A priori estimates of the adjoint problem describing the slow flow of a binary mixture and a fluid in a channel Текст научной статьи по специальности «Математика»

УДК 517.9

A Priori Estimates of the Adjoint Problem Describing the Slow Flow of a Binary Mixture and a Fluid in a Channel

Victor K. Andreev* Marina V. Efimova^

Institute of Computational Modeling SB RAS Akademgorodok, 50/44, Krasnoyarsk, 660036 Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 21.03.2018, received in revised form 08.04.2018, accepted 25.06.2018 We obtain a priori estimates of the solution in the uniform metric for a linear conjugate initial-boundary inverse problem describing the joint motion of a binary mixture and a viscous heat-conducting liquid in a plane channel. With their help, it is established that the solution of the non-stationary problem with time growth tends to a stationary solution according to the exponential law when the temperature on the channel walls stabilizes with time.

Keywords: conjugate problem, inverse problem, a priori estimates, asymptotic behavior. DOI: 10.17516/1997-1397-2018-11-4-482-493.

1. Problem formulation

Many natural and anthropogenic phenomena are described by models of thermal convection. The thermoconcentration flow occurs in inhomogeneously heated multiphase systems with an interface between phases or with a free surface between the liquid and the gas. The role of interfacial convection is great both on small scales, where volumetric effects, such as buoyancy, are insignificant, and under conditions of weightlessness, where the gravitational mechanisms of convective motion are weakened or absent [1-3].

We consider two layers of immiscible liquids between horizontal flat plates. The X axis is directed horizontally, the Y axis is vertically upward. Equations of rigid boundaries are y = 0, y = l2. The equation of the interphase surface is y = li(x,t). The two-dimensional convective fluid flows are described by the Navier-Stokes equations in the Oberbeck-Boussinesq approximation. We introduce the dimensionless independent variables £ = x/li0, n = y/lio, t = vit/l20, u* = p10viUj(miAQ)(-i) is the characteristic velocity, P* = li0Pj(miAQ)(-i) is the modified pressure, Qj = 6j/AO is the characteristic temperature. We can take AO = l\0AA, where AA = max\A20(t) - Ai0(t)| > 0. If A20(t) = Ai0(t), then AA = maxmax \Aj0(y)\ > 0.

j y

C* = /3Cl20C(pf AQ)(-i) is the characteristic concentration of the light component; here vi is the constant kinematic viscosity, AO is the characteristic temperature drop, li0 = max \li(x, 0)\. Then the Oberbeck-Boussinesq equations are written in the following dimensionless form (the

* andr@icm.krasn.ru tefmavi@mail.ru © Siberian Federal University. All rights reserved

sign asterisk is omitted, Uj,vj are the velocity components)

j + M (uj Ujè + Vjj ) + jj = ^ (j + Ujrin ) :

Pio vi

V ' ftl

+ M (UjVje + VjVjn) + j j = ^ j + Vjnn) + Gj (9j + jC), Pio vi fti

9jT + M (Uj j + V9jV ) = X j + 9jVV ), (1)

J'3

Vi

Ct + M (UC + ViCn) = 1 (C^ + Cnn - t(9ix + 9ivv)), j + V jn =

The dimensionless parameters arise in the problem: M = œ1A9l10/(p10v2) is the Marangoni thermal number, Prj = Vj/\j is the Prandtl number, S = v1/D is the Schmidt number, 0 = —aftl/ftf is the separation parameter, v = v2/v1 is the kinematic viscosity ratio, D is the constant diffusion coefficient, aD is the thermal diffusion coefficient Soret, Gj = gftfpw/^ is the Grashof parameters; ftf, ftC are the constant coefficients of the thermal and concentration expansion of the media, ftl = 0; We = a0/œi A9 is the Weber number, w = œ2ftf /œi ftl and Bo = (p2 — pi )gl2/a0 is the Bond number. It is assumed that Bo C 1, We ^ 1 [4], then li^^ = 0 and the interface can be a straight line only. We assume additionally that the motion in the layers is creeping (M C 1). In this case the problem (1) becomes linear. Let us assume that solution of linear systems (1) has the form

Uj = Uj t)& Vj = Vj (n t) ; 9j = Aj (v,T )Ç2 + Bj (v,T ), C = H (v,T )Ç2 + E (V,T ); (2)

Pj = Pj (Ç,v,t).

Substitution (2) in system (1) reduces it to the following equations

AjT = " Ajnn, BjT = " (2Aj + Bjnn ) ; V V

Ht = S (Hnn — 0Ainn), Et = S (2H + Enn — 0(2Ai + Bim)) ;

(3)

^Ujnn — UjT = 2Gj (Aj + fc H )dv + Rj (t ); vi JQj P i

Vjn = —Uj, (Qi = (0,n), Q2 = (1,n)). The functions Pj(&,n,T) have representations

Pj =( Vj Ujnn — UjT) f + hj (t, n), hjn = Vj Vjnn + Gj Bj + § G iE

Pj0 \Vi J 2 Vi fti

Boundary conditions on solid walls are

Ui(0,t)=0, U2(l, t) = 0, A i(0, t) = A io(t), A2(1,t)= A2o(t);

B i (0,t )= B i o(t ), B2(l, t )= B2o(t );

Hn(0, t) — 0Ain(0, t) = 0; et(0, t) — 0Bir,(0, t) = 0.

Conditions on the interface for n = 1 [4] are

Ui = U2, pU2n — U in = —2A i — 2wH, A i = A2, A ^ = kA2n,

B i = B2, B i n = kB2n, Hn — 0A i n =0, En — 0B in = 0.

V

In additionally

J Ui(z, t)dz = 0, J U2(z, t)dz = 0. (4)

Here we have introduced the notation p = p20/pi0, l = l2/li0 > 1, n = pv, k = k2/ki. We supplement the problem with the initial conditions

Uj (y, 0) = 0, Vj (y, 0)=0, Aj (y, 0)= A0(y),

Bj(y, 0) = B0(y), H(y, 0) = H0(y), E(y, 0)= E0(y).

The initial conditions for the velocities are taken to be zero, since we are interested in motion under the action of surface forces and buoyancy forces.

The integral conditions (4) are consequences of the immobility of the n = 1 interface and the mass conservation equations, since from the last equation (3)

r n rl

Vi = - Ui(z, t)dz, V2 = - U2(z,T)dz (5)

J0 Jn

and Vi(1,t) = V2(1, t) = 0.

We note the peculiarity of the problem posed: it is inverse, since the functions Rj (t) must be determined together with Aj(n,T), Bj(n,T), Uj(n,T). First, we define the function A, then we determine H and Uj, the functions E, Bj do not affect the velocity field, and the vertical velocities Vj in the layers are found from the equalities (5). By the representation (2), the temperature on the walls has a minimum at £ = 0 (x = 0) for Aj0(T) > 0, or a maximum for Aj0 (t) < 0, or they alternate. Due to the Marangoni effect, the liquid and mixture can move in different directions.

2. A priori estimates of the function Aj(r,n),H(r,n)

The problem for functions Aj (n,T) is separated. It has the form

ait = Alriv, 0 <n< 1, t G [0,T], Pri

A2t = pr^ A2vv , 1 <n<l, t G [0,T];

a1(1,t ) = a2(1,t ), Aiv (1,t ) = kA2V (1,t );

Ai(0,T ) = aw(t ), a2(1,t ) = a20 (t ).

(6)

(7)

In addition,

Ai(n, 0)= A0(n), A2(n, 0)= A2(n), (8)

where Aj0(t), t g [0,T] and Al(n), 0 < n < 1, A2(n), 1 < n < l are the known functions. The matching conditions for solutions of problem (6)-(8) are satisfied:

A°(1)= A°(1), A°n (1) = kA*n (1), Aw (0) = A0(0), A2o(l) = A°(l).

We replace the unknown functions

Ai(n, t) = A(n, t) + Aw(t)(n - 1)2, 0 < n < 1,

A2(nT)= Mv,t)+ A20((T-I- 1)2 , 1 < n < l.

(9)

Then the functions Aj (n,T) are solutions of the conjugate initial-boundary-value problem

At = P-Avr, + 2A$Tl - A'io(t)(n - 1)2 -

1 ' (10)

- ^A'VV + fi(n,T), 0 <n< 1, t G [0,T],

A = v A 2vA2o(t) (t)(n - 1)

A2t = Pr2 A2nn +(l - 1)2Pr2 - A20 (l - 1)2 (11)

^1V-Â2VV + f2(n,T), 1 < n < l, t G [0,T], Pr2

a'(1,t )= â2(1,t ), A'n (1,t ) = kA2V (1,t ), a'(0,t )=0, Â2(l,T ) = 0, (12)

A'(n, 0)= A0(n) - Ai(0)(n - 1)2 - A°(n),

A2(n, 0)= A<0(n) - - A0(n). (13)

The prime denotes differentiation with respect to t in the right-hand sides of equations (10), (11).

We multiply equation (10) by Pr'A', equation (11) by kvPr2A2, integrate them over the domains of definition and add the results. Taking into account the boundary conditions (12), we obtain the identity

>(t) + i A\vdn + k I Andn = Pri I f'A'dn + kv-iPr2 I f2Â2dn, (14)

2

— W (I ) + Ain+ k A2nd

dT .10 J1 J0 Ji

W(t) = P2Ll! A2(n, t)dn + kpr I )dV. (15)

Since [5]

J0 Ai dn + i A22dn < Mi^fo Aln dn + k ^ A2n dr^j

with a finite minimal positive constant Mi depending on k and l, then the left-hand side of (14) is greater than or equal to

W + 2SW, S ). (16)

dT ' Mi kPr2 J v 7

The right-hand side of (14), using the Holder inequality, does not exceed

(^Pri jQ + i^kv-1Pr2 ^ fld^J

vW (t ) = g(t )y/ W (t ). (17)

From (16), (17) we obtain the inequality

1

W(t) ^vW +1 0 g(t)eStdt^j p Ai(n))2dn+kp21 aM)2dn,

Wo = T J0 ^

where the initial values A1(r), A^ (r) are defined by the equalities (13).

It turns out that one can obtain the estimates | Aj (r,T) |. To do this, note that along with (14) there is also another identity for the problem (10)-(13)

/1_2 fl_2 1 d f1_2 fl_2

AlTdr + kv-1 Pr2 J A2tdr +2 dT J A1ndr + k J A2ndr

= Pri f fiAiTdr + kv-1Pr2 f ¡2A2tdr, 01

from which it follows that

J A2t dr + k J At dr < J (All) 2 dr + k J (A°2n) 2 dr+

+Pr1 f i ffdrdr + kv-1Pr2 i f f22drdr = F(t) JO JO JO J1

with F(t) bounded on [0, Tj. Since

—2 fv- - —2 fl— — A1(r,T ) = 2 A1(r,T )A1n (r,T)dr, A2(r,T ) = -2 A2(r,T )A2V (r,T )dr, jO Jv

using the Holder inequality, the definition of the function W(t) (15), the estimate (19) and substitutions (9) we obtain

(19)

/ 8 \ 1/4

| A!(V,T) Aw(r) | + ( —W(T)F(T)) ,

/ Pi' \i/4 (20)

| A2(V,T) A2o(T) I ^^^W(T)F(T)) ,

uniform in n G [0,1] and n G [1, l] respectively. In (20) the quantity F(t) is given by formula (19) and W(t) is estimated from above by the right-hand side of (18). Therefore, the quantities | Aj(n, t) | are bounded for t g [0, T], if they are Aj0(t) and Aj0(T), j = 1,2.

Further, we need estimates of the derivatives n uniform in AjT (n, t). To this end, we differentiate with respect to t the equations (6) and the boundary conditions (7), assuming the existence of Aj'0(T). Then the problem for the functions AjT coincides exactly with the problem for Aj with changed initial data A1t (n, 0) = A°m/Pr\, A2t (n, 0) = vA°vv/Pr2, and the right-hand sides of the last two boundary conditions (7) are A'w(t), A'20(t). Therefore, we obtain estimates of the form (20):

/ 8 )l/4

| AlT(V,T) |<| A[O(T) | + — W!(T)F!(T)

,

/ PV \K4 (21)

| A2t(r,T) |<| A2o(t) I ^^^W1(t)F1(t)) ,

where W1(t) satisfies the inequality (18) with Aj0(T) replaced by Aj0(T), Aj0(T) by Aj0(T). A similar change must be made in the expression for F(t) to obtain F1(t) (more precisely, in the functions fj(r,T) from (10), (11)). In addition, the initial data in W10 and F1(t) should be replaced by Pr-1A0vv - A'10(0)(r - 1)2 and vPr2-1A°0vv - A'20(0)(l - 1)-2(r - 1)2, respectively.

2 V -r ^ '6XP

(24)

We introduce the substitution Hi = H — ^Ai to obtain an estimate of the functions H(n, t). Then H1(n,T) is a solution of the initial boundary value problem

Hit = S Hinn — ^AiT, 0 <n< 1, t e [0, T];

Hi(n, 0) = H0(n) — Ml(n) = H0(n), 0 <n< 1; (22)

Hin(0,t)=0, Hin(1,t)=0, t e [0,T]. The problem for the known AiT(n,T) and H0(n) has the solution [6]

Hi(n, T) = f1 H0(OG(n, T d — 4 if1 p)G(n, €,T — (23)

Jo Jo Jo

G(n, £,t) = 1 + 2 cos nnn cos exp (--t j

n= 1 ^ '

g {exp [ S(n — £ + 2nf ] exp [ S(n + £ + 2n)

From the representation (23) and the boundedness of Ai^(e,p) (estimate (21)), it follows that \H1(n,T)| is bounded for all n e [0,1] and t e [0,T]. \H(n,T)| is also bounded because H = H1 + 0A1 and by (20). In addition, the derivative HT (n, t) is continuous and bounded for n e [0, 1],t e [0,T].

3. A priori estimates of the functions Uj (r,n) and Rj(t)

We consider the problem for the definition of functions Uj(n,t), Rj(t):

Uinn — UiT = 2Gi P (Ai (z,T) + H (z, t )) dz + Ri(t), 0 <n< 1, (25)

o

vU2nn — U2t = 2G2J^ A2(z,t )dz + R2(t ), 1 < n < l, (26)

Ui(n, 0) = 0 (0 <n< 1), U2(n, 0) = 0 (1 < n < l), (27)

Ui(1,T ) = u2(1,t ), Un (1,t ) — Uin (1,t ) = —2ai(1,t ) — 2wH (1,t ), (28)

ui(0,t ) = U2(l,T )=0, (29)

J Ul(z, r)dz = 0, J U2(z,r)dz = 0. (30)

We introduce the notations

r n

Fi(n,r) = —2Gif (Ax(z,r)+ H (z, r )) dz, 0 <n< 1,r G [0,T], Jo

F2(n,r) = -2G2 J A2(z,r)dz, 1 < n < l, r G [0,T], (31)

F3(r) = -2Ai(1,r) - 2uH(1,t), r G [0,T],

Fi, F2, F3 are continuous and differentiable on their domains of definition by what has been proved above.

Let's make a replacement

Ui (n, t)= U' (n, t) - F3 (t) (2n3 - 3n2 + n) (32)

then the second boundary condition (28) becomes homogeneous for the functions U'(n,T), U2(n,T). The conditions (27)-(30) for these functions also remain homogeneous. Equation (25) for U'(n,T) takes the form

Uinn-U'n = R'(t)-F'(n,T)+6F3(t) (2n - 1)-F3(t) (2n3 - 3n2 + n) - R'(t)-F'(n,T). (33)

In addition, the first initial condition (27) will change U'(n, 0) = F3(0)(2n3 - 3n2 + n) — - Uio(n). _

We multiply equation (33) by U', (26) by pU2, then integrate over n and add the results. Using homogeneous boundary conditions (27)-(30) for U', U2, we obtain the identity

— + i U l dz + pi U2z dz = i F iU idz + pf F2U2dz, (34)

dT Jo Ji Jo Ji

ei(t) = 2 0 Ul(z,T)dz +P I U22(z,T)dz.

Since for Ui, U2 the following inequalities hold Friedrichs theorem, from (34) we obtain the inequality

E + 2SEi < 2gi(t)^EU (35)

dT

S = 2^1,^), Gi(T ) = -

^ F^ 7 + £ F2dz^j 7

(36)

whence the estimate

Ei(t) < + £ Gi(Ti)e2Sri dT^ e^, Ei(0) = 1 £ vwdz.

Hence the norm of the functions Ui(n,T), U2(n,T) in the space L2 is bounded for t e [0,T]

i U2dz < 2E1(t), i U2dz < -E1(t). (37)

J0 Ji P

Similarly to (19) we have the inequality

i u\z dz + pi U22z dz < i u\0z (z)dz + i i F\dzdT + pi i F^dzdT = Q(t ). (38)

J0 Ji J0 J0 J0 J0 Ji

Using (37), (38) and replacing (32), we obtain the estimate

\Ui(n,T)| < (8Ei(t)q(t))i/4 + 6\f3(t)|, (39)

uniform for n e [0,1], t e [0, T]. Similarly,

( 8 \ i/4

\U2(n,T)\ < [—Ei(t)Q(t) . \PP J

To obtain the estimates \Rj(t)|, t g [0, T], it is necessary to estimate \UjT(n,T)|. We differentiate equations (25), (26) and conditions (27)-(30) with respect to t. Taking into account the notation (31), we obtain a problem for Yj(n,T) = UjT(n,T)

Yinn - YlT = -FlT + RlT, 0 <n< 1; vY2nn - y2t = -F2T + R2T , 1 <n<l;

Yi(1,r) = Y2(1,r), pY2n(1,r) - Yin(1,r) = F3T, Yi(0,n) = 0, Y2(l,r )=0,

J Yi(z,r)dz = 0, J Y2(z,r)

(40)

(41)

0.

The initial data for t = 0 for equations (40) follow from (25), (26):

Yi(n, 0) = Fi(n, 0) — Ri(0) = Yi0(n), Y2(n, 0) = F2(n, 0) — R2(0) = Y20(n)

(42)

Integrating (25) with respect to n from 0 to 1, (26) from 1 to l, we find the unknown quantities Rj (0) with allowance for (27), (29)

Ri(0)= I Fi(z, 0)dz, R2(0) = -\ I F2(z, 0)dz. Jo l - 1 Ji

l—1

For the initial boundary value problem (40)-(43), the identity

(43)

dE2 + J Y2dz + ¡i i Yldz = J FiTYidz + p i F2tY2dz - F3TYi(1, r), (44)

2

The right-hand side of (44) does not exceed

E2 (r) = 1 0 Y2dz + Pp i Y22dz.

(45)

i

¿- 0 Fl dz + £2 t Y2 dz + P f F2t dz + f Y22dz + 1F2T + 1jo Y2Z dz (46)

2e

io

2

o

2&

2i

2

i

2

for any £i > 0, e2 > 0. Choose £i < 1, e2 < 2v(l — 1) 2). Using inequalities of Friedrichs for Y1,Y2, from (45), (46), we derive the inequality

E +2SiE2 < F(r), dr

(47)

1 fi fl 1 F(r > = ^1 F22T dz + ÛL Fl dz +2 F

¿i = min

1 - £i;■

2i 2v

2

3t ,

- £2

We obtain the estimate

E2(r) <

(l - 1)2

E2(0)+ [ F(t)e2Sltdt

o

-2b\T

(48)

E2(0) = 1 0 (Yi0(z))2 dz + P £ (Y0(z))2 dz,

with the functions Y0(z) from (42), (43). Hence

I Ul(z,t)dz < 2E2(r), I U2T(z,t)dz < -E2(t). (50)

Jo Ji P

We obtain the boundedness of the norms UjT on its domains of definition with respect to n for

all t € [0, T] by the estimates (20), (21), the properties of the functions H1(n,T)(23), F1(n,T),

F2(n,T), F3(t) (31), inequalities (50).

We multiply equation (25) by n — n2 and integrate from 0 to 1, after some transformations

we find

ri(t ) = 6

U1(1,t) - i (z - z2)UlTdz + ( (z - z2)F1dz Jo Jo

Then we multiply equation (26) by z2 — (l + l)z + l and integrate from 1 to l, we find

r2(t )

(1 - l)3

/(1 - l)U2(1, t) - J z(z - l - 1)U2tdz + J [z2 - (l + 1)z + l] F2dz

(51)

. (522)

All the terms in the pair parts (51) and (52) are bounded for all t € [0, T]. This follows from the obtained estimates \Uj(n,T)|, \Aj(n,T)|, \H(n,T)| and their derivatives with respect to t.

4. The asymptotic behavior of the solution for t ^ x>

The problem (6)-(7), (25), (26), (28)-(30) has a stationary solution [7]; we denote it by As(n), Hs(n), Us(n), Rs. It corresponds to the boundary data As,j0 = const, where As(n), Hs(n) are linear, Us(n) are polynomials of the fourth order, and Rs are constants. Suppose that the functions Aj0(r) are defined and continuous with the derivatives A'j0(r), Aj0(r) for all t > 0. We obtain the conditions under which the solution of the nonstationary problem for y = 0 tends to a stationary solution and establishes estimates of the rate of convergence with the help of the obtained a priori estimates Aj, AjT, Uj, UjT and formulas (23), (24), (51), (52). To this end, we introduce the differences

(53)

Nj (n,T) = Aj (n,T) — As(n), M (n,T) = H (n,T) — H s(n),

Kj (n,T) = Uj (n,T) — Us (n), Lj (t ) = Rj (t ) — R°.

The functions Nj is the solution of the conjugate problem (6)-(8) with A\0(t) replaced by Nw(t) = Aw(t)—A\o, A2o(t) into N2o(t) = A20(t)—A2o, and A0(n) on N0(n) = A0(n) — As(n). Therefore estimates of the form (20) are valid for

\Nj(n,T)\ < \Nw(t)\ + (8ejW(t)F(t))1/4 , (54)

with £1 = (Pr1)-1, e2 = (k2Pr2) 1. Using the simple inequality (a + b)2 < 2(a? + b2), inequality (18) and the definition of G(t) from (17), we obtain the estimate of

W (t) ^ v/w0 where (cm.(13), (18))

0 + 71 Jo [\Nw(t)\ + \N2o(t)\ + |Nlo| + \N2o(t)|] estdt)e

-2St

Wo = PT I [Ni(n) - Nio(0)(n - 1)2]2 dn + 2k-Pr2 J ' i

N2o(n) - N2o(0)

(n - 1)2 (l -1)2

(55)

dn, (56)

71 = max

VPr~i

Pn 2 k Pr

5 Vk-1v3(l - 1)3Pr2 '( )V2v T'2

6

T

Therefore, if the integrals

/ |WJ-o(t)|eÄidt = \Ajo(t) — Asj0\eStdt, \N'j0(t)\eStdt = \A'j0(t)\eSt dt, (57)

Jo Jo Jo Jo

10 JO

converge, then for all t ^ 0

W(t) < c1e-2ÖT, c1 = const (58)

Further, all constants of the form (56) will be denoted by c2, c3, ... . For the function F(t) from (19) we have the estimate

\F(t) < c2 + c3 T \N2o(t) + N2o(t) + (Nio(t)f + (N2o(t)f} dt. (59)

0

Since the integrals (57) converge, then \Ajo(T) — ASjo\ < hj(t)e-Sr, \Ajo(T) < pj(t)e-Sr with nonnegative functions hj (t), pj (t) ^ 0 by t ^ x> and

/ hj(t)dT ^ <x, pj(t)dT ^

oo

The boundedness of \F(t)\ < c4 follows from (59) for all t > 0. Considering (53), (54), (58) and (59) we find

\Nj(n,T)\ = \Aj(n,T) — AS(n)\ < c^+je-ST/2 (60)

for all n e [0,1], (j = 1), n e [1, l], (j = 2) and t > 0. From inequalities (21) we obtain

\NjT(n,T)\ = \AjT(n,T)\ < c5+je-"T/2. (61)

Next we proceed to obtain estimate M(n,T) = H(n,T) — Hs(n) from (53). First we note that the mean value of H0(n) can be considered zero:

f1 H0(e)de = 0.

o

Given this assumption, M represent from (23) and (24) as follows:

M(n,T)= 4Ni(n,T) + fi [H0(e)+ Hs(n) — 4Ai(e)] G(n,e,T)de— (62)

o

—4 Ni(e,T )de — 4 Ni^(e,p)G(n,e,T — ^d^, Jo Jo Jo

where G(n,e,T) = G(n,e,T) — 1, G(n,e,T) is given by (24). Using inequalities (60), (61) for j = 1, from (62) we obtain the estimate

\M(n,T)! = \H(n,T) — Hj(n)\ < cge-UlT, ui = min [ -(63)

In deriving (63) we took into account the inequalities

n2«ie-n2«2s < C 1(ai,<*2,£) ae-a4t ^ C2(a3,a*)

.

1 CT ,-1

for a.k > 0, k = 1,..., 4, Cl, C2 are bounded constants, t > e > 0. The estimate of MT is found from equality

1 r 1 -

Mt = S H— H*(£) — VA?(0] Gnn(n, Z, T)d£—

,V -■ (64)

0 , N1^(Z,p)G

nn(n, Z,T — p)didp. s Jo Jo

It follows from an equation of the form (22) on M and formula (62), then \MT(n,T)\ = = \Ht(n,T)\ < O1oe-^1T.

We proceed to estimate the functions estimated Kj(n,T), KjT(n,T) from (53). We have the inequality (39) for K1, where F3(t) = —2N1(1,t) — 2uM(1,t), the function E1(t) satisfies the inequality (36), and Q(t) is determined from (38). In view of (60), (61), (63), (64) we obtain

\K1(n,T)\ = \U1(n,T) — US(n)\ < O1oe-^2T, ^ = min(^1,S1).

Here ¿i is defined by the equality (48). Similarly, from (32) we obtain \K2(n,T)| = \U2(n,T)--U2(n)\ ^ c11e-SlT, and from (49), (50) we derive estimates \KjT(n,T)\ = \UjT(n,T)\ < < c11+je-SlT provided that condition (57) is satisfied and the integral

/ \N''o(t)\eSt dt = \Ajo(t)\eStdt. (65)

oo

is converges. Thus, if convergence of the integrals (57), (65) takes place, then the nonstationary solution converges to the stationary one in accordance with the exponential law. The same applies to the functions Rj (t), which defined by formulas (51), (52).

The work received financial support from RFBR (project 17-01-00229).

References

[1] A.Nepomnyashii, I.Simanovskii, J.-C.Legros, Interfacial Convection in Multilayer System, Springer, New York, 2006.

[2] R.Narayanan, D.Schwabe, Interfacial Fluid Dynamics and Transport Processes. SpringerVerlag, Berlin, 2003.

[3] R.Kh.Zeytovnian, Convection in Fluids, Springer, Dordrecht, 2009.

[4] V.K.Andreev, V.E.Zakhvataev, E.A.Ryabitskii, Thermo capillary Instability, Nauka, Novosibirsk, 2000 (in Russian).

[5] V.K.Andreev, On Inequalities of the Friedrichs type for Combined Domains, Journal Siberian Federal University. Mathematics and Physics, 2(2009), no. 2, 146-157 (in Russian).

[6] A.D.Polianin, Handbook of linear partial differential equations for engineers and scientists, Boca Raton, London, 2002.

[7] M.V.Efimova, On one two-dimensional stationary flow of a binary mixture and viscous fluid in a plane layer, Journal Siberian Federal University. Mathematics and Physics, 9(2016), no. 1, 30-36.

CXJ

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Априорные оценки сопряженной задачи, описывающей совместное движение жидкости и бинарной смеси в канале

Виктор К. Андреев Марина В. Ефимова

Институт вычислительного моделирования СО РАН Академгородок, 50/44, Красноярск, 660036 Сибирский федеральный университет Свободный, 79, Красноярск, 660041

Россия

Для линейной сопряженной начально-краевой обратной задачи, описывающей совместное движение бинарной смеси и вязкой теплопроводной жидкости в плоском канале, получены априорные оценки решения в равномерной метрике. С их помощью установлено, что решение нестационарной задачи с ростом времени стремится к стационарному решению по экспоненциальному закону, если температура на стенках канала стабилизируется со временем.

Ключевые слова: сопряженная задача, обратная задача, априорные оценки, асимтотическое поведение.