A multipoint initial-final value problem for a linear model of plane-parallel thermal convection in viscoelastic incompressible fluid Текст научной статьи по специальности «Математика»

ОБЗОРНЫЕ СТАТЬИ

MSC 35К70, 60Н30 DOI: 10.14529/mmpl40301

A MULTIPOINT INITIAL-FINAL VALUE PROBLEM FOR A LINEAR MODEL OF PLANE-PARALLEL THERMAL CONVECTION IN VISCOELASTIC INCOMPRESSIBLE FLUID

S. A. Zagrebina, South Ural State University, Chelyabinsk, Russian Federation, zagrebina_sophiya@mail.ru

The linear model of plane-parallel thermal convection in a viscoelastic incompressible Kelvin-Voigt material amounts to a hybrid of the Oskolkov equations and the heat equations in the Oberbeck-Boussinesq approximation on a two-dimensional region with Benard’s conditions. We study the solvability of this model with the so-called multipoint initial-final conditions. We use these conditions to reconstruct the parameters of the processes in question from the results of multiple observations at various points and times. This enables us, for instance, to predict emergency situations, including the violation of continuity of thermal convection processes as a result of breaching technology, and so forth.

For thermal convection models, the solvability of Cauchy problems and initial-final value problems has been studied previously. In addition, the stability of solutions to the Cauchy problem has been discussed. We study a multipoint initial-final value problem for this model for the first time. In addition, in this article we prove a generalized decomposition theorem in the case of a relatively sectorial operator. The main result is a theorem on the unique solvability of the multipoint initial-final value problem for the linear model of plane-parallel thermal convection in a viscoelastic incompressible fluid.

Keywords: multipoint initial-final value problem; Sobolev-type equation; generalized

splitting theorem; 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

Lu = Mu + f (1)

and nonlinear

Lu = Mu + N (u) + f (2)

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 € H, j = M, ^

—to < a <t0 <r\ < ... < Tj < Tj+\ < ... <b < +to,

where Pj are relative spectral projectors (we discuss them in Section 4), 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].

n=1

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 P0 and P\ 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

n=1

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 a-boundedness of an operator only in the case that the L-resolvent of M has a removable 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 a-boundedness in the case that the L-resolvent of M has a pole at infinity. Then, relatively strongly p-sectorial

p

p

Namely, Dudko studied [1] the case that both operators are closed and the spaces U and F coincide;

p

sectorial operators; Keller found [21] sufficient, and in some cases necessary, conditions for the

p

p

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],

(A - V2)vt = vV2v — (v ■ V)v — Vp + gjS, V St = 5VS - v ■VS + y ■ v,

0,

(4)

v

modeling thermal convection in a viscoelastic incompressible Kelvin-Voigt material [23]. Here v = (v\,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 € R, v € R+ and 5 € R+ characterize the elasticity, viscosity, and thermal conductivity of the fluid; g € R+ is the free fall acceleration; finally, y = (0,. . . , 0,1) € Rn and x = (x\,x2, ■ ■ ■ ,xn). WThen one of the horizontal components of the velocity vanishes, (4)

becomes

/х 2 д (ф, Аф) дв

(А-А)АJL = vАОф- ™-*>+ а-,

дв = яАИ- д(Ф,в) + йдф И

dt д(х, у) дх ’

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 A. 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

(A — А)Афі = —v А2ф — авх + C, ві = 5Ав — вфх + Z (6)

for the linear mathematical model of plane-parallel thermal convection in viscoelastic incompressible fluid in the region Q = (0, a) x (0, b) Є R2 with Bernard’s boundary conditions

ф(х, 0, t) = Аф(х, 0, t) = ф(х, b, t) = Аф(х, b, t) = 0, (7)

в(х, 0,t) = в(х,Ь,і)=0, (8)

the functions ф and в are periodic in х with period a. (9)

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.

p

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

p

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). WTe 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). WTe should note that the author already discussed

( L, p)

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 € L(U; F) and a closed linear operator M € Cl(U; F) whose domain is dense. Introduce the L-resolving set pL(M) = [p € C : (pL — M)-1 € L(F; U)} and the ^^^rum aL(M) = C\ pL(M) of M. Provided that pL(M) = 0, we can introduce the right and left

K,r>(M) = n RLk(M) lL„,(m) = n lL„(M)

k=0 k=1

(L,p)-resolutions of M. Here RL(M) = (pL — M)-1L and LL(M) = L(pL — M)-1, while

pk € pL(M) for k = 0,... ,p.

Definition 1. [15] An operator M is railed p-sectorial relatively to an operator L with p € [0}UN (or briefly, (L, p)-sectorial) whenever there exist constants K € R+, a € R, and 0 € (n/2,n) such that

SL&(M) = [p € C : \arg(p — a)\ < 0, p = a} C pL(M); (10)

furthermore,

max -

rL»,v)M)........> LLu,v)(M) _ \ — (ii)

L(U)

L

J(n ,p)(

K

l(f) ) p , ,

n \pk—a\

k=0

for arbitrary pk € SL@(M) for k = 0,... ,p.

Remark 1. If M is an (L, p)-sectorial operator and b > a then the operator Ml = M — bL is also ( L, p ) a

SLe(M ) = SL(M).

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

Lemma 1. If M is an (L,p)-sectorial operator then there exist R > 0 and C > 0 such that \\(pL — M)-1||£(F;U) < C\p\pfor all p € SL(M) \ [p € C : \p\ < R}.

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

M M (L, 0)

(ii) If M is an (L, p)-bounded operator and to is a pole of order at most p € N of the L-

M M ( L, p )

( L, p ) 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;

(Hi) ker rL^p)(M) n \mR(^pp)(M) = [0} and ker LL^p)(M) n imLL'up)(M) = [0};

(iv) the operator M—1 € L(F0;U0) exists.

On assuming that M is an (L, p)-sectorial operator, recall the notation H = M—1L0 and G = LoM—1. Denote by U1 the closure of the linear subspace im^L^p)(M). Denote by U the

closure of the linear subspace UP+rimR^^p)(M) in the norm of U. Denote by F1 the closure of the

linear subspace imLj-'^p)(M), and by F the closure of the linear subspace F^imL^ p)(M) in the F

( L, p ) M

H G p

(ii) lim (pRL(M))p+1u = u for every u € U1 and lim (pLL(M))p+1f = f for every

f € F1;

(Hi) U = U0 ® U1 an dF = F0 ® F1.

Denote the projector onto ^parallel to U0 by P = s- lim (pRL(M ))p+1 and the projector

onto F1 parallel to F0 by Q = s- lim (pLL(M))p+1.

i—1

2. Degenerate Analytic Resolving Semigroups of Operators

On assuming that U and F are Banach spaces, take an operator L € L(U; F) and an operator

M € Cl(U; F). The equation

Lu = Mu

reduces to the equivalent pair of equations

RL(M )U = (pL — M )-1Mu, (13)

LL(M )f = M (pL — M )-1f. (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 € C(V). Refer as a solution to (15) to a vector function

v(t) € C^(R+; V) satisfying this equation for t > 0 and continuous at 0.

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

(i) VsV1 = Vs+t for all s,t> 0;

(ii) for every v0 € V the function v(t) = Vtv0 is a solution to this equation.

A semigroup [Vt : t > 0} is called analytic whenever it can be analytically continued to some

sector £ C C including the ray R+ that is, there exists an analytic mapping V* : £ ^ L(V) enjoying properties (i) and (ii) of the previous definition (with s,t € £), coinciding with V* on the positive semi-axis. In addition, [Vl : t > 0} is c^led um/ormZy bounded whenever \\Vt\\£(V) < const fa a 11 t € 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 (14)) which is analytic in the sector

£ = [t € C : \ argt\ < 0 — n/2 with t = 0},

0

by the integrals

Ut = 2tJ RL(M)eitdp (Ft = 2-J LL(M)eitdp) (16)

r r

of Dunford-Taylor type, where t € R+ and the contour T C S^^M) satisfies \ arg p\ 0 as

p ^ to and p € r.

Lemma 4. If M is an (L,p)-sectorial operator then lim Utu = u for every u € imRf p)(M)

t—0+ (i >p)

and tlirn Ft f = f for every f € imL°^ p)(M)).

Lemma 5. If [Vt : t > 0} is an analytic semigroup then ker Vtl = ker Vt2 for all t1,t2 > 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.

U* F*

ker U* = [^ € U : UV = 0 3t € R+}, ker F* = [^ € F : F= 0 3t € R}.

Put U0 = ker U* and F0 = ker F*. Denote by L0 the restriction of L to U0, and by M0 the restriction of M to U0 n 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 RL(M) and ker Ft D ker L]°(M) for every t > 0.

The kernel of an analytic semigroup is obviously a subspace. Denote by L0 (M0) the restriction of L (M) to ker U* (ker U* n dom M').

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

L0 € £(ker U*; ker F*), M0 : ker U* n dom M ker F*.

Denote by @0) (Ml) the L0-spectrum of M0.

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

Corollary 1. If M is an (L,p)-sectorial operator then the operator M-1 € £(ker F*; ker U*)

exists.

Theorem 2. If M is an (L,p)-sectorial operator then ker U* = U0 and ker F* = F°.

Definition 4. Refer as the image of a semigroup [Vl : t > 0} to the set

imV* = [v € V : v = lim Vfv}.

t—0+

Lemma 8. Every analytic semigroup [Vt : t > 0} satisfi es ker V* n imV * = [0}.

Lemma 9. If [Vt : t > 0} is a strongly continuous and uniformly bounded semigroup then

imV* = imVt .

t>0

Theorem 3. If M is an (L,p)-sectorial operator then imU* = U1 and imF* = F1-

Put Ut = Uand Ft = FtU.

IU IF

U* F*

im U* = [u € U : lim Utu = u}, im F* = [f € F : lim Ftf = f}.

t—0+ t—0+

U1 = im U* F1 = im F* L1 L U1 M1

M U1 dom M

Corollary 2. If M is an (L,p)-sectorial operator then

P = s- lim U*, Q = s- lim F*

t——0 + t——0 +

The operators

P = s- lim U e L(U), Q = s- lim Fl e L(F),

t^0+ t^0+

whenever they exist, are called the units of the semigroups {U* : t > 0} and {F* : 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,p0,p1, ...,pp € Sq(M) we have

\\Rl>p)(M)(XL - M)-1 Mu\\u <

const(u)

\M П \^k\ k=0

for arbitrary u € dom M (respectively, there exists a dense linear subspace F of F such that

||M(XL - m)-1ll{iip)(M)f |If < const(/)

\MH \vk\ k=0

for arbitrary / €F)-

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 operator and to is a pole of order at most p then M is ( L, p )

M ( L, p)

of the semigroups {U* : t > 0} and {F* : t > 0}) exist. Furthermore, the operators P € L(U) and Q € L(F) satisfy

L € L(ker P;ker Q) n L(imP;imQ), M €Cl(ker P; ker Q) nCl(imP;imQ).

M ( L, p )

the additional requirements that the spaces U and F are reflexive (the Yagi-Fedorov theorem). M ( L, p )

U0 ® U1 = U (F0 ® F1 = F). (17)

M ( L, p )

(i)Vu € U LPu = QLu;

(ii) yu € dom M Pu € dom M and MPu = QMu.

Recall that Lk = L^ and Mk = M|dom while dom Mk = dom M nUk for k = 0,1.

M ( L, p) M0 €

Cl(U0; F0) is a bijective operator and M1 € Cl(U1; F1).

3. Existence of the Inverse Operator

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

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

R* = 2— J (pL — M )-1e^*dp, (18)

r

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

Lemma 10. If M is an (L,p) -sectorial operator then the family {R* : t > 0} defined in (18) is analytic in the sector {t € C : | arg t| < 0 — n/2}.

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

(i) it> 0 R*L = U* and LR* = F*;

(ii) is,t> 0 Rs+* = UsR* = R*Fs.

M ( L, p )

(i) it > 0 R* = PR* (R* = R*Q);

(ii) (J imR* = U1 (it > 0 ker R* = F°)-

*>0

R*

as t decreases: imRs C imR* for s > t > 0 follows from claim (ii) of Lemma 11.

M (L, p) (L, p)

sectorial on the left and

const

iX,p0, ...,pp € SL(M) l|R(Lx ,p)(M)(XL — M)-1 |£(F;U) < —p------------------.

|X| n lpk1 k=0

(L, p) M (L, p)

Remark 7. If the operator L-1 € L(F; U) exists and the operator T = ML-1 (or S = L-1M) is sectorial then M is a strongly (L,p^^^^^^^^^ operator. WTe can take L[dom M] as a dense linear

O

FF

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

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

M ( L, p )

the operator L-1 € C(Fl; U1) exists. (19)

M ( L, p )

im L1 = F1

The restriction {U\ : t > 0} ({F* : t € R+}) of the semigroup {U* : t > 0} ({F* : t > 0}) to the subspace U1 (F1) is a nondegenerate analytic semigroup.

Keep the above notation S1 = L- 1M1 and T1 = MiL-1

Corollary 6. In the hypotheses of Theorem 5, the operator S1 € C/(U1) (T1 € Cl(F1)) is

an infinitesimal generator of the semigroup {U\ : t > 0} ({F* : t € R+}).

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

Corollary 7. In the hypotheses of Theorem, 5, the operator S1 (T1) is sectorial; furthermore, aL(M) = a(S1) = *(Ti).

4. Generalized Splitting Theorem

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

n

Tj(M), n Є N; furthermore, j

j=0

Dj С C

aL(M) = [J af(M), n Є N; furthermore, gi-(M ) = 0

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

In addition, Dj П Oq(M) = 0 and Dk П Di = 0 for all j, k,l = 1,n,k = l.

Theorem 6. If M is an (L,p)-sectorial operator and (20) holds then there exist projectors Pj Є L(U) and Qj Є L(F) for j = 1,n, which are of the form

(20)

Pj = Y~i RL(M)dp, Qj = — LL(Mw, j = 1,n. (21)

J Г т J Г т

Corollary 8. The hypotheses of Theorems 4 and 6 yield PjP = PPj = Pj and QjQ = QQj = Qj.

n

Put Po = P — Pj- Corollary 8 implies that P0 € L(U) is a projector.

j=i

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

(i) L0 € L(U0; F0) and M0 € Cl(U0; F0) and moreover, the operator M—1 € L(F°U0) exists;

(гг) Li € L(U1; F1) and Mi € C/(U1; F1)-

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

Corollary 10. If M is an (L,p)-sectorial operator, while (17) and (19) hold, then G = M—1L0 € L(U0) is a degree p nilpotent operator, while S = L—1M1 € C^U1) is a sectorial operator.

Theorem 7. If M is an (L,p)-sectorial operator and (17), (19), and (20) hold then U = PjU + P0U = Uj + U0, Fb = QjF1 + Q0F1 = Fjj + F0; furthermore, we can express Ujj and Fj as

Ujt = ^ jT RL(M)^dp, Fj = 2- Г Li(M)e^dp, j = in (22)

where Tj, j = l,n is defined in (20).

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

P jUt = (2ni)-1 J j RL(M)RL(M)evtdpdv =

= (2«)_1 (IRi(M)dP + ^ I RL(M)evtdv'j = Uj. j = 1.«.

□

lr v — 1 Jr JFj 1 — v Jr

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

(v — 1)RL(M )Rlv(M ) = rL(M ) — Rlv(M ).

This also implies that PjP = PPj = Pj.

Put im Pj = U1j and im Qj = F1j few j = 0, n. By construction,

n n

U1 = 0 U1^d F1 = © F1j.

j=0 j=0

Denote by Lj (Mj) the restriction of L (M) to Uj (domM n U^^^or j = 0, n. By analogy with Corollary 9, we can easily show that Lj € L(Uj; Fj) and Mj € Cl(Uj; Fj) for j = 0, n. Furthermore, by (19) the operators L-1 € L(Fj; Uj) for j = 0, n exist. Also, it is not difficult to show by analogy with Corollary 10 that S0 = L-1M0 € Cl(U0) is a sectorial operator, while Sj = L-1Mj : Uj ^ Uj 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 € L(U; F) and M € 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 € R+ (rj < Tj+1), Uj € U for j = 0, n, rnd f € 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 € C 1((0, rn);U) n C([0, rn];U) satisfying (24) as its solution; refer to a solution u = u(t) to (24) as a solution to problem (23), (24) whenever lim P0(u(t) — u0) = 0

t^ro+

and Pj (u(rj) — uj) = 0 for j = 1, n.

M

an (L,p)-sectorial operator, while (17), (19), and (20) hold, the problem reduces to

Gu° = u0 + M-1f0, (25)

u1j = Sjulj + L-1 f1j, j = 0,n (26)

where f0 = (I — Q)f and f1j = Qj f, while u0 = (I — P)u rnd u1j = Pju, for j = 1, n.

Lemma 14. If M is an (L,p)-sectorial operator, while conditions (17), (19), and (20) are fulfilled, then for every vector function

f0 € Cp([0, Tn]; F0) n Cp+1((0, Tn); F0)

there exists a unique solution to (25); furthermore, it is of the form

u0(t) = — £ Gq M-1f 0(q)(t). q=1

Proof. Substituting u0 = u0(t) into (25), we verify that a solution exists. The successive differentiation of the homogeneous equations (25),

0 = Gpu0(p) = ... = Gu0 = u0,

justifies uniqueness.

□

Lemma 15. In the hypotheses of Lemma 14, for all uj € Urn, d f1j € C ([0,Tn]; F1j) 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

f t

u1j (t) = U— u3 + / Ujj^L—Qjf(s)ds.

JTj

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

9w(s,t) = L dUj-S V(s) I L U t-sdv(s) =0

= Lj^^v(s) + LjUj ~o= °.

tTj

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

□

Theorem 8. If M is an (L,p)-sectorial operator, while (17), (19), and (20) hold, then for every vector function with f0 € Cp([0,Tn]; F0) n Cp+1((0, Tn); F0) and f1 € C([0,Tn]; F1) there exists

a unique solution to problem (23), (24); furthermore, it is of the form

n

u(t) = u0(t) + ^ 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 = vA-0 — adx + dt = 5 Ad — fi^x + ( (27)

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

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

d(x, 0, t) = d(x,h,t) = 0, (29)

the functions ^ and d are periodic in x with period l. (30)

Put U = V x W and F = G x H where V = {v € W|(Q) : v satisfies (28), (30} and W = G = H = L2(Q). Define L and M as

L =((A —0A)An • M =

vA2 \

ox

d

Vnx 5A /

dx

It is obvious that L € L(U; F), while M € Cl (U; F) with

dom M = V x {w € W22(Q) : w satisfies (28) and (29)}.

M (L, 0)

Laplace operator A on Q satisfying (28) and (30). It is convenient to split these eigenfunctions into three families:

^ f 2nmx . nn^l ^ f . 2nkx . nl^l ^ f . njxl

F1 = I cos sin-^ L F2 = sin —— sin L F3 = <^ sin ^ ,

where j k, l> m, n € N Henceforth we denote the normalized functions in each family by _]nni _ki, and _3, while the corresponding eigenvalues by Anm, Ajy, and A3. To construct the operator (lL — M)-1 we apply Fourier’s method: expand the functions v, w, g, and h into Fourier series with respect to the functions {_]nn} U {_ki} U {_3} and insert the resulting series into the system

l(A — A)Av — vAv — awx = g, (1 — 5A)w — (3vx = h.

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

\1r/\ \1\ \1l1 nrm 2 1

Amn[1(A Amn) V Amn\vmn a a wmn gmn,

Ah [1(A — Ah) — vAki\vki + ankwli = gh,

a3[i(a — j — vA3]vf = g3,

/ 1 anm 2 ;1

(l 5Amn)wmn P a wmn hr

(i— 5Ah)wh + pnkvii = hh

(31)

)wki + p—vki = hki,

(l — 5A3 )w3 = h3.

To solve this system, observe firstly that without loss of generality we may take k = m and l = n. Observe in addition that Amn = Amn] therefore, put Amn = Amn = Amn. Solving (31), we obtain the L-resolvent of M as the square matrix A = \ \Aij\\6j=1 whose entries we can express as

A11 = Xy AmnAmn[l(A — Amn) — vAmn] _m^ _mni A15 = Xy Amna anm (•, _mn) _mni

m,n m,n

A22 = ^2 AmnAmn[l(A — Amn) — vAmn] __nn) _mn, A24 = — ^2 Amna ^anm {_mn) _mn,

m,n m,n

A33 = £ x_ [u([,fy f- vA■ ] ’ A42 = J2AmnPa-nm (•, _mn) flnn, j jLPV JJ Ji m,n

A44 ^2 Amn(l 5Amn) fm^ fmn, A51 Amnpa nm fmn fmn,

m,n m,n

A55 = ^2 Amn(l — 5Amn) fmn) fmn, A66 = Y2 . _ \\ ' .

m,n j 15 Aj

Here Ann = Amn[i(A — Amn) — vAmn](i — 5Amn) + afa-2n2m2 and Aj = A3, while all remaining matrix entries are equal to the zero operator O. This implies, firstly, that the L-spectrum of M is

UL(M) = / ----- -----+ £mnl U {5Amn — £mn} U \ T----U {5Aj } . (^2)

t A Amn ) t A Aj J

Here

\^mn\

/

2

m2

Amn(A Amn)

as m,n ^ to, and since Amn ~ —m2 — n2 as m,n ^ to, it follows that there exists a sector of

the required opening angle which includes aL(M). Secondly, for sufficiently large \i\ outside this sector we have

max {\\R1L(M)\\£(u),\\lL(m)\\£(f)J < const \l\-1.

This justifies

Lemma 16. For all a, f,A,v € R an d 5 € R+ the opera tor M is (L, 0)-sectorial.

UF

Yagi-Fedorov theorem imply that condition (17) is fulfilled. Furthermore,

(i) U0 = F0 = {0} U1 = U, and F1 = F if A = Amn and A = Aj;

(ii) U0 = F0 = ker L = span {co1(_j, 0)} U1 = {u € U : {u,_j} = 0} and F1 = {f € F : (g, _j} = 0} = im L if A = Amn and A = Aj;

(iii) u0 = F° = kerL = span{co1 (_lmn-,0), 0)} f1 = {f € F : {g,_'mn} =

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

= 2nma lv 1АтП ((w, фхтп)ф2тп + (w, ф2тп)ф1п)}г fA = \mn and A = j

L1-1

ij \Hj-

matrix A = \ \Aij||27-=1 with

A = I ф1т^ ф1пп + I фгп^ фгпп + I (',ф]) <fj

т,п Amn(A — Amn) т,п Amn(A — Amn) j Aj (A — Aj)

A21 = Vmu, A21 = Oj A22 = Ij

where

{

O if A = Ann,

anma v Amn((', _mn) _mn + (', _mn) _mn) ^ A = Amn.

The prime on the sum symbol indicates that the terms with A = Amn or A = Aj are absent. This justifies

Lemma 17. Conditions (17) and (19) are fulfilled for all a,f,A,v € R, and 5 € R+.

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

Theorem 9. For all a, f,A,v € R, 5, Tj € R+ uj € U for j = 0, n, and £,( € C 1([0, Tn]; L2(Q)) there exists a unique solution to problem (23) for (27) with boundary conditions (28)-(30).

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

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Received May 14, 20Ц

МНОГОТОЧЕЧНАЯ НАЧАЛЬНО-КОНЕЧНАЯ ЗАДАЧА ДЛЯ ЛИНЕЙНОЙ МОДЕЛИ ПЛОСКОПАРАЛЛЕЛЬНОЙ ТЕРМОКОНВЕКЦИИ ВЯЗКОУПРУГОЙ НЕСЖИМАЕМОЙ ЖИДКОСТИ

С.А. Загребина

Линейная модель плоскопараллельной термоконвекции вязкоупругой несжимаемой среды Кельвина - Фойгта представляет собой гибрид системы уравнений Осколкова и уравнения теплопроводности в приближении Обербека - Буссинеска, заданных в двумерной области с условиями Бенара. Целью нашего исследования является разрешимость этой модели с так называемыми многоточечными начально-конечными условиями. Такие условия используются для восстановления параметров изучаемых процессов по результатам многочисленных наблюдений с различных точек и в различные моменты времени, что позволяет, например, прогнозировать аварийные ситуации, в том числе нарушение непрерывности процесса термоконвекции в результате нарушения технологии и т.п.

Ранее для моделей термоконвекции изучалась разрешимость задач Коши и начально-конечной, кроме того, была рассмотрена устойчивость решений задачи Коши. Многоточечная начально-конечная задача для этой модели изучается впервые. Кроме того, в данной работе приводится доказательство обобщенной теоремы о расщеплении в случае относительно секториального оператора. Основной результат статьи - теорема об однозначной разрешимости многоточечной начально-конечной задачи для линейной модели плоскопараллельной термоконвекции вязкоупругой несжимаемой жидкости.

Ключевые слова: многоточечная начально-конечная задача; уравнение соболевского типа; обобщенная теорема о расщеплении; линейная модель плоскопараллельной термоконвекции вязкоупругой несжимаемой жидкости.

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Софья Александровна Загребина, кафедра «Дифференциальные и стохастические уравнения, Южно-Уральский государственный университет (г. Челябинск, Российская Федерация) , zagrebina_sophiya@mail.ru.

Поступила в редакцию Ц мая 2014 г.