f q u o t i e n t algebras.
STABILITY AND EXACTNESS
169
The aim o f t h i s s e c t i o n i s t o p r e s e n t a b a s i c s t a b i l i t y r e s u l t c o n c e r n i n g t h e r e g u l a r weak s o l u t i o n s f o r a r b i t r a r y p o l y n o m i a l n o n l i n e a r PDEs by con= s t r u c t i n g l a r g e classes o f c h a i n s o f q u o t i e n t a l g e b r a s ( 2 4 ) o r ( 9 3 ) , Chap= t e r 3, e x t e n d i n g those p a r t i c u l a r ones used i n Chapter 4. I n t h i s way, t h e r e s u l t s c o n c e r n i n g t h e r e s o l u t i o n o f s i n g u l a r i t i e s p r e s e n t e d i n Chapter 4 w i l l a l s o be improved. We s h a l l o n l y deal w i t h t h e case o f i n f i n i t e smoothness. The r e s u l t s o b t a i n e d can e a s i l y be r e f o r m u l a t e d i n t h e f i n i t e l y smooth case.
A s i m p l e and u s e f u l t o o l f o r e s t a b l i s h i n g t h e mentioned s t a b i l i t y p r o p e r t y o f t h e r e g u l a r weak s o l u t i o n s i s t h e s u f f i c i e n t c o n d i t i o n on t h e s e s o l u = t i o n s p r e s e n t e d i n Tbeorem 9, S e c t i o n 4, Chapter 4, as w e l l as t h e s u f f i = c i e n t c o n d i t i o n on C , R - r e g u l a r i d e a l s presented i n Theorem 20, S e c t i o n 10, Chapter 3. Supposg t h e PDE i n ( 6 ) i s Cm-smooth and t h e d i s t r i b u t i o n S E U'(Q)\C"(Q) i s a C - r e g u l a r weak s o l u t i o n f o r t h e PDE i n ( 6 ) which admits a sequence S" s a t i s f y i n g t h e c o n d i t i o n s o f functions s
(18)
s
=
(19)
w
=
(20)
lW i s a v a n i s h i n g i d e a l i n (C"(Q))
(21)
(v"
< 5, ' >
T(D)s
-
u(f)
E
t zw)n(ucm(n)
V"
+ R.S) 1
=
N
o_
Chapter 4 )
where (see ( 7 3 . 2 ) ,
lw = t h e i d e a l i n (C"(Q))N generated by IDP,
(22)
Ip E Nnl
Then, we denote by iDs t h e s e t o f a l l v a n i s h i n g i d e a l s 7 i n (Cm(Q))N, such t h a t
(23)
lw c 7
(24)
(vm +
i)n(ucm(n) + R.S) 1 =
2
Theorem 2 Under t h e above c o n d i t i o n s , t h e f o l l o w i n g p r o p e r t i e s h o l d :
1)
The s e t of i d e a l s i D s i s c h a i n complete.
2)
Each i d e a l 1 E i D s i s Cm, R - r e g u l a r , where R =
3)
F o r each i d e a l I E i D s t h e r e e x i s t C m - r e g u l a r i z a t i o n s
(25)
(Invm>@)
satisfying the conditions
1
R s.
170
E . E . Rosinger
4)
Each Cm-regula r i z a t i o n (V,S')win (25) y i e l d s t h e following resolu= t i o n of s i n g u l a r i t i e s f o r t h e C -regular weak solution S of t h e PDE in ( 6 ) :
4.1)
S = s + l R ( V , S ' ) E AR(V,S'), \d R E 1, where s i s given i n (18), s E S ' and does
4.2)
S s a t i s f i e s t h PDE in ( 6 ) in t h e usual a l g e b r a i c sense, with multi= p l i c a t i o n in A ( V , S ' ) and the p a r t i a l d e r i v a t i v e operators
not depend
on R
R
Dp : A"(V,S')
+
Ak(V,S'), p
E
N n y IpI Q m , w i t h II, k E
R,
R-k am
Proof 1)
I t follows e a s i l y from (23) and ( 2 4 )
2 ) and 3 ) follow from the s u f f i c i e n t condition on Coo, R-regular i d e a l s presented in Theorem 20, Section 10, Chapter 3.
4)
I t follows from 2 ) and 3 ) i f account i s taken of Theorem 7 , Section 4, Chapter 4.
Remark 2
+
From the point of view of t h e s t a b i l i t of the (?-regular weak solution S , Theorem 2 above presents a specia i n t e r e s t . Indeed, the s e t of ideals iD, being chain complete, t h e application of Zorn's lemma w i l l y i e l d t h e fol 1 owing property \d I E
-
iDs :
1 E iD, maximal:
ic 1 Then, t h e cha ns of quotient algebras
see ( 2 4 ) , Chapter 3 )
AR
associated t o l a r e gr maximal i d e a l s E iDS will o f f e r b e t t e r s t a b i l i t y s o l u t i o n S. properties f o r t e C -rG@TFweak
-8
6.
S t a b i l i t y of Solutions with Jump Discontinuities f o r Resoluble Systems of PDE S
Suppose t h e m-th order Coo-smooth polynomial nonlinear system of PDEs in (89) Chapter 4 , i s resoluble ( s e e (149),Chapter 4 ) .
STABILITY AND EXACTNESS Suppose u ,ut : R system of-PDts.
Ra a r e two
+
cm-smooth s o l u t i o n s o f t h e mentioned
F i n a l l y , suppose t h a t t h e f u n c t i o n u : R
+
U ( X ) = u-(x)
(31)
171
-
(u+(x)
+
Ra d e f i n e d b y
u-( x ) ) H ( x ) , x E 0,
where H i s t h e H e a v i s i d e f u n c t i o n ( 9 4 ) , Chapter 4, a s s o c i a t e d w i t h t h e C -smooth h y p e r s u r f a c e r i n ( 9 2 ) , Chapter 4, i s a weak s o l u t i o n o f t h e mentioned system o f PDEs, t h e r e f o r e s a t i s f y i n g on r t h e necessary and s u f = f i c i e n t j u n c t i o n c o n d i t i o n s (108) o r (151), Chapter 4. We s h a l l a l s o assume t h a t u- and u+ a r e em-independent on 5, Chapter 4 ) .
r
(see S e c t i o n
Then s = ( S ~ , . . . , S , ) E ( ~ " ) ~ c o n s t r u c t e d i n ( 9 9 ) , Chapter 4, has t h e proper= tY
(32)
ua =
(33)
wB = TB(D)s
,a>
-
, V 1< a < u(fB)
a
v",
E
V 1< B
<
b
We s h a l l denote 1 , = the ideal i n
(34)
(c"(R)) N generated b y I D P w B l 1 G B
Q
b,p
E
F i n a l l y , we denote b y JDS
t h e s e t o f a l l v a n i s h i n g i d e a l s I i n ( c " ( R ) ) ~ , such t h a t (35)
7,
(36)
(v" + r ) n
c 1
+
(uCm((,)
1
R.S) =
2
I n view o f t h e p r o o f o f Theorem 13, S e c t i o n 6, Chapter 4, i t i s easy t o see t h a t
JD, # 4 Theorem 3 Under t h e above c o n d i t i o n s , t h e f o l l o w i n g p r o p e r t i e s h o l d :
1)
The s e t o f i d e a l s JD, i s c h a i n complete.
2)
Each i d e a l 7
3)
F o r each i d e a l 7
(37)
(rn
E
JD, E
i s Cm JD,
, R-regular,
1
R = R s.
there e x i s t Cm-regularizations
7 , SOT)
satisfying the conditions
where
N n}
E.E.
172
Rosinger
(38) (39)
(40) (41) 4)
4.1) 4.2)
Each C m - r e g u l a r i z a t i o n ( V , S ' ) i n (37) y i e l d s t h e f o l l o w i n g r e s o l u = t i o n o f s i n g u l a r i t i e s f o r t h e weak s o l u t i o n u o f t h e system o f r e = s o l u b l e PDEs i n ( 8 9 ) , Chapter 4: R R uU = sa + I ( L S ' ) E A ( V , S l ) , 11 1 S a w i t h 1 S a Q a, do __ n o t depend on R .
Q
a,
R E
fl, where sa
E S',
u s a t i s f i e s each o f t h e PDEs o f t h e system ( 8 9 ) , Chapter 4, i n t h e usual a l g e b r a i c sense, w i t h t h e m u l t i p l i c a t i o n i n Ak(V,Sl) and t h e p a r t i a l d e r i v a t i v e operators
Dp : AR(V,Sl)
+
Ak(V,Sl),
p
E
Nny I p I S m y w i t h R , k
E
fly a-k
2 m
Proof S i m i l a r t o t h e p r o o f o f Theorem 3, S e c t i o n 5. Obviously, a corresponding v e r s i o n o f Remark 2, S e c t i o n 5, w i l l a p p l y t o Theorem 3 above.
0
OF SEQUENTIAL AND WEAK SOLUTIONS FOR POLYNOMIAL NONLINEAR POEs
0. I n t r o d u c t i o n The g e n e r a l i t y p r o p e r t i e s o f s e q u e n t i a l , i n p a r t i c u l a r weak o r d i s t r i b u t i o n s o l u t i o n f o r polynomial n o n l i n e a r PDEs b e i n g s e t t l e d i n t h e way s p e c i f i e d i n S e c t i o n 0, Chapter 3, t h e i n t e r p l a y between t h e s t a b i l i t y and exactness p r o p e r t i e s o f t h e mentioned s o l u t i o n s w i l l be d e a l t w i t h i n t h e p r e s e n t chapter. As s p e c i a l l y p o i n t e d o u t i n S e c t i o n 1, Chapter 1, one o f t h e b a s i c reasons f o r s t u d y i n g s e q u e n t i a l , weak o r d i s t r i b u t i o n s o l u t i o n s o f polynomial non= l i n e a r PDEs w i t h i n t h e framework o f q u o t i e n t algebras i s t h e n e c e s s i t y o f d e a l i n g w i t h t h e problem o f s t a b i l i t y o f these s o l u t i o n s . T h e r e f o r e , a p r i m a r y a t t e n t i o n w i t h i n t h e p r e s e n t c h a p t e r w i l l be g i v e n t o t h e s t a b i l i t y p r o p e r t i e s o f t h e s o l u t i o n s f o r n o n l i n e a r PDEs. We s h a l l o n l y deal w i t h t h e case o f chains o f q u o t i e n t a l g e b r a s ( 2 4 ) , Chap= t e r 3. The r e s u l t s p r e s e n t e d f o r t h a t case, can e a s i l y be r e f o r m u l a t e d f o r t h e chains o f q u o t i e n t a l g e b r a s ( 9 3 ) , Chapter 3, as i n d i c a t e d i n S e c t i o n 4. The r e s u l t s c o n c e r n i n g s t a b i 1 it y and exactness w i 11 be p r e s e n t e d on two l e v e l s . F i r s t , i n Sections 1-4, t h e problems w i l l be d e a l t w i t h on a gene= r a l l e v e l , i . e . r e s u l t s concerning t h e s t a b i l i t y and exactness o f any se= q u e n t i a l s o l u t i o n f o r any polynomial n o n l i n e a r PDE c o n s i d e r e d w i t h i n t h e chains o f q u o t i e n t a l g z a s ( 2 3 ) , Chapter 3, w i l l be presented. Then, i n S e c t i o n s 5 and 6, r e s u l t s concerning t h e s t a b i l i t y o f t h e weak s o l u t i o n s f o r t h e classes o f polynomial n o n l i n e a r PDEs s t u d i e d i n Chapter 4 w i l l be pre= sented.
1. B a s i c Facts and Remarks Concerning S t a b i l i t y Suppose g i v e n a C m - r e g u l a r i z a t i o n ( V , S ) . Then, t h e s t a b i l i t y p r o p e r t y o f s e q u e n t i a l weak o r d i s t r i b u t i o n s o l u t i o n s o f polynomial nonTinear PDEs, c o n s i d e r e d w i t h i n t h e framework o f t h e c h a i n s o f q u o t i e n t a l g e b r a s c o n t a i n = i n g t h e d i s t r i b u t i o n s (see ( 2 4 ) , Chapter 3)
(1)
A R ( V Y s ) = AR(V,S)/IR(V,S),
i s d i r e c t l y expressed b y t h e
(2)
IR(V,S), R
E
size o f
il E
8,
the ideals
R
t h a t i s , b e t t e r s t a b i l i t y means l a r g e r i d e a l s ( 2 ) , and i n e x t r e m i s , maximal s t a b i l i t y means maximal i d e a l s ( 2 ) .
163
164
E.E.
Rosinger
R e c a l l i n g now t h e way g i v e n i n (17-20), Chapter 3, t h e i d e a l s ( 2 ) were c o n s t r u c t e d , i t i s easy t o see t h a t these i d e a l s i n c r e a s e t h e i r s i z e , when= ever V o r S do so. However, g i v e n a C m - r e g u l a r i z a t i o n ( V , S ) , t h e s i z e o f S cannot be increased, i n view o f t h e f o l l o w i n g : Lemma 1 -~ If (V1,S1) and (V2,S2) a r e Cm-regularizations,
t h e n S1
C
S2 i m p l i e s S1 = S2.
Proof I t f o l l o w s e a s i l y from (21.2 , S e c t i o n 3, Chapter 3, which can a c t u a l y be w r i t t e n under t h e e q u i v a l e n t form m
S
=
vmQ S
0
Nevertheless, given a C m - r e g u l a r i z a t i o n ( V , S ) , t h e f o l l o w i n g changes i n S are possible. Lemma 2 I f ( V , S ) i s a C m - r e g u l a r i z a t i o n and S 1 i s a v e c t o r subspace i n Smsuch t h a t
then
(V,Sl)
i s also a
Cm-regularization.
Proof Obviously ( V , S l ) w i l l s a t i s f y (21.2) and (21.3) i n D e f i n i t i o n 1 , S e c t i o n 3, Chapter 3. Concerning t h e r e l a t i o n (21.1) i n t h e mentioned d e f i n i t i o n , we n o t i c e t h a t T ( V ) n SI c I ( V ) n = Z(V) n ( V
@ S)
(v
@ sl)
c Z(V)
ns
=
= Q
the l a s t i n c l u s i o n r e s u l t i n g from t h e f a c t t h a t V
c Z(V)
0
On t h e o t h e r hand, i n c r e a s i n g V i s a l s o n o t a t r i v i a l t a s k , i n view o f t h e r e l a t i o n (see ( 2 2 ) , Chapter 3 ) :
(3)
V C
#
V"
which i m p l i e s t h a t t h e maximum s i z e f o r V , g i v e n b y V", i s i n a c c e s s i b l e and we s h a l l have t o s e t t l e f o r V h a v i n g maximal s i z e s o n l y .
165
STABILITY AND EXACTNESS
Therefore, once again the necessit of dealing with various chains of quo= t i e n t algebras (24) or ( 9 3 ] d e d from the inexistence of canonical chains of the mentioned types, 2.
Maximal Stability
I n view of Lemma 1, we shall suppose given a fixed vector subspace S c Sm which s a t i s f i e s (see ( 2 1 . 2 ) and (21.3), Chapter 3) the conditions:
(4)
S" = V"
(5)
%"(n)
0
S
=
and study thg s t a b i l i t y of golutions as only spaces V c V which give C -regularizations such vector subspaces V , for instance V = p. Section 1, we shall be interested in maximal
depending on the vector sub= ( V , S ) . Obviously, there exist However, as mentioned in V.
Suppose now given a polynomial nonlinear PDE (6)
C ci(x) l ~ i Q h l < j < k i
and the corresponding PDO (7)
T(D) =
C
l < i < h
with the order (8)
m
=
max{ l p . . ] 1J
1
ci(x)
u(x)
DpiJ
=
f(x), x
E
il
.n
l < j < k i
1 < i < h, 1
Dpij
Q j Q
kil
We shall suppose that the PDE in ( 6 ) i s 0-smooth, that i s , c i , f (see Section 0 , Chapter 4 ) .
E
C"(n)
I t will be useful t o deal with the s t a b i l i t y of solutions for the PDE in ( 6 ) within the framework of the following rather general notion of solution.
Definition 1 A sequence of functions s E ( C m ( n ) )N i s caLled an S-sequential solution of the PDE i n (6), only i f there exists a C -regularization ( V , S ) , such that (9)
s E AR(V,S),
V R
E
W
-
u(f)
E
IR(V,S), 'd 1 E \
(10) w
=
T(D)s
The meaning of the above notion of solution results from:
-Proposition 1 N If s E ( C " ( 0 ) ) i s an 2-sequential solution o f the m-th order PDE in (6), then, for a certain C -regularization ( V , S ) , the following properties hold: 1) S
=
s
t l'l(V,S)
E A'(V,S),
Y 1E
fl
166
2)
E . E . Rosinger
S satisfies th plication i n A
i n t h e usual a l g e b r a i c sense, w i t h R (PDE U S )i nand(6), the p a r t i a l d e r i v a t i v e operators
Dp: A'(U,S)
Ak(VS)
+
,
p
ENn,
IpI
my with R , k g
m,
multi=
R-k> m
Proof I t f o l l o w s e a s i l y from ( 9 ) , ( 1 0 ) above and Theorems 2 and 3, S e c t i o n 3, Chapter 3.
n
The g e n e r a l i t y o f t h e above n o t i o n o f s o l u t i o n r e s u l t s from: Proposition 2 Ifa d i s t r i b u t i o n S E P ' ( R ) \ C m ( n ) i s a Cm-regular weak s o l u t i o n o f t h e PDE i n ( 6 ) , the! t h e r e e x i s t s a sequence o f f u n c t i o n s s E S and a v e c t o r sub= space S C S s a t i s f y i n g ( 4 ) and (5), such t h a t
(11)
s
(12)
s i s an S - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 )
=
*>
Proof I t f o l l o w s e a s i l y from (71-73), Chapter 4.
NOW, i n o r d e r t o p r e s e n t t h e r e s u l t on t h e maximal s t a b i l i t y o f S-sequential s o l u t i o n s o f t h e PDE i n ( 6 ) , i t i s convenient t o i n t r o d u c e t h e f o l l o w i n g d e f in i t i on. Definition 2
A sequence o f f u n c t i o n s s E ( C ~ ( R ) )i ~s c a l l e d a maximally s t a b l e S-sequen= t i a l s o l u t i o n o f t h e PDE i n ( 6 ) , o n l y i f t h e r e e x i s t s a c m - r e g u l a r i z a t i o n (V,s),w i t h V maximal, and s a t i s f y i n g ( 9 ) and ( 1 0 ) . Theorem 1 Any s - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 ) i s maximally s t a b l e . Proof Denotemby RV t h e s e t o f a l l t h e v e c t o r subspaces V C Vm f o r which (V,s) i s a C - r e g u f a r i z a t i o n . We s h a l l show t h a t R Vs i s c b a i n complete. Indeed, i n view o f (21.1), Chapter 3, a v e c t o r subspace U CmV belongs t o RVS , o n l y i f I( V ) n S = Q , where I( V ) i s t h e i d e a l i n ( C (f2))N g e n y a t e d b y V. But, i t i s easy t o see t h a t I( V ) i s t h e v e c t o r subspace i n ( C (f2))N generated b y V. (Cm(n))N. Therefore, RVS w i l l indeed be c h a i n complete. NOW, i n view o f ( 9 ) and ( l o ) , as w e l l as Z o r n ' s Lemma, t h e p r o o f i s com= pleted.
0
Remark 1 R e s u l t s on t h e s t r u c t u r e o f t h e maximal v e c t o r subspaces V w h i c h d e f i n e t h e m a x i m a l l y s t a b l e s e q u e n t i a l s o l u T E 6 - 3 n d whose e x i s t e n c e i s g r a n t e d by Theo= rem 1, w i l l be presented i n t h e n e x t chapter.
STABILITY AND EXACTNESS 3.
167
I n t e r p l a y between S t a b i l i t y and Exactness
As seen i n Sections 4 and 5, Chapter 1, t h e exactness p r o p e r t y o f sequen= t i a l , weak o r d i s t r i b u t i o n s o l u t i o n s f o r polynomial n o n l i n e a r PDEs c o n s i = dered w i t h i n t h e framework o f t h e c h a i n s o f q u o t i e n t a l g e b r a s ( 2 4 ) , Chapter size o f the ideals 3, o r ( 1 ) above, i s d i r e c t l y expressed by t h e -
(13)
ZR(V,S) , R
E
R
E
h'
and algebras (14)
A'(Y,S)
,R
t h a t i s , b e t t e r exactness means s m a l l e r i d e a l s ( 1 3 ) and l a r g e r a l g e b r a s ( 1 4 ) . I n e x t r e m i s , maximal exactness means minimal i d e a l s ( 1 3 ) and maximal a l g e = bras (14). NOW, i n view o f (17-20), Chapter 3, where t h e i d e a l s ( 1 3 ) and a l g e b r a s ( 1 4 ) were d e f i n e d , i t i s easy t o see t h a t t h e i d e a l s (13) decrease i n t h e i r s i z e t o g e t h e r w i t h V , w h i l e i n a c o n f l i c t i n manner, t h e a l g e b r a s ( 1 4 ) i n c r e a s e t h e i r size together w i t h V . a t concerns S , i t was seen i n Lemma 1, Sec= t i o n 1, t h a t n e i t h e r i n c r e a s e n o r decrease i s p o s s i b l e .
+
More p r e c i s e l y , suppose t h e PDO i n ( 7 ) i s considered t o a c t as f o l l o w s (see p c t . 2 and 4, Remark 3, S e c t i o n 3, Chapter 3): T(D) : AR(V,S)
+
Ak(V,S)
w i t h 2 , k E 4 , k < R , R-k > m. Then, maximal s t a b i l i t y f o r t h e s e q u e n t i a l , w&ak o r d i s t r i b u t i o n s o l u t i o n s o f t h e m)E i n (6)maximal ideals 7 ( V , S ) , t h a t i s , maximal V . On t h e p t h e r s i d e , maxima7 exactness f t h e same s o l u t i o n s m e a m m a 1 i d e a l s Z ( V , S ) , and maximal a l g e b r a s A ( V , S ) , t h a t i s , i n t h e same t i m e minimal and maximal V .
R
The way t h e above c o n f l i c t i s s e t t l e d r e q u i r e s i n each p a r t i c u l a r case o f s e q u e n t i a l , weak o r d i s t r i b u t i o n s o l u t i o n considered f o r t h e PDE i n ( 6 ) , a s p e c i f i c a n a l y s i s o f t h e s t a b i l i t y and exactness i n t e r e s t s i n v o l v e d . 4.
Remarks i n t h e F i n i t e l y Smooth Case k2 Suppose, i h e PDE i n ( 6 ) i n C -smooth, f o r a c e r t a i n g i v e n k2 E N, t h a t i s , c., f E C 2(Q). We s h a l l be i n t e r e s t e d i n t h e s t a b i l i t y o f s e q u e n t i a l s a l u t i o n o f t h e in-order PDE i n ( 6 ) , g i v e n b y sequences o f f u n c t i o n s s €(Ck1(Q))N, w i t h k l E N, kl > m. The framework w i l l be g i v e n t h i s t i m e by t h e chains o f q u o t i e n t algebras ( 9 3 ) , Chapter 3. More p r e c i s e l y , we s h a l l c o n s i d e r t h a t t h e PDO i n ( 7 ) a c t s as f o l l o w s (see Remark 5, S e c t i o n 5, Chapter 3): T(D) : A'( V,S) * Ak( V , S )
where ( V , S ) i s a C " - r e g u l a r i z a t i o n , k Q min {kl-m,k23.
w h i l e R, k
E
N, R-k > m y "kl,
I t i s easy t o see t h a t t h e remarks i n S e c t i o n 1, remain v a l i d , under t h e i r corresponding form, f o r C" - r e g u l a r i z a t i o n s and t h e c h a i n s o f q u o t i e n t a l = gebras ( 9 3 ) , Chapter 3, w h i c h t h e y generate.
168
E.E.
Rosinger
Therefore, given a v e c t o r subspace S c So which s a t i s f i e s t h e c o n d i t i o n s
so
(15)
v"
=
@S
Uco(n)
y
cs
kl N we c a l l a sequence o f f u n c t i o n s s E ( C (a)) an S - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 ) , o n l y i f t h e r e e x i s t s a C " - r e g u l a r i z a t i o n ( V , S ) such that
, tf
A'(v,s)
(16)
s
(17)
w = T(D)s-u(f)
E
R E N, R G kl E
k I (V,S), V k
E
a-k
N,
> m,k < min{kl-m,k2}
I n t h a t case, one o b v i o u s l y o b t a i n s t h a t S = s
+ I
R
( V , S ) E A ( V , S ) , U R E N, R
< kl
and S s a t i s f i e s t h e PDE i n ( 6 ) , i n t h e usual a l g e b r a i c sense, w i t h t h e m u l t i p l i c a t i o n i n Ak( V , S ) and t h e p a r t i a l d e r i v a t i v e o p e r a t o r s
DP : A ~ ( v , s ) f o r R, k
E
N, R-k
+
> my a
A ~ ( v , s ) , ~E Nn , I P I
Q
kl,
Q
my
k G min{kl-m,k21
k Moreover, ifa d i s t r i b u t i o n S E P ' ( R ) \ C " ( R ) i s a C ' - r e g u l a r weak s a l u t i o n o f t h e PDE i n ( 6 ) , then, t h e r e e x i s t s a sequence o f f u n c t i o n s s E S and and a v e c t o r subspace S c So s a t i s f y i n g (15), such t h a t S = < s, s i s an S - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 ) . kl N Now, c a l l i n g a sequence o f f u n c t i o n s s E ( C (n)) a maximally s t a b l e S - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 ) , o n l y i f t h e r e l a t i o n s (16) and ( 1 7 ) h o l d f o r a c e r t a i n C " - r e g u l a r i z a t i o n ( V , S ) , w i t h V maximal, one can e a s i l y o b t a i n t h e corresponding v e r s i o n o f Theorem 1. F i n a l l y , i t i s easy t o see t h a t t h e remarks i n S e c t i o n 3 can a l s o be adap= t e d t o t h e case o f f i n i t e smoothness. 5.
S t a b i l i t y o f Regular Weak S o l u t i o n s
Two general classes o f weak s o l u t i o n s f o r polynomial n o n l i n e a r PDEs were s t u d i e d i n Chapter 4 t i r s t , i n S e c t i o n 4, Chapter 4, t h e r e g u l a r weak s o l u t i o n s f o r a r b i t r a r y polynomial n o n l i n e a r PDEs were i n t r o d u c e d as an e x t e n s i o n o f t h e p i e c e w i s e smooth weak s o l u t i o n s o f s i m p l e polynomial n o n l i n e a r PDEs s t u d i e d i n S e c t i o n s 1-3, Chapter 4. Then, i n S e c t i o n 6, Chapter 4, t h e weak s o l u t i o n s w i t h d i s c o n t i n u i t i e s across hypersurfaces were s t u d i e d i n t h e case of r e s o l u b l e systems o f polynomial n o n l i n e a r PDEs, systems e x t e n d i n g t h e t y p e o f e q u a t i o n s o f magnetohydrodynamics and general r e l a t i v i t y , s t u d i e d i n S e c t i o n 5, Chapter 4.
.
The r e s u l t s on t h e mentioned weak s o l u t i o n s presented i n Chapter 4, con= cerned t h e r e s o l u t i o n o f t h e i r s i n g u l a r i t i e s and were o b t a i n e d b y construe= t i n g a r t i c u l a r i n s t a n c e s o f t h e chains o f q u o t i e n t algebras (24) o r (93), C h a p t b 3 T 5 T u c h a way t h a t t h e weak s o l u t i o n s s a t i s f i e d t h e polynomial n o n l i n e a r PDEs i n t h e usual a l g e b r a i c sense, w i t h t h e m u l t i p l i c a t i o n and p a r t i a l d e r i v a t i v e o p e r a t o r s i n t h e c h a i n s o f q u o t i e n t algebras.
STABILITY AND EXACTNESS
169
The aim o f t h i s s e c t i o n i s t o p r e s e n t a b a s i c s t a b i l i t y r e s u l t c o n c e r n i n g t h e r e g u l a r weak s o l u t i o n s f o r a r b i t r a r y p o l y n o m i a l n o n l i n e a r PDEs by con= s t r u c t i n g l a r g e classes o f c h a i n s o f q u o t i e n t a l g e b r a s ( 2 4 ) o r ( 9 3 ) , Chap= t e r 3, e x t e n d i n g those p a r t i c u l a r ones used i n Chapter 4. I n t h i s way, t h e r e s u l t s c o n c e r n i n g t h e r e s o l u t i o n o f s i n g u l a r i t i e s p r e s e n t e d i n Chapter 4 w i l l a l s o be improved. We s h a l l o n l y deal w i t h t h e case o f i n f i n i t e smoothness. The r e s u l t s o b t a i n e d can e a s i l y be r e f o r m u l a t e d i n t h e f i n i t e l y smooth case.
A s i m p l e and u s e f u l t o o l f o r e s t a b l i s h i n g t h e mentioned s t a b i l i t y p r o p e r t y o f t h e r e g u l a r weak s o l u t i o n s i s t h e s u f f i c i e n t c o n d i t i o n on t h e s e s o l u = t i o n s p r e s e n t e d i n Tbeorem 9, S e c t i o n 4, Chapter 4, as w e l l as t h e s u f f i = c i e n t c o n d i t i o n on C , R - r e g u l a r i d e a l s presented i n Theorem 20, S e c t i o n 10, Chapter 3. Supposg t h e PDE i n ( 6 ) i s Cm-smooth and t h e d i s t r i b u t i o n S E U'(Q)\C"(Q) i s a C - r e g u l a r weak s o l u t i o n f o r t h e PDE i n ( 6 ) which admits a sequence S" s a t i s f y i n g t h e c o n d i t i o n s o f functions s
(18)
s
=
(19)
w
=
(20)
lW i s a v a n i s h i n g i d e a l i n (C"(Q))
(21)
(v"
< 5, ' >
T(D)s
-
u(f)
E
t zw)n(ucm(n)
V"
+ R.S) 1
=
N
o_
Chapter 4 )
where (see ( 7 3 . 2 ) ,
lw = t h e i d e a l i n (C"(Q))N generated by IDP,
(22)
Ip E Nnl
Then, we denote by iDs t h e s e t o f a l l v a n i s h i n g i d e a l s 7 i n (Cm(Q))N, such t h a t
(23)
lw c 7
(24)
(vm +
i)n(ucm(n) + R.S) 1 =
2
Theorem 2 Under t h e above c o n d i t i o n s , t h e f o l l o w i n g p r o p e r t i e s h o l d :
1)
The s e t of i d e a l s i D s i s c h a i n complete.
2)
Each i d e a l 1 E i D s i s Cm, R - r e g u l a r , where R =
3)
F o r each i d e a l I E i D s t h e r e e x i s t C m - r e g u l a r i z a t i o n s
(25)
(Invm>@)
satisfying the conditions
1
R s.
170
E . E . Rosinger
4)
Each Cm-regula r i z a t i o n (V,S')win (25) y i e l d s t h e following resolu= t i o n of s i n g u l a r i t i e s f o r t h e C -regular weak solution S of t h e PDE in ( 6 ) :
4.1)
S = s + l R ( V , S ' ) E AR(V,S'), \d R E 1, where s i s given i n (18), s E S ' and does
4.2)
S s a t i s f i e s t h PDE in ( 6 ) in t h e usual a l g e b r a i c sense, with multi= p l i c a t i o n in A ( V , S ' ) and the p a r t i a l d e r i v a t i v e operators
not depend
on R
R
Dp : A"(V,S')
+
Ak(V,S'), p
E
N n y IpI Q m , w i t h II, k E
R,
R-k am
Proof 1)
I t follows e a s i l y from (23) and ( 2 4 )
2 ) and 3 ) follow from the s u f f i c i e n t condition on Coo, R-regular i d e a l s presented in Theorem 20, Section 10, Chapter 3.
4)
I t follows from 2 ) and 3 ) i f account i s taken of Theorem 7 , Section 4, Chapter 4.
Remark 2
+
From the point of view of t h e s t a b i l i t of the (?-regular weak solution S , Theorem 2 above presents a specia i n t e r e s t . Indeed, the s e t of ideals iD, being chain complete, t h e application of Zorn's lemma w i l l y i e l d t h e fol 1 owing property \d I E
-
iDs :
1 E iD, maximal:
ic 1 Then, t h e cha ns of quotient algebras
see ( 2 4 ) , Chapter 3 )
AR
associated t o l a r e gr maximal i d e a l s E iDS will o f f e r b e t t e r s t a b i l i t y s o l u t i o n S. properties f o r t e C -rG@TFweak
-8
6.
S t a b i l i t y of Solutions with Jump Discontinuities f o r Resoluble Systems of PDE S
Suppose t h e m-th order Coo-smooth polynomial nonlinear system of PDEs in (89) Chapter 4 , i s resoluble ( s e e (149),Chapter 4 ) .
STABILITY AND EXACTNESS Suppose u ,ut : R system of-PDts.
Ra a r e two
+
cm-smooth s o l u t i o n s o f t h e mentioned
F i n a l l y , suppose t h a t t h e f u n c t i o n u : R
+
U ( X ) = u-(x)
(31)
171
-
(u+(x)
+
Ra d e f i n e d b y
u-( x ) ) H ( x ) , x E 0,
where H i s t h e H e a v i s i d e f u n c t i o n ( 9 4 ) , Chapter 4, a s s o c i a t e d w i t h t h e C -smooth h y p e r s u r f a c e r i n ( 9 2 ) , Chapter 4, i s a weak s o l u t i o n o f t h e mentioned system o f PDEs, t h e r e f o r e s a t i s f y i n g on r t h e necessary and s u f = f i c i e n t j u n c t i o n c o n d i t i o n s (108) o r (151), Chapter 4. We s h a l l a l s o assume t h a t u- and u+ a r e em-independent on 5, Chapter 4 ) .
r
(see S e c t i o n
Then s = ( S ~ , . . . , S , ) E ( ~ " ) ~ c o n s t r u c t e d i n ( 9 9 ) , Chapter 4, has t h e proper= tY
(32)
ua =
(33)
wB = TB(D)s
,a>
-
, V 1< a < u(fB)
a
v",
E
V 1< B
<
b
We s h a l l denote 1 , = the ideal i n
(34)
(c"(R)) N generated b y I D P w B l 1 G B
Q
b,p
E
F i n a l l y , we denote b y JDS
t h e s e t o f a l l v a n i s h i n g i d e a l s I i n ( c " ( R ) ) ~ , such t h a t (35)
7,
(36)
(v" + r ) n
c 1
+
(uCm((,)
1
R.S) =
2
I n view o f t h e p r o o f o f Theorem 13, S e c t i o n 6, Chapter 4, i t i s easy t o see t h a t
JD, # 4 Theorem 3 Under t h e above c o n d i t i o n s , t h e f o l l o w i n g p r o p e r t i e s h o l d :
1)
The s e t o f i d e a l s JD, i s c h a i n complete.
2)
Each i d e a l 7
3)
F o r each i d e a l 7
(37)
(rn
E
JD, E
i s Cm JD,
, R-regular,
1
R = R s.
there e x i s t Cm-regularizations
7 , SOT)
satisfying the conditions
where
N n}
E.E.
172
Rosinger
(38) (39)
(40) (41) 4)
4.1) 4.2)
Each C m - r e g u l a r i z a t i o n ( V , S ' ) i n (37) y i e l d s t h e f o l l o w i n g r e s o l u = t i o n o f s i n g u l a r i t i e s f o r t h e weak s o l u t i o n u o f t h e system o f r e = s o l u b l e PDEs i n ( 8 9 ) , Chapter 4: R R uU = sa + I ( L S ' ) E A ( V , S l ) , 11 1 S a w i t h 1 S a Q a, do __ n o t depend on R .
Q
a,
R E
fl, where sa
E S',
u s a t i s f i e s each o f t h e PDEs o f t h e system ( 8 9 ) , Chapter 4, i n t h e usual a l g e b r a i c sense, w i t h t h e m u l t i p l i c a t i o n i n Ak(V,Sl) and t h e p a r t i a l d e r i v a t i v e operators
Dp : AR(V,Sl)
+
Ak(V,Sl),
p
E
Nny I p I S m y w i t h R , k
E
fly a-k
2 m
Proof S i m i l a r t o t h e p r o o f o f Theorem 3, S e c t i o n 5. Obviously, a corresponding v e r s i o n o f Remark 2, S e c t i o n 5, w i l l a p p l y t o Theorem 3 above.
0