A multipoint initial-final value problem for a linear model of plane-parallel thermal convection in viscoelastic incompressible fluid

Many phenomena and processes in economics, physics, and technology, like, for instance, plane-parallel thermal convection in viscoelastic incompressible fluid, are modelled by linear

Lui = Mu + f

and nonlinear

Lui = Mu + N ( u ) + f

Sobolev-type equations [33]. The interest in Sobolev-type equations, which nowadays form a large subfield of nonclassical equations of mathematical physics [30], has been increasing recently; see the wonderful historical survey in [31].

The goal of our study is the solvability of (1) with the so-called multipoint initial-final conditions (see [5] for instance and reference therein)

Pj ( u ( Tj ) — Uj ) = 0 , Uj E U, j = 0 , n, —то < a <  t о <  t 1 < ... < Tj < Tj +1 < ... < b <  + to ,

where Pj are relative spectral projectors (we discuss them in Section 1). while uj are arbitrary vectors in a Banach space U. These conditions are used to reconstruct the parameters of the processes in question from the results of multiple observations at various points and times. This enables us to, for instance, predict emergency situations, including the violation of continuity of thermal convection processes as a result of breaching technology, and so forth.

We should note that problem (1), (3) in the case n = 1 (the initial-final value problem) has been studied quite actively in various aspects. In particular, there are results concerning the optimal control of the solution to these problems [9], including Sobolev-type equations of high order [6].

The history of problem (1), (3) in the case n = 1 starts on the one hand in [12], where it is called Verigin’s problem, and on the other hand, independently, in [32], where it is called the conjugation problem. However, in both cases instead of relatively spectral projectors P 0 and P 1 we consider spectral projectors of the operator L on assuming it to be selfadjoint. The first results in this direction are presented in [19], which treats a particular case of problem (1), (3) with, moreover, more rigid conditions on the L -spectrum of M than here. Problem (1), (3) is considered in [2] with the same conditions on the L -spectrum of M as in [19]; however, in this case the possibility of greater freedom in relatively spectral conditions is mentioned. We should note that there were attempts to study [18] the solvability of a particular case of problem (2) for nonlinear Sobolev-type equations (3) in the case n = 1, but as yet these studies have not been pushed further. In addition, if in (3) we put n = 0 then problem (3) reduces to the Showalter-Sidorov problem [20], which has already played an important role in a number of models with applications to economics [7] and technology [29].

Our approach rests on the theory of relatively p -sectorial operators and degenerate analytic resolving semigroups of operators. Sviridyuk [14] pioneered the concept of a relatively sectorial operator. He showed that relative sectoriality of an operator naturally generalizes the concept of sectoriality [28]. However, it soon turned out that relative sectoriality generalizes the concept of relative ст -boundedness of an operator only in the case that the L -resolvent of M has a, removtrble singularity at infinity. In order to fill this embarrassing gap, Bokareva introduced [16] the concept of relative p -sectoriality of an operator, generalizing the concept of relative ст -boundedness in the case that the L -resolvent of M has a pole at infinity. Then, relatively strongly p -sectorial operators on the right (on the left) [16] and relatively strongly p -sectorial operators [24], [25] were introduced. Subsequently, relatively p -sectorial operators were studied in various situations. Namely, Dudko studied [1] the case that both operators are closed and the spaces U and F coincide; Efremov studied [17] optimal control problems for Sobolev-type equations with relatively p- sectorial operators; Keller found [21] sufficient, and in some cases necessary, conditions for the existence of bounded solutions to these equations; Kuznetsov [22] began to search for relatively p- sectorial operators among elliptic operators; Yakupov used [26] relatively p -sectorial operators to study the phase spaces of certain problems in the hydrodynamics of viscoelastic fluid.

Consider now a precursor of (2): a hybrid of Oskolkov’s system [11] and the heat equation in the Oberbeck-Boussinesq approximation [8],

J ( A - V 2) vt = vV ² v - ( v • V ) v -Vp + gYS, V • v = 0 , |            St = 5VS - v •VS + y • v,

modeling thermal convection in a viscoelastic incompressible Kelvin-Voigt material [23]. Here v = ( v i , v2, ••• ,vn ) with vi = vi ( x,t ) and n = 2 or 3 is the vector function representing fluid velocity; the scalar functions S = S ( x,t ) and p = p ( x,t ) represent the temperature and pressure of the fluid; the parameters A E R, v E R+, a nd 5 E R+ characterize the elasticity, viscosity, and thermal conductivity of the fluid; g E R+ is the free fall acceleration; finally, y = (0 , • • •, 0 , 1) E R ⁿ and x = ( x 1 ,x2, ••• ,xn ). When one of the horizontal components of the velocity vanishes, (4)

becomes

( А - А)А д- — v А² ф -

^ = ⁵ А 9 -dt           о ( x, у)

д ( ф, А ф ) ^{д ( Х,}У )

∂θ

+ ^а"гГ, ∂x

+ вдф

∂x

which models plane-parallel thermal convection in a layer of viscoelastic incompressible Kelvin-Voigt material.

For (4) Sviridyuk considered the first initial-boundary value problem [13] and showed that it is solvable for arbitrary values of А . Then jointly with Yakupov [26] he described the morphology of the phase space of the Cauchy-Benard problem for (5). Sukacheva and Matveeva studied [27] the non-autonomous case of this problem. Subsequently they considered a generalized model of thermal convection [10], established the local solvability of the Cauchy problem for it, and found the solution numerically using the modified Galerkin method. We should also mention the studies [4] of the stability of solutions to the Cauchy-Benard problem for (5) in a neighborhood of the origin. The existence of stable and unstable invariant manifolds in the problem was established basing on the Hadamard-Perron theorem. Note also that [3] showed the unique solvability of the initial-final value problem for the linearized model of thermal convection (4).

This article is devoted to a qualitative study of the multipoint initial-final value problem

( А — А)А Lt — —v А ² ф — aQx + £, 9t — 5 А 9 — вфх + Z

for the linear mathematical model of plane-parallel thermal convection in viscoelastic incompressible fluid in the region И — (0 , a ) x (0 , b ) G R² with Benard’s boundary conditions

ф ( x, 0 , t ) — А ф ( x, 0 , t ) — ф ( x, b, t ) — А ф ( x, b, t ) — 0 ,                      ( । )

9 ( x, 0 , t ) — 9 ( x, b, t ) — 0 ,

the functions ф and 9 are periodic in x with period a.

In the first three sections we collect auxiliary facts of Sviridyuk’s theory [15] of relatively p sectorial operators and degenerate resolving semigroups of operators, adapted to our situation. In Section 1 we introduce the concept of a relatively p -sectorial operator. In Section 2 we consider degenerate resolving semigroups of operators and the construction of units of semigroups of operators. In Section 3 we consider conditions for the existence of the inverse operator. In Section 4 we prove a generalized splitting theorem for the spaces and actions of operators. There we construct relatively spectral projectors, which in this case are units of the semigroups of operators, on assuming relative p -sectoriality. In Section 5 we study the multipoint initial-final value problem for Sobolev-type equations with a relatively p -sectorial operator M. The main result of this section is a theorem on the unique solvability of problem (1), (3). In Section 6 we apply these abstract results to the linear model of plane-parallel thermal convection in viscoelastic incompressible fluid. There we reduce the stated problem to the abstract equation (1). We verify the ( L, 0)-sectoriality of M. The main result of this section is the theorem on the unique solvability of the multipoint initial-final value problem (3). We should note that the author already discussed [34] a generalized decomposition theorem in the case of strongly ( L,p )-radial operator. We make all arguments in real Banach spaces, but, while addressing spectral questions, introduce their natural complexifications. All contours are oriented counterclockwise and bound the region lying to the left as they are traversed.

1.    Relatively p-sectorial Operators

On assuming that U and F are Banach spaces, consider a continuous linear operator L E L(U; F) and a closed 1 inear operator M E Cl(U;F) whose domain is dense. Introduce the L-resohung set pL(M) = {p E C : (pL — M)-1 E L(F; U)} and the L-spectrum yL(M) = C \pL(M) of M. Provided that pL(M) = 0, we can introduce the right and left pp

RL» ) ( M ) = П RL ( M ) - "^{1 L} L ) ( ^M ) = П LLk ( ^M ) k =0                               k =1

( L,p>resolutions of M. Here RL ( M ) = ( pL — M ) ^{- 1} L and LL ( M ) = L ( pL — M ) ^- 1. while pk E pL ( M ) fc>r k = 0 ,... ,p.

Definition 1. [15] An operator M is called p-sectorial relatively to an operator L with p E { 0 }U N (or briefly. ( L,p)-scctoriaf whenever there exist constants K E R+- a E R. aiid 0 E ( п/ 2 , п ) such that

SL, e( M ) = {p E C : | arg( p — a ) | <  0 , p = a} C p^L ( M );               (10)

furthermore,

^max{|| ^RL,P ^{)( M} ) 11 l (u) , || ^LL,P ^{)( M} ) 11 l ₍ f)} <

K

p

|µk - a| k=0

(И)

for arbitrary pk E SLe ( M ) fc>r k = 0 ,... ,p.

Remark 1. If M is an ( L,p )-sectorial о уcerator and b > a then the operator M = M — bL is also ( L,p )-sectorial. Furthermore, we can choose the constant a in 1 to be 0. Assume henceforth that S o ^L e( M ) = SL ( M ).

Remark 2. If the operator L E L (U; F) has continuous inverse then the sectoriality of the operator L^{- 1} M E Cl (U) implies the ( L,p )-sectoriality of M E Cl (U; F). and the ( L, 0)-seetoriahty of M implies the sectoriality of L^{- 1} M (or equivalently. of ML^{- 1}).

Lemma 1. Ц M is an ( L,p^-sectorial operatoi’ then there, exist R >  0 aiid C >  0 such that I I( pL — M ) ^{- 1} | L _№u) < CIplp for all p E SL ( M ) \ {p E C : |p| < R}.

Remark 3. (i) If M is an ( L,p )-bounded оperator and to is an order 0 pole of the L -resolvent of M then M is an ( L, 0)-sectorial operator.

• (ii)    If M is an ( L,p )-bounded oycerator and to is a pole of c >rder at most p E N of the L- resok'ent of M then M is an ( L,p )-sectorial operator.

Lemma 2. The following claims hold for every ( L,p)-sectorial operator M:

• (i)    the length of every chain of generalized M-eigenvectors of L is bounded by p;

• (ii)    the set ker RL. _p )( M ) coincides with the M-root space of L;

(in) ^{ker R}L.p )^{( M} ) П im ^RL.p )^{( M} ) = { ⁰ } aiid ^{ker L}L.p )^{( M} ) П im ^LL.p )^{( M} ) = ^{{ 0 }} ;

(iv) the operator M- ¹ E L (F⁰; U⁰) exists.

On assuming that M is an ( L,p )-sectorial operator, recall the notation H = M- ¹ L о and G = L о M- 1. Denote by U¹ the closure of th e linear subspace im RL. p )( M ). Denote by U the closure of the linear subspace U⁰-+im RL. p )( M ) in the norm of U. Denote by F¹ the closure of the linear subspace im LL.p )( M ), and by F the closure of the linear subspace F⁰-+im LL.p )( M ) in the norm of F.

Lemma 3. The following claims hold for every ( L,p)-sectorial operator M:

• (i)    the operators H and G re nilpotent of degree- at most p:

• (ii)    lim ( ^RL ( M )) ^{p +1} u = u for every u E U¹ and lim ( ^LL ( M )) p ⁺¹ f = f for every Ц^ + ^   ^ц                                ц^ + ^ц

• 2.    Degenerate Analytic Resolving Semigroups of Operators

f E F¹:

(Hi) U = U⁰ Ф U¹ aiid F = F⁰ Ф F¹.

Denote the projector onto U¹ parallei to U⁰ bу P = s- lim ( pRL ( M )) ^{p +1} and the projector

Ц^+го onto F1 parallei to F0 bv Q = s- lim (pLL(M))p+1.

ц^+^

On assuming that U and F are Banach spaces. take an operator L E L (U; F) and an operator M E Cl (U; F). The equation

Lu = Mu(12)

reduces to the equivalent pair of equations rL (M)u = (^L — M)-1 Mu,(13)

LL (M) f = M(цL — M)-1 f.(14)

It is convenient to regard (13) and (14) as concrete interpretations of the abstract equations

A v = Bv(15)

defined on a Banach space V with A, B E L (V). Refer as a solution to (15) to a vector function v ( t ) E C^ro (R+; V) satisfying this equation for t >  0 and continuous at 0.

Definition 2. A mapping V* : R₊ re L (V) is called a semigroup of resolving operators (or simply a resolving semigroup) of (15) whenever

• (i)    v^svt = v^{s +} t for all s, t >  0:

• (ii)    for every v 0 E V the funotion v ( t ) = Vtv 0 is a solution to this equation.

A semigroup {Vt : t >  0 } is called analytic whenever it can be analytically continued to some sector E C C including the ray R+. that is. there exists an analytic mapping V* : E re L (V) enjoying properties (i) and (ii) of the previous definition (with s,t E E), coinciding with V* on the positive semi-axis. In addition, {Vt : t >  0 } is called uniformly bounded whenever || Vt|L (v) <  const for all t E R+.

Theorem 1. For every ( L,p(-sectorial operator M there exists a resolving semigroup {Ut : t >  0 } (or {Ft : t >  0 }) of (13) (respectively (Ц)) which is analytic in the sector

E = {r E C : | arg т | < 0 — п/2 wi th т = 0}, where we take 0 from Definition 1, and uniformly bounded. Furthermore, this semigroup is defined by the integrals ut = 1- [ RL (M)e^d^   (Ft = -1O [ LLL (M)e^d^)

2 П i                                2 П i г                               г of Dunford-Taylor type, where t E R+, and the contour Г C SL@(M) satisfies | arg^| ^ 0 as ц re re aiid ц E Г.

Lemma 4. If M is an ( L,p"^-sectorial operator then

lim Utu = u for every u E im R

L

⁽ pop )

( M )

and

lim Ft t—i 0+

^f = ^f for every f ^E i^m LL.p )

( M ))•

Lemma 5. If {Vt : t >  0 } is an analytic .semigroup then ker Vt ¹ = ker Vt ² for all t 1 , t ₂ >  0.

Definition 3. The set ker V• = ker Vt,t > 0 is called the kernel of the analytic semigroup {Vt : t> 0 }.

The preceding statement shows that the kernel is well-defined.

Consider the kernels of the semigroups U• and F*:

ker U• = {ф E U : U^tф = 0 3t E R₊ }, ker F• = {^ E F : Ft^ = 0 3t E R }.

Put U⁰ = ker U• aiid F⁰ = ker F*. Denote by L ₀ the restriction of L tо U⁰. anc 1 by M ₀ the restriction of M tо U⁰ П dom M.

As in the case of holomorphic groups, it is clear from the expressions (16) of the resolving semigroups of (13) and (14) that their elements have nontrivial kernels ker Ut D ker R^ ( M ) and ker Ft D ker L^ ( M ) for every t >  0.

The kernel of an analytic semigroup is obviously a subspace. Denote by L 0 ( M 0) the restriction of L ( M ) to ker U• (ker U• П dom M ).

Lemma 6. If M is an ( L,p"^-sectorial operator then

L 0 E L (ker U• ;ker F• ) ,  Ml 0 : ker U• П dom M ^ ker F•.

Denote by aL ( MI) tlie L t0-spectrurn of Ml 0.

L/Wx

Lemma 7. If M is an ( L,p)-sectorial operator then aL ( M ) contains no finite points.

Corollary 1. If M is an ( L,p)-sectorial operator then the operator MI^ 1 E L (ker F• ;ker U• ) exists.

Theorem 2. If M is an ( L,p)-sectorial operator then ker U • = U⁰ a nd ker F • = F⁰.

Definition 4. Refer as the image of a semigroup {Vt : t > 0} to the set imV• = {v E V : v = lim Vtv}. ti0+

Lemma 8. Every analytic semigroup {Vt : t >  0 } satisfies ker V• П im V• = { 0 }.

Lemma 9. If {Vt : t > 0} is a strongly continuous and uniformly bounded semigroup then imV• = ^J imVt . t> 0

Theorem 3. If M is an ( L,p)-sectorial operator then im U • = U¹ a nd im F • = F¹.

Put [ 7 t = Ut I.-, a nd Ft = Ft

UF

• Consider the images of the semigroups U• and F•;

im U• = {u E U : lim Utu = u}, im F• = {f E F : lim Ftf = f}.

ti 0+                                        ti 0+

Put U¹ = im U• aiid F¹ = im F•. Denote by L 1 the restriction of L 10 U¹. anc 1 by M 1 the restriction of M 10 U¹ П dom M.

Corollary 2. If M is an (L,p"(-sectorial operator then z^z                                            z^z ,                 z^z                                             z^z ,

P = s- lim Ut,  Q = s- lim Ft.

t—f 0+                t—f 0+

The operators

P = s- lim Ut G L (U),  Q = s- lim Ft G L (F), t—0+                       t—0+ whenever they exist, are called the units of the semigroups {Ut : t > 0} and {Ft : t > 0}. It is not difficult to see that the units of semigroups are projectors.

Definition 5. An operator M is called strongly (L,p(-sectorial on the right (on the left) whenever it is (L,p)-sectorial and for X, цо, Ц1,-iPp G SL(M) we have const( u)

p

|λ|        |µk| k=0

\\RL^p )( M )( XL - M ) - ¹ Mu\ u <

о for arbitrary u G dom M (respectively. there exists a dense linear subspace F of F such that const( f )

p

|λ|        |µk| k=0

\M ( XL - M ) ^{- 1} LL^p )( M ) f \ F <

о for arbitrary f GF).

Remark 4. (i) If M is an ( L, a )-bounded operator and to is a removable singular point of the L -resolvent of M then M is a strongly ( L, 0)-sectorial operator on the right and on the left.

• (ii)    If M is an ( L,a )-bounded oywrator and to is a pole of c >rder at most p then M is a strongly ( L,p )-sectorial operator on the right and on the left.

Theorem 4. If M is a strongly ( L,p(-sectorial operator on the right (on the left) then the units of the semigroups {Ut : t >  0 } and {Ft : t >  0 } ) exist. Furthermore. the operators P G L (U) and Q G L (F) satisfy

L gL (ker P ;ker Q ) П L (im P ;im Q ) , M GCl (ker P ;ker Q ) nCl (im P ;im Q ) .

Remark 5. Theorem 4 also holds in the case that M is an ( L,p )-sectorial operator, but under the additional requirements that the spaces U and F are reflexive (the Yagi-Fedorov theorem).

Corollary 3. If M is a strongly ( L,p(-sectorial operator on the right (on the left) then

U⁰ Ф U¹ = U (F⁰ Ф F¹ = F) .

Corollary 4. If M is a strongly ( L,p^-sectorial operator on the right and on the left then

• (г) Vu G U LPu = QLu:

3.    Existence of the Inverse Operator

(ri.) Vu G dom M Pu G dom M and MPu = QMu.

Recall that Lk = L | u _k a nd Mk = M |_{dom M}k, while dom Mk = dom M n U^k for k = 0 , 1.

Corollary 5. If M i.s a strongly ( L,p)-sectorial operator on, the right and on, the left then, M0 G Cl (U⁰;F⁰) i.s a bigeetirc operator and M₁ G Cl (U¹;F¹).

On assuming that U and F are Banach spaces, take L E L (U;F) aiid M E Cl (U;F).

We now indicate conditions for the existence of the operator L- ¹ E L (F¹;U¹). To this end, we use an integral of Dunford-Taylor type to define the family of operators {Rt : t >  0 } as

Rt

=     /( ^L - M ) ¹ e^dp-,

2 ni

г where the contour Г satisfies (16), while M is an (L,p)-sectorial operator, and so the integral converges.

Lemma 10. If M is an ( L,p)-sectorial operater then the family {R^l : t >  0 } defined in (18) is analytic in the sector {т E C : | arg т| <  0 — п/ 2 }.

Lemma 11. In the hypotheses of Lemma 10, we have

• (г)    Vt> 0 RtL = Ut шid LRt = Ft

• (и)    Vs,t> 0 Rs ⁺ t = U^sRt = RtF^s.

Lemma 12. If M is a strongly ( L,p^-sectorial operator on the right (on the left) then

• (г)   Vt >  0 Rt = PR ( Rt = RfQ ):

  U im Rt = U¹ ( Vt >  0 ker Rt = F⁰). t>o

  
  

  («)

  
  

We can also observe that, as in the case of semigroups, the images of the operators Rt increase as t decreases: im R^s C im Rt for s > t >  0 follows from claim (ii) of Lemma 11.

Definition 6. An operator M is called strongly ( L,p)-sectorial whenever it is strongly ( L,p )-sectorial on the left and

• VX,^ о ,-..,Op E SL ( M ) \\R^p )( M )( XL — M ) ^{- 1} hr ( FU ) <  -^0^ |λ|        |µk|

  .

  
  

k =0

Remark 6. Every strongly ( L,p )-sectorial operator M is strongly ( L,p )-sectorial on the right.

Remark 7. If the operator L-1 E L(F; U) exists and the operator T = ML-1 (or S = L-1M) is sectorial then M is a st rough" (L,p)-sectorial operator. We can take L [dom M ] as a dense linear О subspace F of F.

Remark 8. If M is an ( L, a )-bounded оperator and to is an inessential singular point then M is a strongly ( L,p )-seetorial operator.

Lemma 13. If M is a strongly ( L,p)-sectorial operator then the family of operators {Rt : t >  0 } defined in (18) is uniformly bounded.

Theorem 5. If M is a strongly (L,p)-sectorial operator then the operator L-1 eL(F1;U1) exists.                           (19)

Remark 9. Condition (19) holds provided that M is a strongly ( L,p )-sectorial operator or (17) is fulfilled and im L 1 = F¹ (Banach's Theorem).

The restriction {Ut : t >  0 } ( {Ft : t E R ₊ } ) of the semigroup {Ut : t >  0 } ( {Ft : t >  0 } ) to the subspace U¹ (F¹) is a nondegenerate analytic semigroup.

Keep the above notation S 1 = L- ¹ M 1 and T 1 = M 1 L 1 1.

Corollary 6. In the hypotheses of Theorem 5, the operator S 1 E Cl (U¹) ( T 1 E Cl (F¹)) is an infinitesimal generator of the semigroup {U* : t >  0 } ( {F* : t E R + } ).

The Hille-Yosida-Feller-Miyadera-Phillips theorem immediately yields

Corollary 7. In the hypotheses of Theorem 5, the operator S 1 ( T 1) is sectorial; furthermore, a^L ( M )= a ( S 1 ) = a ( T 1).

4.    Generalized Splitting Theorem

On assuming that U and F are Banach spaces, take L E L (U; F) aiid M E Cl (U; F) so that M is an ( L,p )-sectorial operator. In addition, assume that

n aL(M) = ^ ajL(M), n E N; furthermore, aL(M) = 0

j =0

lies in a bounded region Dj C C

__________ >

with piecewise smooth boundary dDj = Г j C C , j = 1 , n.

In addition. Dj П aL ( M ) = 0 aiid Dk П D/ = 0 for all j,k,l = 1 , n, k = l.

Theorem 6. If M is an ( L,p ^-sectorial operator and (20) hi elds then there, exist projectors Pj E L (U) ш id Qj E L (F) ftrr j = 1 , n. which are of the form

Pj = ^ [ RL ( M ) d^, Qj = Л [ L l ( M ) d^’ j = 1 ,n.            ⁽²¹⁾

^{2 n}i Г j                     ^{2 n}i г j

Corollary 8. The hypotheses of Theorems f and 6 yield PjP = PPj = Pj a nd QjQ = QQj = Qj.

n

Put P 0 = P — 52 Pj- Corollary 8 implies that P 0 E L (U) is a projector.

j =1

Corollary 9. If M is an ( L,p^-sectorial operator then

• (i)    L 0 E L (U⁰; F⁰) aiid M₀ E Cl (U⁰; F⁰). and moreovei\ the operator M- ¹ E L (F⁰;U⁰) exists: (it) L 1 E L (U¹;F¹) aiid M 1 E Cl (U¹;F¹).

Assume now that, apart from (20), conditions (17) and (19) are fulfilled.

Corollary 10. IfM i.s an ( L,p)-sectorial operator, while (17) and (19) hold, then G = M 0 ¹ L 0 E L (U⁰) is a degree p nilpotent operator, while S = L- ¹ M 1 E Cl (U¹) is a sectorial operator.

Theorem 7. If M is an ( L,p^-sectorial operator and (17), (19), and (20) hold then

U* = Pj U* + P _D U* = Uj + Ut, F * = Qj F * + Q 0 F * = Fj + F* ;

furthermore, we can express Uj* a nd Fj as

Uj =     / RL(M)eL*d^’ j 2 ni JTj l where. Гj. j = 1 ,n is defined in (20).

Fj = 1m [ LL ( M ) e^Ltd^, j = 1 ,n j    2 ni J_Tj ^l

Proof. Indeed, since the analytic semigroup Uj extends to an analytic group, it follows that U0 = Pj. Hence.

= ⁽2 ⁿi ⁾ - 7г, /г

PjUt

RL ( M ) r V ( M ) e^v d^dv

= (2 ni ) ¹

[ f--v J_T RL ⁽ M ' e^V,dv ) = j j = 1 ,n,

ΓjΓ by the residue theorem and the analog of Hilbert’s identity for the L-resolutions

( v - Ц ) r L ( M ) r L ( M ) = R^ ( M ) - r L ( M ) .

This also implies that PjP = PPj = Pj.

□

Put im Pj = U^{1 j} aiid im Qj = F^{1 j} for j = 0 , n . By construction.

U¹ = ф U¹ j <4ul F¹ = ф F^{1 j}. j =0                j =0

Denote by Lj IM^ the restriction of L (M ) tо U j (dom M О U j ) for j = 0 , n . By analogy with Corollary 9, we can easily show that Lj E L (U j ; F j ) and Mj E Cl (U j ; F j ) for j = 0 , n . Furthermore, by (19) the operators L- ¹ E L (F j ; U j ) for j = 0 , n exist. Also, it is not difficult to show by analogy with Corollary 10 that S о = L- ¹ M о E Cl (Uo) is a sectorial оperator. while Sj = L- ¹ Mj : U j ^ U j for j = 1 , n are bounded operators.

5.    Multipoint Initial-Final Value Problem for Sobolev-Type Equations with a Relatively p-sectorial Operator

On assuming that U and F are Banach spaces, take L E L (U; F) aiid M E Cl (U; F) so that M is an ( L,p )-sectorial operator. In addition, assume that conditions (17), (19), and (20) are fulfilled.

Taking Tj E R+ (t, < Tj +1), uj E U for j = 0 , n , and f E C^ (R+;F), consider the problem

Pj ( u ( Tj ) - Uj )=0 , j = 0 ,n,                            (23)

for the linear Sobolev-type equation

Lu = Mu + f.                              (24)

Refer to a vector function u E C 1((0,Tn);U) О C([0,Tn]; U) satisfying (24) as its solution', refer to a solution u = u(t) to (24) as a solution to problem (23). (24) whenever lim Po(u(t) — uo) = 0 t→τ0+ and Pj(u(Tj) — uj) = 0 fc)i‘ j = 1, n.

We are now ready to prove the unique solvability of problem (23) for (24). Since M is an ( L,p )-sectorial operator, while (17), (19), and (20) hold, the problem reduces to

Gu! ⁰ = u ⁰ + M0 ¹ f ⁰ ,                                 (25)

u, ¹ j = Sj u ¹ j + L -j¹ f ¹ j ,j = 0 7n                             (26)

where f ⁰ = (I — Q ) f arid f ^{1 j} = Qjf . while u ⁰ = (I — P ) u and u ^{1 j} = Pju. for j = 1 , n.

Lemma 14. If M is an (L,p"^-sectorial operator, while conditions (П), (19), and (20) are fulfilled, then for every vector function f0 E Cp([0,Tn];F0) О Cp+1((0,Tn);F0)

there exists a unique solution to (25); furthermore, it is of the form u0(t) = - f GqM-1 f 0(q)(t). q=1

Proof. Substituting u ⁰ = u ⁰( t ) into (25), we verify that a solution exists. The successive differentiation of the homogeneous equations (25),

0 = Gpu0( p) = ... = Gut0 = u0, justifies uniqueness.

□

Lemma 15. In the hypotheses of Lemma If, for all uj E U a nd f1 j E C ([0, Tn ]; F1 j) there exists a unique solution to problem uj (Tj) = Pjuj = 0 for the equation with index j in (26); furthermore, it is of the form u1 j (t) = Ujt Tj uj +

τj

Uj^-sLj Qj f ( s ) ds.

Proof. By substitution, we verify that uj = uj ( t ) is a solution to this problem. Suppose that vj = vj ( t ) for t E [0 , Tn ] is another solution to this problem. Construct the vector function w ( s,t ) = LjUjt-sv ( s )• By construction.

dw ( s, t ) ∂s

= Lj jv ( s ) + LjU-dMs * ∂s            ^j ∂s

= 0 .

Hence, w ( Tj,t ) = w ( t, t ). that is. Uj ^Tj.

□

Theorem 8. If M is an ( L,p"^-sectorial operator, while (17), (19), and (20) hold, then for every vector function with f ⁰ E C^p ([0 ,T_n ]; F⁰) C l C^{p +1}((0 ,T_n ); F⁰) at id f ¹ E C ([0 , Tn ]; F¹) there exists a unique solution to problem (23), (2f); furthermore, it is of the form

u ( t )= u ⁰( t )+ ff uj ( t ) .

j =1

6.    The Linear Model of Plane-Parallel Thermal Convection in Viscoelastic Incompressible Fluid

Consider the linear model

( A - A)A ^t = v A ^ - a9x + f, 9t = 5 A 9 - в^х + Z

of plane-parallel thermal convection in viscoelastic incompressible fluid in the region И = (0 , a ) x (0 , b ) E R² with Benard’s boundary conditions

^ ( x, 0 , t ) = A ^ ( x, 0 , t ) = ^ ( x, h, t ) = A ^ ( x, h, t ) = 0 ,                   (28)

9 ( x, 0 , t ) = 9 ( x, h, t ) = 0 ,	(29)
the functions ^ and 9 are periodic in x with period l.	(30)

Put U = V x W and F = G x H. where V = {v E W 24(Q) : v satisfies (28). (30 } and W = G = H = L 2(Q). De fine L and M as

∂

L =

∂x

It is obvious that L E L(U; F). wlrile M E Cl(U; F) with domM = V x {w E W22 (Q) : w satisfies (28) and (29)}.

In order to prove that M is an ( L, 0)-sectorial operator, consider the eigenfunctions of the Laplace operator A on Q satisfying (28) and (30). It is convenient to split these eigenfunctions into three families:

F 1 = Icos2 nmr _sin nny\   F 2 = (sin^ nk? sin niy t    F , = (sin П П} .

a b        ab       b where j. k. l. m. n E N. Henceforth we denote the norm;;dized functions in each family by фmn. Ф k^ a nd ф3, while the corresponding eigenvalues by X mn, X ki, a nd Xj. To construct the operator

( цL - M )

1 we apply Fourier's metln)d: expand the functions v. w. g. and h into Fourier series with respect to the functions {фmn}и{фki}и{ф3} and insert the resulting series into the system ц (X — A)A v — vAv — awx = g, (ц — 6 A) w — evx = h.

Applying a series of orthogonal projectors yields blocks of six equations:

^Xmn ^{[ ц ( X  X}mn )   ^vXmn ^] vmn

-

πm _{2     1}

a wmn gmn.

a

^X ki ^[ ц ^{( X} - ^X ki ) ^{— vX} k ^] vki ⁺ ^—wk i = g ² 1, ^{X j}[ ^{ц ( X} - ^{X j)} - ^vXj^ vj = gj.

^{( ц}  ^^Xmn ) ^wmn

-

πm _{2     1}

в a wmn hmn,

⁽ Ц ^{— 5X}kl ) w ² l ^{+ enkv}kl = ^hkl.

( ц - 5Xj ) wj = hj.

To solve this system, observe firstly that without loss of generality we may take k = m and l = n. Observe in addition that X mn = X mn ; thereto re, put X mn = X mn = X_mn . Solving (31), we obtain the L -resolvent of M as the square matrix A = ||Aij|| ^-=1 whose entries we can express as

A 11 = £ A mn X

A 22

m,n

— A mnX m,n

A 33

= £      (". Фз) Фз j X3 [ц ( X — X3 )

• mn ^{[ ц ( X X}mn ) ^vXmn ^] ( •, ^фmn ) ^фmn, A 15 = £ ^A mn^a ^XmTK ( •.^mn ) ^фmn, m,n

• mn ^{[ ц ( X — X}mn ) ^{— vX}mn ^] ( ", ^фmn ) ^фmn, A 24 = ^— £ ^A mna ^anm (•, ^фmn ) ^фmn, m,n

-

—, A 42 = VA mnea vXj ]       ^

m,n

^nm ( ". ^фmn ) ^фmn,

A 44

A 55

^2 ^A mn ^{( ц 6X}mn ) ( •,^фmn /i ^фmn, A 51 m,n

= £ ^A mn ^{( ц — 6X}mn ) ( ", ^фmn) ^фmn, A 66 m,n

- £ ^a mn ^ea ^{1 n}m (;,^ф mn) ф mn. m,n

=

⟨ ·, φj ⟩ φj

jµ

- δλj .

Неге Аmn = Xmn[ц(Л — Xmn) — vXmn](ц — 5Xmn) + ава 2п2m2 aiid Xj = Xj. while all remaining matrix entries are equal to the zero operator O. This implies, firstly, that the L-spectrum of M is

a^L ( M ) = |

νλmn

λ

λmn

+ ε_mn ∪ {δλ_mn - ε_mn} ∪ _λ ^ν _- ^λj _λj ∪ {δλj}.

Here

εmn

∼

m2

X_mn ( X — X_mn )

as m, n ^ to, and since Xmn ~ —m2 — n2 as m, n ^ to, it follows that there exists a sector of the required opening angle which includes ^L(M). Secondly, for sufficiently large |ц| outside this sector we have max {llRL(M)||£(u), ||LL(M)^(f), } < const |ц|-1.

This justifies

Lemma 16. For all a,0,X,v E R a nd 5 E R+, the operator M is ( L, 0)-sectorial.

Let us now verify (17) and (19) Since U and F are reflexive spaces, Lemma 16 and the Yagi-Fedorov theorem imply that condition (17) is fulfilled. Furthermore,

• (i)    U⁰ = F⁰ = { 0 } , U¹ = U, and F¹ = F if X = Xmn and X = Xj'

• (ii)    U⁰ = F⁰ = ker L = span { col( фj, 0) } . U¹ = {u E U : (u, фj) = 0 }. ar id F¹ = {f E F : (g, фj) = 0 } = im L if X = Xmn aiid X = Xj;

(in) U⁰ = F⁰ = ker ^l = span ^{{ coi(} ф mn, ^{0) , coi(} ф m„, ^{0) }} . F¹ = ^{f ^e F^: g,Ekmn =

0 ,k =  1 , 2 }  =  im L, U¹  = { u E U : v = v + Vmn ( w ) , (v,фтп) = 0 ,k = 1 , 2 , vmn ( w ) =

=^{2 nm}a- ^{1 v} - ^{1 X}mn ( w ф mnk mn ⁺ w ф т_п)ф mn )} if ^x = xmn arid ^x = j

Condition (19) is also fulfilled; furthermore, we can express the operator L- ¹ as the square matrix A = ||Aj|| ² j =₁ with

A11 = Y m,n

·, φmn φmn

X_mn ( X — Xmn )

+ m,n

·, φmn φmn

X_mn ( X — Xmn )

+

j

⟨ ·, φj ⟩ φj

^Xj ^{( X — X}j ) ,

A21 = Vmn, A21 = O, A22 = I, where

Vmn = ( O if X = X_mn, t ^{anma v X}mn ⁽{ ", ^фmn^ ^фmn ⁺ { ", ^фmn^ ^фmn ) ^ ^X = ^Xmn-

The prime on the sum symbol indicates that the terms with X = X_mn о г X = Xj are absent. This justifies

Lemma 17. Conditions (11) and (19) are fulfilled for all a,e,X,v E R, a nd 5 E R+.

By (32), the L -spectrum ct ^l ( M ) оf M is discrete. This means that the hypotheses of Theorem 6 hold as well; moreover, they do for every closed contour y E C bounding a region which contains finitely many points of ct ^l ( M ) and is disjoint from ct ^l ( M ). Therefore, the hypotheses of Theorem 8 hold, and so we have

Theorem 9. For all a, e,X,v E R, 5, Tj E R+, Uj E U for j = 0 , n, and ^,Z E C ¹([0 , T_n ]; L ₂(H)) there exists a unique solution to problem (23) for (27) with boundary conditions (28)-(30).

The author is grateful to Professor G. A. Suiridyuk for fruitful discussions and interest in this work.