ANALYSIS OF THE CLASS OF HYDRODYNAMIC SYSTEMS Текст научной статьи по специальности «Математика»

DOI: 10.14529/mmph220305

ANALYSIS OF THE CLASS OF HYDRODYNAMIC SYSTEMS

0.P. Matveeva1, T.G. Sukacheva1,2

Novgorod State University, Velikiy Novgorod, Russian Federation

2 South Ural State University, Chelyabinsk, Russian Federation

E-mail: [email protected], [email protected]

Abstract. The solvability of the Cauchy-Dirichlet problem for the generalized homogeneous model of the dynamics of the high-order viscoelastic incompressible Kelvin-Voigt fluid is considered. In the study, the theory of semilinear equations of the Sobolev type was used. The indicated problem for the system of differential equations in partial derivatives is reduced to the Cauchy problem for the indicated type of the equation. The theorem on the existence of the unique solution of this problem, which is a quasi-stationary trajectory, is proved, and its phase space is described.

Keywords: Sobolev type equation; phase space; viscoelastic incompressible fluid.

1. Formulation of the problem

System of equations

nm -1

(1 -œV2h = nV2v-(v-V)v + XIAm,sV2wm,s -Vp,

m=1 s=0

0 = V- v,

dwm,0 + 7— W

-fr-- = V + amWm,0 , m = 1, Г,

dw

= SWm,s-1 +amWms , S = 1> nm - 1 ®m e №-, Am,s e № + ,

describes the homogeneous generalized model of the dynamics of the high-order viscoelastic incompressible Kelvin-Voigt fluid [1].

The functions v = (v1;...,vn),vt = vt(x,t), xeW have the physical meaning of the fluid flow

velocity, the function p = p(x,t) describes the pressure. Here, WcRn, n = 2,3,4 is a bounded domain with boundary 3W of the class C¥ . The parameters ve R+ and œ e R correspond to the viscous and elastic properties of the liquid. The parameters Am s define the pressure retardation time. For the system (1), the Cauchy-Dirichlet problem is considered

|v( X, 0) = V0( X), p( X,0) = Po ( X), WmS ( X,0) = Wm s ( x) "x Î W,

(2)

lv(x,i) = 0, Wms(x,t) = 0 "(x,t)e3WxR, m = 1, r, s = 0, nm -1.

Разрешимость задачи (1), (2) рассматривается в рамках теории полулинейных уравнений соболевского типа [2, 3].

The solvability of problem (1), (2) is considered within the framework of the theory of semilinear Sobolev type equations [2, 3].

2. The solution of the problem

The proof of the existence theorem for the unique solution of the problem is that the Cauchy problem for the semi-linear Sobolev type equation [4] is first studied, and then the original problem is reduced to it.

Consider the Cauchy problem

u(0) =u0 (3)

for the semi-linear Sobolev type equation

L U = M(u). (4)

Here U and F are Banach spaces, operators L e L(U; F) and M e C¥ (U; F).

Definition 1. The solution of the problem (3), (4) is the vector function

u e C¥ ((-t0;to);U), t0 = t0(u0) > 0, satisfying equation (4) and condition (3).

Problem (3), (4) is solvable not for all initial data from the Banach space U, and if the solution of this problem exists, then it may be not unique.

Definition 2. A Banach Ck - manifold B is called the phase space of equation (4), if "u0 e B there is a unique solution u — u(t) of the problem (3), (4) on the interval (—10, to).

Definition 3. The solution u — u(t) of the problem (3), (4), for which Lu0 ° 0 "t e (—10; t0), where u0 — P u, is called a quasi-stationary trajectory of the equation (4).

Here u — u0 + u1, u0 e U0, u1 e U \ U — U0 © U . P is the projection of the Banach space U onto U 0.

Let the operator L be bi-splitting, its kernel kerL and image im L be complemented in the spaces

u1

U and F respectively. Denote by M^ e L(U;F) the Frechet derivative of the operator M at the point

u0 e U and introduce into consideration the chains M^ — associated vectors of the operator L , which we will choose from some complement coim L to the kernel ker L in the Banach space U. Consider the condition

(C1). Regardless of the choice coim L any chain M^ — associated vectors of any vector <pe kerL\ {0} contains exactly p elements.

Denote by L the restriction of the operator L to coim L . By virtue of the Banach closed graph theorem the operator L: coim L ® im L is a toplinear isomorphism. Let U0 — ker L and construct sets U0 — Aq [Uq], q — 1, p, where A — L-LMu . Sets UQ c coim L are linear spaces, the image FQ — Mu [U0] is a linear space, and F^ nim L — {0} (under condition (C1)).

Let us introduce one more condition

(C2). F0 © im L = F.

Equation (4) can be rewritten in the form

Lu — M^ u + F(u), (5)

where F — M — M^ e C¥ (U; F) by construction. Having influenced the equation (5) successively by the projectors Qq : F ® fQ (fQ — M^UQ], q — 1, p) and I — Q we obtain the equivalent system

fL u0 — Muo uQ + Fo(u),

LuQ — Muq uQ—i + Fp—i (u), (6)

0 — Muq u0p + Fp (u),

L u1 — (I — Q)M(u),

where uQ eUQ, Fq — Qq F(u) + Qq M^ u1, q — 1^, u1 eU1.

Lemma 1. Let the operators L e L(U;F), M e C¥ (U;F), and L be a bi-splitting operator, and conditions (C1) and (C2) be satisfied. Then the equation (4) is equivalent to the system (6).

Remark 1. Under the conditions of Lemma 1 the operator M^ (L, p) is bounded at the point u0

[5, 6].

Let us find solutions to the problem (3), (4). To obtain quasi-stationary trajectories from the set of possible solutions to the problem (3), (4), we introduce one more condition.

Let us consider the set U = {u e U: u0 = const, q = 1, p}. U is a complete affine manifold, modeled by the subspace U0 © U1. Let the point u0 e U, by OUg c U we denote the neighborhood of the point

un.

(C3). Fq (u) ° 0 "u e Ouo, q = 1, p. Theorem 1. Let

(i) the conditions of Lemma 1 are satisfied;

(ii) point u0 e B, where B = {u e U :Q0 M(u ) = 0};

(iii) condition (C3) is satisfied

Then there is a unique solution of the problem (3), (4), which is a quasi-stationary trajectory, and u(t) e B "te (-t0,t0).

As in [7], we pass from the system (1) to the system

r nm-1

(1 - œV2)vt = VV2V - (v • V)v + £ £ ^m,s V2Wm,s - p,

m=1 s=0

0 = V(V- v),

dw

m,0

dt

dw„

= v + a w

l-L ,,

m m,0

m = 1, r,

dt

SWm,s—1 +amWm,s , s = 1, nm — 1, a Î M_ , Am s Î M +

We will be interested in the local unique solvability of the problem (7), (2). Let us reduce problem (7), (2) to problem (3), (4). For this we set

U = ®ioUh F = ©f=oFi, K = ni +«2 + - + nr,

(7)

(8)

Where

Uo= Hi x Hp x Hp,

Fo= Hax Hp x Hp ;

Ut = H2 nH1 = HixHp,

F = l2 = hs x hp , i = 1, Here H д. is the subspace of the solenoidal vectors of the space

H2 nHl, H2=(W22(W))n, H 1=(W21(W))n; H2 is orthogonal (in the sense L2(W) = (L2(W))n) complement to Hi ; Ha and H2 - are the closures of the subspaces H2 and H;2 in the norm L2

respectively, Hp = Hp S: L2(W) ® H s is the orthoprojector along Hp. Then Se L(H2 nH*), and imS = H 2, £erS = Hp. The element of space U is a vector u (x, t) , it has the form

u(Xt) = (us, UP, up, W1,0, Wr,0, wriwrir X

where us = Sv, up = (I -S)v, up = p , ^ = ns -1, s = 1, r and

/ 0 000 00ч

U(0) = (% , ui0 , up0 , W1,0, Wo,0, W1Д, W1,^ , Wr,1, WHr ),

where

us0 = Sv0 , ui0 = ( 1 _ S)v0, upa = Po, Wi00 = Wi0 (x,0X i = 1, r; W° = Wi, (x,0X

"i0 ■■!> "j "j

i = Lr, j = ÎJr; u(x, t) = 0 "(x, t) eôWx M. Operators L , M : ^ ® F are defined by formulas

L =

( SA^S 0 0 0

0

0 0

ПАЖП 0 0

00 0 I

0

00

0 ^ 0 0 0

(9)

where n = I - S, A„, = 1 - cV . Note that L is the order matrix K + 3.

M (u) = M1u+M2(u),

(10)

where Ml is a matrix of the form

(yA vA 0 A10A • • Ar 0 A Aii A • • Ац A • • Arir

vA vA -I A10A • • Ar0A Aii A • ■ Ац A • ■ Arir a

SC ПС 0 0 0 0 0 0

I I 0 a1 0 0 0 0

I I 0 0 ar 0 0 0

0 0 0 I 0 a1 • 0 0

0 0 0 0 0 0 a1 • 0

V 0 0 0 0 0 0 0 • ar у

Here A = SA, A = PA; C(us + up) = V(V-(us + up)).

The operator M2 has the form M2=(SB(us + up) + fs, PB(us+ up) + fp, 0, ..., 0)T, where

B(u„ + up) = -((uff + up) • V)(uff + up).

Lemma 2. Let spaces U, F be defined by formulas (8), and let n = 2, 3, 4, and operators L , M :U ® F be defined by formulas (9), (10). Then: (i) operator L e L(U; F), and if ce-1 $ s(-V2), then £erL = {0}x(0}xHp x{0}...{0}, imL = HsxHpx(0}xF1 xFK ; (ii) operator Me C¥(U;F).

The statement (i) of Lemma 1 is obvious, and the statement (ii) is checked directly.

mu = M1 + M3,

(SBs SBp 0 ^

where operator M1 is defined above, and M3 =

PBs PBp 0 v 0 0 0у

(11)

Here Bs(Bp) is the Frechet to us(up). Obviously,

partial derivative of the operator B at the point us+u

"n > 3 "ueU Mun) ° 0 [7].

We have reduced the problem (7), (2) to the problem (3), (4).

Next, we check the feasibility of conditions (C1)-(C3). Denote by Acs the restriction of the operator SAcS to H^.

Lemma 3. Let the conditions of Lemma 2 be satisfied, and kerAcs ={0}. Then each vector je kerL \ {0} has exactly one M'u is associated vector, regardless of the point u eU.

Proof. Let the vector j = (0, 0, jp, 0, ..., 0) e kerL, jp = 0 . Find the vector yeU such that Ly = M'uj. From (9) and (10) we have

0

AœsVa = 0 PAœ¥p = -jp •

(12)

We get ys = 0, yP = -PAa}jp , the component yp of vector y is arbitrary, and the remaining K components of the vector y are equal to zero. M'yy £ imL, since CyP = 0, if yP = 0 [6]. •

The condition (d) is satisfied forp = 1. Now let's check condition (C2). Denote by Aa^P the restriction of the operator PA^n to HP .

Lemma 4. Under the conditions of Lemma 3, the operator A;P : HP ® Hp is a toplinear isomorphism.

By virtue of Lemma 2, the operator L from (9) is bisplitting. Let U° = kerL,

coimL = HS xHp x(0}xU1 x...xUK. Let's describe lineals

F00= m;[U00] = {0}xHp x{0}x{0}x...x{0} = {0}xHKx{0}x{0}x...x{0}cimL,

K K

Ui0 = L"1 [ F00 ] = SA-1 [Hp ] x A^p[Hp ] x {0} x {0}x...x{0} =

K

SA'1 A-P[Hp]xHP x{0}x{0}x...x{0} c coimL

K

by Lemma 4; Fi0 = M'Uq [U°] = SBA,-1 [Hp]xPBA~l[Hp]xCA;-1 [Hp]x{0}x...x{0}.

K

Let C be the restriction of the operator C to Hp. Since there is an operator C-1 [6], then by Lemma 4

F10= £¿0A;A-PC-1[Hp]xPBA;1 A~JpC-1[Hp]xHp x{0}x...x{0} c imL .

Here and above B0 is the Frechet derivative of the operator B at the point us0 + up0, and the operator L_1 is defined from (9).

The operators P0 and P1 have the form of matrices

(0 0 0 0 ••• 0^

0 0 0 0 0 0 П 0 0 0 0 0

0 0 0 0

0

f 0 i/2 0 0 0 П 0 0 0 0 0 0

v0 0 0 0 ••• 0,

where P1 = SA- A-Pn; and the operators are Q0 and Q1 respectively

(0 0 Q13 0

f 0 0 0 0 • • 0 >1

Q021 П Q023 0 • • 0

0 0 0 0 • • 0

0 0 0 0

0 0 Q2 0 0 П 0 0 0

23

0 0 0 0

(13)

(14)

where Q13 = SB0A~JA-PC, Q23 = ni?0A-A-PC"1n, Q021 = -PA-A-^S , Q023 = -Q021Q^3 - Q23. The operators Pk e L(U) and Qk e L(F) , k = 0, 1, defined in (13), (14) are projectors, and

imPk = U°k, im Qk = F^ , k = 0, 1 and P0P1 = P1P0 = 0, Q0Q1 = Q1Q0 = 0. ker Q1 = imL and

0

Fj © im L = F, condition ^2) is satisfied.

к

0

0

To check condition (C3) we construct the set

U = { u e U : P1u = const} = { u e U : up = const} .

The condition (C3) consists of the only equality

QiM (u) = (Q13C (ua + up), Q23C (ua + up), C (uff + up), 0,..., 0)T =0,

which is executed if up = 0 . If U = {u e U : up = 0}, then the condition (C3) is implemented.

Let us construct the set B. According to Theorem 1, B = {u e U : Q0M (u ) = 0}. For up =0 Q0M(u) = 0 ^ (Q021S + n)B(us) -up =0 , and

Q021S + n = A~pnA-1S + A-pnA^n = ApA? , (15)

then

B={u e U: A~pPA~l(B(ua)) = up, up =0 ,e HS , u, e HS xHp , i = 1f}. (16)

Notice, that

PA'1 AS + S = PA"1 (SAa + PAa )S = 0.

nA,-1 AasS = -AxpnAx S , AtepnA,-1 AasS = -nAiE S , Aasp nA,-1S = nAte A£eSS = O)^ .

Theorem 2. Let the conditions of Lemma 3 be satisfied Let u0 e B (16). Then, for some t0 = t0 (u0) , there exists the unique solution of the problem (7), (2), which is a quasistationary trajectory, u = (us, 0, p, w10, ..., wr0, w11, w1^, ..., wr1, wrl ) of classC¥((-t0,t0);U), and such that u e B for all t e (-t0, t0).

References

1. Oskolkov A.P. Initial-boundary value problems for equations of motion of Kelvin-Voight fluids and Oldroyd fluids. Proc. Steklov Institute of Mathematics, 1989, Vol. 179, pp. 137-182. (in Russ.).

2. Sviridyuk G.A., Sukacheva T.G. Cauchy problem for a class of semilinear equations of Sobolev type. Siberian Mathematical Journal, 1990, Vol. 31, no. 5, pp. 794-802. DOI: 10.1007/BF00974493

3. Sukacheva T.G. On a Certain Model of Motion of an Incompressible Visco-Elastic Kelvin-Voigt Fluid of Nonzero Order. Differ. Equ., 1997, Vol. 33, no. 4, pp. 557-562.

4. Matveeva O.P. Kvazistatsionarnye traektorii zadachi Teylora dlya modeli neszhimaemoy vyazkouprugoy zhidkosti nenulevogo poryadka (Quasistationary trajectories of the Taylor problem for the model of the incompressible viscoelastic liquid of the nonzero order). Bulletin of the South Ural State University. Series "MathematicalModelling, Programming and Computer Software ", 2010, no. 5, pp. 39-47. (in Russ.).

5. Sukacheva T.G., Matveeva O.P. Zadacha Teylora dlya modeli neszhimaemoy vyazkouprugoy zhidkosti nulevogo poryadka (The Taylor Problem for a Zero-Order Incompressible Viscoelastic Fluid model). Differetsial'nye uravneniya, 2015, Vol. 51, no. 6, pp. 771-779. (in Russ.).

6. Sviridyuk G.A., Sukacheva T.G. Nekotorye matematicheskie zadachi dinamiki vyazkouprugikh neszhimaemykh sred. VestnikMaGU, 2005, no. 8, pp. 5-33. (in Russ.).

7. Matveeva O.P., Sukacheva T.G. Homogeneous Model of Non-Zero Order Incompressible Viscoelastic Fluid. Bulletin of SUSU. Series "Mathematics. Mechanics. Physics", 2016, Vol. 8, no. 3, pp. 22-30. DOI: 10.14529/mmph160303

Received June 23, 2022

Information about the authors

Matveeva Olga Pavlovna is Cand. Sc. (Physics and Mathematics), Associate Professor, Algebra and Geometry Department, Novgorod State University, Velikiy Novgorod, Russian Federation, e-mail: [email protected].

Sukacheva Tamara Gennadyevna is Dr. Sc. (Physics and Mathematics), Professor, Algebra and Geometry Department, Novgorod State University, Velikiy Novgorod, Russian Federation; Leading Researcher, Laboratory of Nonclassical Sobolev Type Equations, South Ural State University (NRU), Chelyabinsk, Russian Federation, e-mail: [email protected].

Bulletin of the South Ural State University Series "Mathematics. Mechanics. Physics" _2022, vol. 14, no. 3, pp. 45-51

УДК 517.958 DOI: 10.14529/mmph220305

АНАЛИЗ ОДНОГО КЛАССА ГИДРОДИНАМИЧЕСКИХ СИСТЕМ

1 12 О.П. Матвеева , Т.Г. Сукачева

1 Новгородский государственный университет им. Ярослава Мудрого, Великий Новгород, Российская Федерация

2 Южно-Уральский государственный университет, г. Челябинск, Российская Федерация E-mail: [email protected], [email protected]

Аннотация. Рассмотрена разрешимость задачи Коши-Дирихле для обобщенной однородной модели динамики вязкоупругой несжимаемой жидкости Кельвина-Фойгта высокого порядка. При исследовании использована теория полулинейных уравнений соболевского типа. Указанная задача для системы дифференциальных уравнений в частных производных сводится к задаче Коши для указанного типа уравнения. Доказана теорема о существовании единственного решения указанной задачи, которое есть квазистационарная траектория, описано ее фазовое пространство.

Ключевые слова: уравнение соболевского типа; фазовое пространство; вязкоупругая несжимаемая жидкость.

Литература

1. Осколков, А.П. Начально-краевые задачи для уравнений движения жидкостей Кельвина-Фойгта и жидкостей Олдройта / А.П. Осколков // Труды матем. ин-та АН СССР. - 1988. - № 179. - С.126-164.

2. Свиридюк, Г.А. Задача Коши для одного класса полулинейных уравнений типа Соболева / Г.А. Свиридюк, Т.Г. Сукачева // Сиб. матем. журнал. - 1990. - Т. 31, № 5. - С. 109-119.

3. Сукачева, Т.Г. Об одной модели движения несжимаемой вязкоупругой жидкости Кельвина-Фойгта ненулевого порядка / Т.Г. Сукачева // Дифференциальные уравнения. - 1997. -Т. 33, № 4. - С. 552-557.

4. Матвеева, О.П. Квазистационарные траектории задачи Тейлора для модели несжимаемой вязкоупругой жидкости ненулевого порядка / О.П. Матвеева // Вестник ЮУрГУ. Серия «Математическое моделирование и программирование». - 2010. - № 16(192), Вып. 5. - С. 39-47.

5. Сукачева, Т.Г. Задача Тейлора для модели несжимаемой вязкоупругой жидкости нулевого порядка / Т.Г. Сукачева, О.П. Матвеева // Дифферециальные уравнения. - 2015. - Т. 51, № 6. -С. 771-779.

6. Свиридюк, Г.А. Некоторые математические задачи динамики вязкоупругих несжимаемых сред / Г.А. Свиридюк, Т.Г. Сукачева // Вестник МаГУ. - 2005. - № 8. - С. 5-33.

7. Матвеева, О.П. Однородная модель несжимаемой вязкоупругой жидкости ненулевого порядка / О.П. Матвеева, Т.Г. Сукачева // Вестник ЮУрГУ. Серия «Математика. Механика. Физика». - 2016. - Т. 8, № 3. - С. 22-30.

Поступила в редакцию 23 июня 2022 г.

Сведения об авторах

Матвеева Ольга Павловна - кандидат физико-математических наук, доцент, кафедра алгебры и геометрии, Новгородский государственный университет им. Ярослава Мудрого, г. Великий Новгород, Российская Федерация, e-mail: [email protected].

Сукачева Тамара Геннадьевна - доктор физико-математических наук, профессор, кафедра алгебры и геометрии, Новгородский государственный университет им. Ярослава Мудрого, г. Великий Новгород, Российская Федерация; ведущий научный сотрудник, лаборатория неклассических уравнений соболевского типа, Южно-Уральский государственный университет, г. Челябинск, Российская Федерация, e-mail: [email protected].