The multipoint initial-final value condition for the Navier–Stokes linear model Текст научной статьи по специальности «Математика»

MSC 47D06, 35K70

DOI: 10.14529/mmpl50111

THE MULTIPOINT INITIAL-FINAL VALUE CONDITION FOR THE NAVIER - STOKES LINEAR MODEL

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

A.S Konkina, South Ural State University, Chelyabinsk, Russian Federation, alexandra.konkina@yandex.ru

The Navier - Stokes system models the dynamics of a viscous incompressible fluid. The problem of existence of solutions of the Cauchy - Dirichlet problem for this system is included in the list of the most serious problems of this century. In this paper it is proposed to consider the multipoint initial-final conditions instead of the Cauchy conditions. It should be noted that nowadays the study of solvabilityof initial-final value problems is a new and actively developing direction of the Sobolev type equations theory. The main result of the paper is the proof of unique solvability of the stated problem for the system of Navier -Stokes equations.

Keywords: relatively p-seetorial operators; the multipoint initial-final value condition; the Navier - Stokes linear model.

Assume that Q C Rn, n = N \ {1}, is a bounded domain with boundary dQ of class CIn the cylinder Q x R+ consider the Dirichlet problem

v(x,t)=0, (x,t) e dQ x R+ (1)

for the system of equations

vt = vV2v -Vp, V • v = 0, (2)

which models a linear approximation of dynamics of a viscous incompressible fluid. Transfer system (2) to

vt = vV2v - p, V(V • v) = 0, (3)

replacing Vp ^ p [1]. This approach is different from the proposed in classical monographs [2, 3]. It is based on [4] and was developed in [1, 5, 6]. This article focuses on study of solvability of multipoint initial-final problem for the classical linear Navier - Stokes model (1), (3).

1. Multipoint Initial-Final Conditions. Assume that U and F are Banach spaces, consider a continuous linear operator. L e L(U; F) and a closed linear operator M e Cl(U; F) whose domain is dense in U Let M be (L,p)-swtorial, p e {0} U N. Consider problem (1), (3). It can be reduced to a linear Sobolev type equation

Lu = Mu + f. (4)

( L, p ) M

existence of degenerate analytic semigroups of operators

Ut = i (pL - M)-1Le^dp and Ft = f L(pL - M)-le^td^, 2ni Jr 2ni Jr

UF

ker U' = U0, ker F' = F0 and image imU' = U1, imF' = F1 of these semigroups. It is easy to show that U0 ® U1 = U0 ® U1 = U0 ® U^ F0 ® F1 = F0 ® F1 = F0 ® F1- We need a stronger statement

U0 ® U1 = U (F0 ® F1 = F), (A1)

( L, p) M p e

{0} U N U F

Denote by Lk (Mk) the restriction of L (M) on Uk (domM n Uk), k = 0,1. If operator M is strongly (L,p)-secioria/ on the right (left), p € {0} U N, then Lk € L(Uk; Fk) Mk € Cl(Uk; Fk), k = 0,1, and there exists an operator M-1 € L(F0;U0), and projector P = s — ^lirn U\ (Q =

s — ^lirri Ff) splitting space U (F) according to (Al), so that U1 = imP (F1 = imQ). Introduce another condition:

there exists an operator L-1 € C(Fl; U1), (^2)

(L, p) M p € {0} U N

(L, p) M p € {0} U N

(L, p) M p € { 0} U N

(L, p) M p € {0} U N

references therein). Then the operator G = M-1L0 € L(U0) is nilpotent of degree p, and the operator S = L-1M1 € Cl(U1) is sectorial.

M

aL(M) = У aj-(M), n € N, where af(M) = 0 is contained in a bounded j=0

domain Dj С С with piecewise smooth boundary dDj = Г С С. Moreover,

(A3)

Dj П ^L(M) = 0 and Dk П Di = 0 for all j, k,l = l,n,k = l. Construct a relatively spectral projectors [8] Pj € L(U) and Qj € L(F), j = l,n, given by

Pj = I (»L - M)-iLd/j., Qj = 2" I L(^L - M)-ldv., j = Щ (5)

and it turns out that under the condition of (L, p)-sectoriality of operator M and conditions (Al), (A2), PjP = PPj = Pj and QjQ = QQj = Qj, j = l,n. Therefore, in this case there is a

n

projector Po = P - £ Pj, Po € L(U).

j=i _

So, suppose that conditions (Al) - (A3) hold, commit Tj € R+ (rj < rj+i), Uj € U, j = 0,n, and consider the multipoint initial-final value condition

Pj (u(Tj) - Uj) = 0, j = 0,n (6)

for linear Sobolev type equation (4). Vector function u € Ci((r0,rn);U)ПC([r0,rn];U), satisfying equation (4), is called a solution; the solution u = u(t) of (4) is called a solution of multipoint initial-final value problem (4), (6), if in addition lim P0(u(t) - u0) = 0 and Pj(u(rj) - uj) = 0,

j =T7n.

The study of solvability of the initial-final value problems for nonclassical equations of mathematical physics, including higher order, is actively developing as well as the problem of optimal control of these solutions (see for example, [lj and references therein).

2.Unique Solvability of Abstract Problem. Assume that U and F are Banach spaces, take L € L(U; F) and M € Cl(U; F) such that M is an (L,p)-sectorial operator and assume that (Al) - (A3) hold.

Theorem 1. If M is an (L,p)-sectorial operator, and. (Al) - (A3) hold. Then

n n n n

ut = ^ Pjut + P0U = U + u0» pt = E Qjpt + Q0pt = E Fj + F0, j=i j=i j=i j=i

Вестник ЮУрГУ. Серия «Математическое моделирование 133

и программирование»(Вестник ЮУрГУ ММП). 2015. Т. 8, № 1. С. 132-136

S.A. Zagrebina, A.S Konkina

mereover, we can express Uj and Fj in the form

Uj = — f (uL - M)-1Le^du, Fj = — f L(uL - M)-1 e&dp, j = I~n. (7) J 2ni Jr. J 2ni Jr.

j =- (uL - M)-lLe^du, Fj - 1 T'-T

J 2ni JTj J

Denote by U1j = im P j and F1j = im Qj, j = 0,n. By construction

U1 = 0 U1j and F1 = 0 Fj

j=0 j=0

Denote by Lj (M^ the restriction of L (M) on U1j (domM n U1j), j = 0~n. It is easy to show that the operators Lj £ L(U1j; F1j), Mj £ Cl(U1j; F1j), j = 0, n, and by virtue of (A2) there exists an operator L-1 £ L(F1j;U1j), j = 0,n. Also, it is easy to show that the operator S0 = L-lM0 £ Cl(U0) is sectorial, and the operator Sj = L-1Mj : U1j ^ U1j, j = 1,n is bounded. Now we are ready to prove the unique solvability of problem (6) for equation (4) that is due to (L, p)-sectoriality of operator M, conditions (Al) - (A3) is reduced to the form

Qui0 = u0 + M-1f0, (8)

u1j = Sju1j + L-1f1j, j = 0,n (9)

where f0 = (I - Q)f, f1j = Qjf, u0 = (I - P)u, u1j = Pju, j = 0~n.

Theorem 2. [8] If M is an (L,p)-sectorial operator and (Al) - (A3) hold. Then for any vector function f0 £ Cp([t0, rn]; FP)nCp+1((r0,rn);; F°), f1 £ C([t0,t„]; F1) there exists a unique solution of problem (4), (6), which also has the form

p n / ft \

u(t) = QqM0-1f(q)(t) + Y, Uj-Tjuj + Utj-sL-jQjf (s)ds I .

q=1 j=1 V Jrj J

3. Specific Interpretation. Assume that Q C Rn, n = N \ {1}, is a bounded domain with boundary dQ of class C^. For the reduction of problem (1), (3) to the homogeneous equation (4) (f = 0) we need the functional spaces from [4J. Assume that H2 and H (HCT and Hn) are subspaces

of solenoidal and potential vector functions of space H2 = (W22(Q)n w1(Q))n (L2 = (L2(Q))n). Formula A = diag {V2,..., V2} defines a linear continuous operator with discrete finite-negative spectrum a(A), thicken only to -m. Denoted by Aa(n) the restriction of A on H2(n).

Lemma 1. (Solonnikov - Vorovich - Yudovich theorem). Operator Aa(n) £ L(H^n), HCT(n)), moreover a(A#(n)) = &(A) and A = Aa£ + Ann.

Herein n £ L(H2, H) designates a projector along H2, £ = I - n.

Lemma 2. (Kapitanski - Pileckas theorem). Formula B : u ^ V(V ■ u) defines the operator B £ L(H2, Hn), with ker B = H2.

Let U = F = H x Hn x H^, where Hp = Hn, vector u £ U have the form u = (ua ,un, up). By formulas

' I O O \ ( vAa O O

L = | O I O I , M = I O vAn -I O O O O B O

define the operator L £ L(U; F), imL = H x H x {0} ker L = {0} x {0} x Hp Mid M £ Cl(U; F), dom M = H2 x Hn x Hp. Thus, the reduction of model (1), (3) to (4) is finished.

Lemma 3. [5] For all v € R+ operator M is strongly (L, 1) -sectorial.

Construct a subspace U° = F° = {0} x H x Hp, U1 = F1 = H x {0} x {0}. Obviously conditions (Al) and (A2) are fulfilled, and

1 = (

M-i / O B-1 Mo = ( —I vAnB-1

where Bn is a restriction of B on H (Lemma 2 implies that Bn : H ^ Hn is a toplinear isomorphism). It is also easy to verify that

M-lLo = ' O —1

/ O —I \ O j

= \ O O

is a nilpotent operator of degree 1.

Denote by a (A) = {Xk}, the spectrum of A, where Xk € R_ are eigenvalues, numbered in nonincreasing order considering their multiplicity, then aL(M) = {v_1Xk}• It is clear that for such set, we can choose the contours r € C, which would satisfy the condition (A3). Construct

/ evXkt(-,^k)Vk O O \

Ut =

v-1Xk )

O O O

v o o o )

j = 0, n.

Then, by Theorem 2 and Lemma 3 we have the following Theorem 3. For all v € R+, Tj € R+ (rj < Tj+1), Uj € U, j = 0, n, there exists a unique solution

n

of problem (6) for model (1), (3), and this solution u = u(t) has the form ua (t) = ^^ Uj Tj uTja,

un = 0 up = 0.

j

j=0

In conclusion, the authors consider it their pleasant duty to express their sincere gratitude to G.A. Sviridyuk for fruitful discussions.

References

1. Zagrebina S.A. The Initial-Finite Problems for Nonclassical Models of Mathematical Physics. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming & Computer Software, 2013, vol. 6, no. 2, pp. 5-24.

2. Ladyzhenskaya O.A. The Mathematical Theory of Viscous Incompressible Flow. N.-Y., London, Paris, Gordon and Breach, Science PubL, 1969.

3. Temam, R. Navier—Stokes Equations. Theory and Numerical Analysis. Amsterdam, N.-Y., Oxford, North Holland Publ. Co., 1979.

4. Sviridyuk G.A. On a Model of the Dynamics of a Weakly Compressible Viscoelastic Fluid.

Russian Mathematics (Izvestiya VUZ. Matematika), 1994, vol. 38, no. 1, pp. 59-68.

p

Mathematics, 1999, vol. 59, no. 2, pp. 298-300.

6. Matveeva O.P., Sukacheva T.G. Matematicheskie modeli vyazkouprugikh neszhimaemykh zhidkostey nenulevogo poryadka [The Mathematical Model of a Viscoelastic Incompressible Fluid of Nonzero Order]. Chelyabinsk, Publishing Center of South Ural State University, 2014. (in Russian)

7. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, Koln, Tokyo, VSP, 2003. DOLIO. 1515/9783110915501

Вестник ЮУрГУ. Серия «Математическое моделирование 135

и программирование»(Вестник ЮУрГУ ММП). 2015. Т. 8, № 1. С. 132-136

S.A. Zagrebina, A.S Konkina

8. Zagrebina S.A. Multipoint Initial-Final Value Problem for the Linear Model of Plane-Parallel Thermal Convection in Viscoelastic Incompressible Fluid. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming & Computer Software, 2014, vol. 7, no. 3, pp. 5-22.

Received September 24, 2014

УДК 517.9 DOI: 10.14529/mmpl50111

МНОГОТОЧЕЧНОЕ НАЧАЛЬНО-КОНЕЧНОЕ УСЛОВИЕ ДЛЯ ЛИНЕЙНОЙ МОДЕЛИ НАВЬЕ - СТОКСА

С. А. Загребина, А. С. Конкина

Система уравнений Навье - Стокса моделирует динамику вязкой несжимаемой жидкости. Проблема существования решений задачи Коши - Дирихле для этой системы вошла в список наиболее тяжелых математических проблем нынешнего века. В данной статье вместо условия Коши предложено рассматривать многоточечные начально-конечные условия. Необходимо отметить, что в настоящее время в теории уравнений соболевского типа, к которым можно отнести систему Навье - Стокса, активно развивается новое направление исследований - разрешимость начально-конечных задач. Основным результатом статьи является доказательство однозначной разрешимости поставленной задачи.

Ключевые слова: относительно р-секториальные операторы; многоточечное начально-конечное условие; линейная модель Навье - Стокса.

Литература

1. Загребина, С.А. Начально-конечные задачи для неклассических моделей математической физики // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2013. - Т. 6, № 2. - С. 5-24.

2. Ladyzhenskaya, О.A. The Mathematical Theory of Viscous Incompressible Flow. - N.-Y., London, Paris, Gordon and Breach, Science PubL, 1969.

3. Temam, R. Navier - Stokes Equations. Theory and Numerical Analysis. - Amsterdam, N.-Y., Oxford, North Holland Publ. Co., 1979.

4. Свиридюк, Г.А. Об одной модели динамики слабосжимаемой вязкоупругой жидкости / Г.А. Свиридюк // Известия вузов. Математика. - 1994. - № 1. - С. 62-70.

5. Свиридюк, Г.А. Об относительно сильной р-секториальности линейных операторов / Г.А. Свиридюк, Г.А. Кузнецов // Доклады Академии наук. - 1999. - Т. 365, № 6. -С. 736-738.

6. Матвеева, О.П. Математические модели вязкоупругих несжимаемых жидкостей ненулевого порядка: монография / О.П. Матвеева, Т.Г. Сукачева. - Челябинск, Издат. центр ЮУрГУ, 2014.

7. Sviridyuk, G.A. Linear Sobolev Type Equations and Degenerate Semigroups of Operators / G.A. Sviridyuk, V.E. Fedorov. - Utrecht; Boston; Koln; Tokyo: VSP, 2003.

8. Zagrebina, S.A. Multipoint Initial-Final Value Problem for the Linear Model of Plane-Parallel Thermal Convection in Viscoelastic Incompressible Fluid // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2014. - Т. 7, № 3. - С. 5-22.

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

Александра Сергеевна Конкина, магистрант, кафедра «Уравнения математической фи:

ция), alexandra.konkina@yandex.ru.

Поступила в редакцию 24 сентября 2014 г.