Strongly continuous operator semigroups. Alternative approach

Let U and F be Banach spaces, operator L E L(U;F). opei'ator M E Cl(U;F), function f (•) : R ^ F. Consider the Cauchy problem u (0) = u о(1)

for the operator-differential equation

L u = Mu + f.(2)

If the operator L is continuously invertible, then the equation (2) can be reduced to a pair of equivalent equations ft = Su + h, g = Tg + f.(3)

Here the operators S = L-1M E Cl(U), dom S = dom M, T = ML-1 E Cl(F), dom T = L[dom M], the function h = L-1 f : R ^ U. It is convenient to consider the equation (3) in the frame of the equation v = Av + z(4)

on the Banach space V. Here A : dom A ^ V, dom A = V, z ( • ) : R ^ V.

The Cauchy problem

v(0) = vо, vо E dom A(5)

for the homogeneous equation v = Av(6)

is completely studied with the help of the semigroups theory. The main result of the classical semigroups theory [1] is a theorem of Hill-Iosida-Feller-Phillips-Miyadera (the HIFPM theorem), establishing a bijection between the resolving semigroup of the homogeneous equation (6) and

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ the operator A. called the infinitesimal general,or of a, semigroup. The criterion for the operator A being the infinitesimal generator of a semigroup (or generating the semigroup) are some conditions on the resolvent R^(A) = (ц1 — A)-1 of the оperator A. Depending on these conditions, operator A generates the analytical group, analytical semigroup or strongly continuous (C0) semigroup.

The theory of degenerate operator semigroups developed by G.A.Sviridyuk and his disciples generalizes these results to the case of the Sobolev type equations [2-6]. It also consists of three parts: analytical groups, analytical semigroups and, finally, strongly continuous semigroups with kernels. We suggest the alternative (in comparison to [7]) method of construction of C g-semigroup for the equation (2). To our opinion, these results are very useful for the numerical modelling of different processes based on the first order Sobolev type equations and to spread methods to the higher order Sobolev type equations [8].

• 1.    Relatively radial operators

Following [2, 7], introduce the L -resolvent set p^L ( M ) = {ц E C : ( цL — M ) ^{- 1} E L ( F ; UI ) } and the L -spectrum a^L ( M ) = C \ p^L ( M ) of operator M. The operator functions ( цL — M ) ^{- 1} , RL ( M ) = ( цL — M ) ^{- 1} L, LL ( M ) = L ( цL — M ) - ¹ are cailed L -resolvent. right and left, L -resohunts of operator M.

Definition 1. The operator M is called radial with respect to operator L (shortly, L-radial), if (i) 3a E R Vц> a ц E p^L ( M )

• (ii)    3K >  0 Vц > a Vn E N

nn

• - {" ыЩ R^L < ^M ' “‘ ( U ) ■"n !Щ L^{L ( M )} "^ ( F • ^{} 5} .      +

Remark 1. Without loss of generality one can put a = 0 in definitonl.

Remark 2. If there exists the operator L^{- 1} E L ( F ; U ), then оperator M is L -radial exactly, when the operator L^- 1M E Cl ( U ) (or. equivalent!y. the operator ML^{- 1} ECl ( F )) is radial.

Set U ⁰ = ker L F = ker LL ( M )). В у L ₀ ( M ₀) denote restrictioii of the operator L ( M ) l,o lineal U ⁰ (dom M ₀ = U ⁰ C l dom M ).

Definition 2. Weakly L-radial operator is an operator M for which condition (i) is satisfied as well as condition (ii) when n = 1 in Definition 1.

Theorem 1. [2] Let the operator M be weakly L-radial. Then:

• (г) any vector p E ker L \ { 0 } does n ot have M-adjoint vectors:

(tt) ker RL ( M ) П iiи R^ ( M ) = { 0 },  ker LL ( M ) П iin LL ( M ) = { 0 }.

(Hi) there exists the operator M- ¹ E L ( F ⁰; U ⁰).

By U ¹ ( F 1) denote the closure of the lineal im RL ( M ) ( iin LL ( M )) by norm of the space U ( F ).

Lemma 1. [2] Let the operator M be weakly L-radial. Then

• (i)    lim цR^L ( M ) u = u Vu EU 1;

^m + ^  ^

• (u) lim цLL ( M ) f = f Vf EF ¹ .

• 2. The resolving operator semigroups

^m + те

By "U (_ F ) denote the closure of the lineal U ⁰-+ im RL^p )( M ) ( F ⁰-+ im LL^p )( M )) by norm of the space U ( F ). Olu’iously. U ¹ ( F ¹) is the sulispace in U ( F ).

Lemma 2. Let the operator M be weakly L-radial. Then U = U ⁰ ©U ¹ , F = F ⁰ ©F 1.

Consider two equivalent forms of the equation (2)

rL (M)u = (aL - M)“1 Mu,(7)

LL (M) f = M(aL - M) -1 f(8)

as concrete interpretations of the equation

Av = Bv,(9)

defined on a Banach space V, where the operators A, B E L ( V )

Definition 3. The vector-function v E C (R+; V ), differenliable on R+ and satisfying (9) is called a solution of the equation (9).

A little away from the standard [1], following [7] define

Definition 4. The mapping V' E C (R+; L ( V )) is called a semigroup of the resolving operators (a resolving semigroup) of the equation (9), if

• (i)    V^sVtv = V^{s +} tv for all s,t >  0 and any v from th.e space V:

(in) v ( t ) = Vtv is a solution of the equation (9) for any v from a tlense in V set.

The semigroup is called uniformly bounded, if

3C> 0 Vt >  0 II Vt^L ( V ) < C.

Theorem 2. Let the operator M be L-radial. Then there exists a uniformly bounded and strogly continuous resolving semigroup of the equation (7) ((8)), treated on the subspace U ( F ), presented in the form:

Ut = s - lim Hf- pt 7    RL (M A, km+^ (k — 1)!        yd/l- 1

(Ft = s - l™-)- (k)k (Lpm)))

km + ^ ( k - 1)! y t y \d^{—- 1}        ) ^ _ k

Proof. Denote the following families of operators:

Ut= ( - 1) k^{- 1} ( k ^ k ( dk^{- 1} pL_(M Л

U^k   ( k - 1)! V t J U ^^k- ¹ R^ ⁽ MУ

^ = t

Note that

Vu E U ⁰     Uk u = 0 .

Since the operator M is L -radial. approximations of are uniformly bounded by a constant. K from Definition 1:

lUtIl(и) < K Vt E R+ Vk E N.(11)

Let us take u E dom M and find the derimlir'e dt uu — dt

d fi-kf:; PXICRRL(M))

dt у (k - 1)! \t) dckl1-1 1 ^_

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ

= т-1^kk (-к/VRL(M) - ^-^rRL(M))

( k - 1)! V    tk ⁺¹ dak^{- 1 L}       tkdak ^L ) _L = k

(-1)k—1 /k\k+1 / dk—1            k dk— 1\

• = ( k - 1)! ( t)    ("dak ■ ^R^ ⁽ M ) ⁺ td^k^{- 1} ^ R^{L (} M )) ^R^ ⁽ M )) L = t u =

П k—1 / k\k+1 jk-1            / L\

=     ! ( ?)   ^ ( ^RL ⁽ M» RM ^{( M} * - I) L = t u =

= tut (tL - M)

Thus, d Uku = Uk fL - t-M) 1 Mu   ^u e dom M.(12)

dtk

Now let u E imRL(M). i.e. u = RL(M)v in some в > 0 ii v EU. Proof, then t—m Utu = u.(13)

Make the change a = ^k.

lim Utu = lim t—i 0+        L—i + ^^

( - 1) ^{- 1}

( k - 1)! ^Ц

- 1

_f , IL ( M ) u da^{k- 1 lV} ’

lim

L— + ^

-_ -       m      m

E (-аЦ”+1 aaRf(m)- z—'1 \ m!        dam L

=

( - 1) ^{m- 1} , ( m - 1)! ^Ц

-

m- 1

:md^ ^rL ( ^M )) u ⁺ _д 1;^т ^a^RL ( ^M ) ^u =

k- 1

lim E

L— + ^ ^-^

⁽ 1) m .¹ am ( - ^ad— r^l ( m ) - ( m - 1)!       m dam ^L

_dm- 1

, m_T rL(M) u+ lim aRL(M)u = dam 1           l—+^

k- 1

lim

L— + ^ ^—^

m- 1      m- 1

m m am dam- г^{( R}L ^{( M ))(} a^RL < ^M ' ^-1 ' u ⁺

lim aRL ( m ) u.

L— + ^  ^L

Due to || (m—QfamdLm-1 (RL(M)) 11^(U) — K ^or апУ a > 0 un4er L-radiality of the operator M. we get k— 1 ( i \m—1

E (m-nyamdlm—-1(RL

— (k -1)kllaRf (m)u - u) |u m 0, a ^ +^TO-

The second limit in (14) is equal to u.

Now for the same vector u consider the difference

t

Ut u - Ut u =

J ds⁽Ut^-sUsu) ds =

t

= / (^U‘^-'^Uss (L -

M^ ) ¹M -k

_Ult-s_Uks _{L -}

t-TMd^-1M)

uds =

t

= J Ut-sUs (s

^^^^^^^^r

t

— s

l

L

— kM -1 M(L

—

TM)

-1

M (Re (M ))²vds

t

= jt-rut (k ^— t^) 0

kRe(M) —— R^ei (M)((eL — M)-¹M)²vds. S s t — S t-s

Taking into account (11), we get

t hUtu — Utu^u < K2/ k — t—s

l    K²

s(t — s)

kl s (t-s)

II((eL — M)-¹M)²vhuds <

< K^-2 fr + j) I((eL — M)-¹M)²vIu• 2 k l У

From (15), (10), (11) and density of im R» (M) in U¹ it follows the existence of the limit

U^t = s — lim Ut, U^teL (U), ||U ^L ,,_;. < K Vt> 0. to                        ^L(U >

Inequality (15) shows that Utu uniformly with respect to t E (0, T] converges to U^tu. Thus, the family {U^t : t > 0} is strongly continuous with respect to t, because due to continuity of right L-resolvents of the operator it follows a strong continuity of the family {Ut : t > 0} lor any k E N. In order to extend the strong continuity of {U^t : t > 0} up to zero, we define an element of the family of operators U⁰ as a, strong limit:

U⁰ = s — lim U^t. t—¥ 0+

Due to (10)

Vu E U⁰U⁰u = lim U^tu = lim lim U^tu = 0 t—— 0+        t—— 0+ к——o

In addition, using the above-mentioned uniform convergence, it can be shown that

Vu E U¹UT⁰u = lim UJ^tu = lim lim Utu = lim lim Utu = u. t—— 0+         t—— 0+ t——^o         t——^o t—— 0+

So. we gel, l,hat, UT⁰ = P¹. Note that

Uh =

(—1)^kl-1(klA^kl (dE-L_rl₍,,g (kl — 1)! ft/ fd^^kl-1 R^⁽ ))

» = kl

(—1) ^l-2 (kl — 2)!

(tt’tdn, - R=M0

ц = ki

...

(^—j)l¹ /^kl\ ^k! / d¹ , e , e (t-1)iA (—nr fy) fdm^(R"^(MTR»^(M» )

^ = ktl

^l (—1)^l-1d^1-1

(l — 1)! d^^1-1

[ Re ( m )]

к

)' .■(Ukt kl

»=ki

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ

From (11) it follows, that

lim l——+oo

(Uk^ k=^{(U k)}

k

. Indeed, al, u G U

U_lk^t

k

u

^^^^^^^^r

(Uk)ku

U

— tx k-m-1           t

E( Uq     (^Ut) (Ui^k

=

^^^^^^^^r

U к ^ u

<

< k-^k-1

tt

Uk u - U t u

U

U

Tending in identity (16) l ^ + to, we obtain

U1

= (U Д.

Let us show that hence and from the strong continuity it follows that {U¹ : t > 0} is a semigroup. Taking rational s = k/l nt = m/n using (17) twice, we get

~  ~ ,    ~ kn ~ lm / ~ 1 ^kn / ~ 1 ^lm / ~ 1  ^kn+lm    ~ kn+lm

U ^sU¹ = U n U n = (u in\  UJ in\  = Ш In)     = U n = U^s+\

For arbitrary real numbers s,t > 0 there exist sequences of rational numbers {s_n : n G N}, {tn : n G N} such I,hat, lim sn = s, lim tn = t. Then.

n→∞      n→∞

Vn G N Usn U¹ⁿ = Usn⁺¹ⁿ.                              (18)

Since,

IUsnU¹ⁿu - U^sU Mu < IUsn|_£(U)IIU¹ⁿu - U¹ u^u + IIUsnU ¹u - U^sU ¹u^u <

< KIU¹ⁿu - U ¹uIu + ||UsnU ¹u - U^sU ¹uIu,

tending n ^ to in (18). (19) and using the strong continuity with respect, I,о t of the operators family {U¹ : t > 0}, we obtain the desired.

Further, let, u¹ = (RL(M))²v lor some в > 0 arid v GU. Then

!Uiu¹- 77¹u¹Iu = lim ||U{u¹ k→∞

- Uku¹ iiu <

< lim Kit2 (1 + 1) |((8L - M)-1M)2v|u = k—o 2  \ k

К 4t2

-2-1((eL - M)^-1M)²vIu,

due to (15).

U{u¹ = (|rL(M)) u¹ = ^7 + ( 1l-M) M^u¹ =

= u¹ + RL(M)RL(M)(eL - M)-¹Mv =

= u¹ + t(L - tM)- 1(L - tM + tM)Re(M)(eL - M)

¹Mv =

= tRL(M)(eL - M)-¹Mv + tRi (M)Re(M)((eL - M)-¹M)²v.

Therefore

LT1

^^^^^^^^r

t

-u¹- Re(M)(eL - M)^-1Mv

<

U

<

UL-I u 1

- Re(M)(eL - M)^-1Mv

+

U

U t - utt !

1_u1

U

<

< tKy((eL - M)-¹M)2v^u +    ^Re(M)((eL - M)^-1M)²V^U.

Tending t ^ 0+ we obtain. tlial

U t - I tlim_ —t—u1 = Re(M)(eL - M)-1 Mv.                   (20)

Act on (20) by the operator Us and get the differentiability on the right of the semigroup at this element u1 = (Re(M))2v at point s > 0. In order to show the differentiability on the left at this point, one can consider the expression us-t - us 1

U^s-t (Ut - I) 1

t

s > t > 0,

proceed to the limit when t ^ 0+, using the uniform boundedness of the semigroup. So, by virtue of(20)

djt(R^(M))²v = JTtR^e(M)(eL - M)^-1Mv.

Act on last identity by the operator R^e (M).

By construction Ut commutes willi the operators R^e(M) and (aL - M)^-1M lor the corresponding u 1, therefore by (20) we obtain

R^(M)djtu¹ = (aL - M)^-1MUtu¹.                     (21)

Obviously. for u⁰E U⁰  U^s(u⁰ + u 1) = U^su 1. Then one can change u¹ bу u = u⁰ + u¹E

U⁰+ im(Re(M))² in identity (21). Thus, the function u(t) = Utu is the solution of the equation (7) for arbitrary u from the dense in U lineal U⁰+ im(Re(M))².

(For the semigroup Ft = s - lim F^. which is cons true Ind by means of the left L-resohnnt. k^^

the proof is identical).

□

The semigroup Ut Ft) at first is defined not on the whole space U (F), but on some subspace U FY Introduce the sufficient condition of their coincidence: U = U F = F-

Theorem 3. [2] Let the space U (F) 6e cefteiive, the operator M be weakly L-radial. Then U = U⁰®U¹ (F = F⁰®F 1).

Definition 5. Operator M is called strongly L-radial on the right (on the left), if it is L-radial and yRe(M)(XL - M)-1 Mu^u < const(u)  Vu E dom M

µ                         λµ

◦

(there exists a dense in F lineal F such, that yM(XL - M)-1 Le(M) f y^ < const(f) Vf eF)

λµ for any X, ц > 0.

Theorem 4. [2] Let the operator M be strongly L-radial on the right (on the left). Then U = U⁰®U¹ (F = F⁰®F 1).

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ

Remark 3. Obviously, that under the condition of strong L-radiality of operator M on the right (on the left) the resolving semigroup of the equation (7) ((8)) is defined on the whole space U (F). and the projector P lim ^RL(M) (Q = s — lim ^LL(M)) is it's unit.

µ→+∞  ^µ             µ→+∞  ^µ