CC BY
70
MSC 35K70 DOI: 10.14529/mmp140401
ON A CLASS OF SOBOLEV-TYPE EQUATIONS
T. G. Sukacheva, Novgorod State University, Velikiy Novgorod, Russian Federation, tamara.sukacheva@novsu.ru,
A.O. Kondyukov, Novgorod State University, Velikiy Novgorod, Russian Federation, k.a.o_leksey999@mail.ru
The article surveys the works of T.G. Sukacheva and her students studying the models of incompressible viscoelastic Kelvin-Voigt fluids in the framework of the theory of semilinear Sobolev-type equations. We focus on the unstable case because of greater generality. The idea is illustrated by an example: the non-stationary thermoconvection problem for the order 0 Oskolkov model. Firstly, we study the abstract Cauchy problem for a semilinear nonautonomous Sobolev-type equation. Then, we treat the corresponding initial-boundary value problem as its concrete realization. We prove the existence and uniqueness of a solution to the stated problem. The solution itself is a quasi-stationary semi-trajectory. We describe the extended phase space of the problem. Other problems of hydrodynamics can also be investigated in this way: for instance, the linearized Oskolkov model, Taylor's problem, as well as some models describing the motion of an incompressible viscoelastic Kelvin-Voigt fluid in the magnetic field of the Earth.
Keywords: Sobolev type equations; incompressible viscoelastic fluids; relatively p-sectorial operators; extended phase spaces.
Introduction
The system
(1 - XV2)vt = vV2v - (v • V)v -Vp - g^O + f,
0 = V- v, (1)
Ot = &V2O - v •VO + v • y
models the evolution of the velocity v = (v1, v2, ..., vn), vi = Vi(x, t), pressure gradient Vp = (pi, p2, ..., pn), pi = pi(x, t) and temperature O = O(x,t) of the simplest non Newton fluid — incompressible viscoelastic Kelvin - Voight fluid [1,2].
The parameters Л £ R, v £ R+ and ж £ R+ characterize elasticity, viscosity and heat conduction of the fluid respectively; g £ R+ is the acceleration of the gravity; y = (0, ..., 0, 1) is the unit vector in Rn; the fee term f = (f1, ..., fn), fi = fi(x,t), corresponds to the external influence on the fluid.
We investigate the solvability of the initial-boundary value problem
v(x, 0) = v0(x), O(x, 0) = O0(x), Vx £ Q;
v(x, t) = 0, O(x, t) = 0, V(x, t) £ дQ x R+ ^
for system (1). Here Q С Rn, n = 2, 3, 4, is a bounded domain with a smooth boundary dQ of class CA.P. Oskolkov [3,4] began to study problem (1), (2) and investigated
the solvability of this problem in case X-1 > — Ai ( X1 is the least eigenvalue of Laplace operator with homogeneous Direchlet condition in the domain Q).
The first initial-boundary value problem (2) for system (1) was considered by G.A. Sviridyuk [5,6], and its modification for the flat parallel current was studied by him in [7]. In these papers the indicated problem was studied when the free term f did not depend on time under other assumptions (less general in Sviridyuk's papers [6,7]) on the corresponding differential operators.
Our aim is to study the solvability of problem (1), (2) when the free term f = f (x,t) is not stationary. We consider this problem in the frame of the Sobolev type equations theory. The base of this theory was created by professor Sviridyuk and now this theory is actively developed by his followers. This problem is investigated on the base of the concept of relatively p-sectorial operators and degenerative semi-groups of operators. The existence theorem of unique solution to this problem is proved and the description of its extended phase space is obtained.
So at first we study the abstract Cauchy problem and then consider problem (1), (2) as its concrete interpretation. We prove the existence theorem of the unique local solution to problem (1), (2). This solution is a quasi-stationary semi-trajectory.
Other models of incompressible viscoelastic fluids may be studied in the same way as problem (1), (2). The examples of these models will be indicated at the end of the paper.
1. Abstract Cauchy Problem for the Semi-Linear Non-Autonomous Sobolev Type Equations
Let U and F be Banach spaces, operator L E L(U; F), i.e. it is linear and continuous, ker L = {0}; operator M : dom M — F is linear and closed and it is densely defined in U, i.e. M E Cl(U; F). Denote by UM be the lineal dom M, endowed with the norm of graph HI • H = \\M • \\F + || • \\U. Assume that F E C™(UM;F), and function f E CF). Consider the Cauchy problem
u(0) = u0 (3)
for the semilinear non-autonomous Sobolev type equation
LU = Mu + F (u) + f (t). (4)
Definition 1. A vector-function u E C^((0,T); UM) is a solution to problem (3), (4) if it satisfies (3), (4) so that u(t) — u0 when t — 0 + .
L
pL(M) = {^ E C : (pL — M)-1 EL(F; U)} and the L-spectrum aL(M) = C \ pL(M) of the operator M.
Definition 2. An operator M is an (L,p)-sectorial one, if there exist the constants
n
a E R, k E R+, 6 E (^,n) such that
(i) SlL^(M) = {pE C : | arg(p — a)I < 6 , p = a } C pL(M);
k
(n) max{ \\ Rlm(m) Wcw, \\ ^ >p)(M) \\lf } ^ yT^^A for all p, p0, p1, ... ,pp E SL a(M).
Here
Rl p) (M) = Wq=oRLq (M), Lfr p)(M) = np=0 LLq (M)
are right and left (L,p)-resolvents of operator M respectively, RL(M) = (¡L — M )-1 L, Lj(M) = L(yL — M )-1 [10].
Definition 3. An operator M is a strongly (L,p)-sectorial one, if it is (L,p)-sectorial and for all ¡ip E SL a(M)
(i) || MR^p)(M) (iL — M )-1f y < -COnP^-1
for all f from some dense in F lineal;
(«) 1 (lL — M)-1l^,(m) B«™ < | i — a ¡ngnr\ ¡q — a\
Remark 1. Ifp = 0, then (L,p)-sectorial and strongly (L,p)-sectorial operator M is called LL
M ( L, p)
Under this assumption the solution to this problem may be non-unique. Consider the following example.
Example 1. [11] Let UM = U = F = R2, and operators L, M and F be given by
L = ( 01 ) ■ M = ( J J ) , F (x) = ( ).
The operator M is strongly (L, l)-sectorial, since
(iL — M )-1 = ( —0? —¡j) , RL(M ) = LL(M )^0 —J) •
Consider the Cauchy problem x(0) = 0, where x = col (x1;x2) for the equation
Lx = Mx + F (x).
There exist two solutions to this problem col (0, 0) and col (t/2,t2/4).
Thus the solution of problem (3), (4) may be non-unique. So we restrict the idea of the solution to equation (4). Moreover it is known [12-14], that solutions to problem (3), (4) exist not for all u0 E UM. Thus we introduce two definitions.
Definition 4. [15]. The set B1 C um x R + is called an extended phase space of equation (4), if for every point u0 E UM such that (u0, 0) E B0 there exists a unique solution to problem (3), (4), and (u(t),t) E Bf.
Remark 2. If & = b x R + , where B C um , then the set B is called a phase space of equation (4).
Deflnition 5. Let the space U = U0 © U1 and ker L C U0. The solution u = v + w, where v(t) E U0, and w(t) E U1 for a 11 t E (0,T), to equation (4) is called a quasi-stationary semi-trajectory, if LV = 0.
Remark 3. The concept of quasi-stationary semi-trajectory generalizes the concept of quasi-stationary trajectory, introduced for the dynamic case [11,13,14].
It is well known that if the operator M is strongly (L,p)-sectorial, then U = U0 ©U1, F = F0 ©F1, where
U0 = y EU : Ut^ = 0 3t E R+}, F0 = EF : F= 0 3t E R+}
are kernels, and
U1 = [u EU : lim Utu = u}, F1 = [f EF : lim Fff = f}
are images of the analytic resolving semigroups
U =2h i RL(M F = 2- J LL(M (5)
(r C Sq a(M) is a contour such that arg ¡j ^ ±6 when ^ of the linear
homogeneous equation Lu = Mu. Let Lk(Mk) be the restriction of the operator L(M) on Uk (Uk n dom M), k = 0, l^enLfc : Uk Fk, Mk : Uk n dom M Fk, k = 0, 1; and restrictions M0 and L1 of operators M and L on the spaces U0 n dom M and U1 respectively are linear continuous operators and they have bounded reverse operators. [10]. So we reduce equation (4) to the equivalent form
Ru0 = u0 + G(u)+ g(t) u0(0) = u0,
(6)
u1 = Su1 + H (u) + h(t) u1(0) = u0,
where uk E Uk, k = 0,1, u = u0 + u1, operators R = M0-1L0, S = L-1M1, G = M-1(I-Q)F, H = L^QF, g = M-1(I-Q)f, h = L^Qf.Rere Q E L(F)(= L(F; F)) is the corresponding projector.
Definition 6. System of equations (6) is called a normal form of equation (4)-
ML equation (4) (f (t) = 0) coincides with the form in [7].
We study quasi-stationary semi-trajectories of equation (4), for which Ru0 = 0. So we
R ker R im R
completable in space U. Denote U00 = ker R, and U01 = U0 0 U00 is a complement of the subspace U00. Then the first equation of the normal form (6) is reduced to the form
Ru01 = u00 + u01 + G(u) + g(t), (7)
where u = u00 + u01 + u1.
Theorem 1. Let operator M be strongly (L,p)-sectorial, and operator R be bi-splitting. If there exists a quasi-stationary semi-trajectory u = u(t) of equation (4), then it satisfies the following relations
0 = u00 + u01 + G(u) + g(t), u01 = const. (8)
Proof. The first relation follows from (7) due to the requirement of quasisteadyness Ru0 = Ru01 = 0. The second relation follows from the identity Ru01 = 0, because in the Banach theorem about the inverse operator the restriction of operator QrR(I — PR) on U01 is a continuously invertible operator. Here QR and Pr are the projectors on im R and ker R respectively, ker PR = U01.
□
Remark 5. The second relation in (8) explains the meaning of the term quasi-stationary semi-trajectories, i.e. such semi-trajectories, which are stationary on some variables. In the other words the quasi-stationary semi-trajectory necessarily lies in the plane (I — PR)u° = const.
Theorem 1 establishes the necessary condition for the existence of quasi-stationary semi-trajectory of equation (4). Now consider the sufficient conditions.
It is known that if the operator M is strongly (L,p)-sectorial then the operator S is sectorial [10]. So it generates an analytic semigroup on U ^ Denote it by {U\ : t > 0} since the operator Uf in fact is the restriction of the operator U* on U1. The fact that U = U0©U1 shows that there exists a projector P E L(U), corresponding to this splitting. It can be shown that P E L(UM) [10]- Then the space UM splits into the direct sum UM = UM © UM so that the embedding UM C Uk, k = 0, 1, is dense and continuous. Symbol A'v denotes the Frechet derivative of an operator A, defined on some Banach space V, at v E V.
Theorem 2. Let operator M be strongly (L,p)-sectorial, operator R be bi-splitting, operator F E Crx(UM; F), vector-function f E CF), and the following conditions be satisfied:
(i) In the neighborhood Ouo C UM of point u0 the following relation holds
0 = U01 + (I — Pr)(G(u00 + U01 + u1) + g(t)); (9)
(ii) Project or PR E L(UM), and operator I + PRG'u0 : U™ ^ UM is a topological linear isomorphism (U™ = UM H U00);
(iii) For the analytic semigroup {U\ : t > 0} the following condition is valid
l l|Uill£(ui;UMM)dt< tt VT E R+. (10)
Then there exists a unique solution of problem (3), (4), which is the quasi-stationary semi-trajectory of equation (4).
Proof. Consider the neighborhood Ouo of point u0. In this neighborhood the first equation (6) turns to
0 = u00 + Pr(G(u00 + U01 + u1) + g(t)) (11)
by condition (i). Further, from (i) by the implicit function theorem there exist neighborhoods Ouoo c U°M (U°M = U00 hum), Oui C UM UM = U1 hum) of points u00 = PR(I — P)u0, and the mapping 8 : Oui ^ Ouoo of class C™ such that
the equation
u
00
8{u\t) (12)
is equivalent to (11).
Now, by (12) the second equation in (6) in the neighborhood of Oui turns to
u1 = Su1 + H (¿(u1) + U01 + u1) + hit), (13)
where operator H((I + ¿)(-) + u01) : Oui — U1 belongs to class C™ by construction.
To prove the unique solvability of the problem u1(0) = u0 for equation (13) we use the Sobolevskiy-Tanabe method, described in [17, Chapter 9]. By (iii), smoothness of operator Hh
if u0 E Um, then for some T E R+ equation (13) has a unique solution u1 = u1(t), t E [0,T) such that u1(t) — u0 for t — 0+ in the topology of U^.
Thus, the solution of (3), (4) in this case has the form u = u1 + ¿(u1) + u001, and this solution is a quasi-stationary semi-trajectory by construction.
□
Remark 6. For any quasi-stationary semi-trajectory of equation (4) relation (9) follows immediately from the first equation in (8).
Remark 7. Condition (10) for the conventional analytic semigroups with the estimate
L(U1;UM) < t 1c0nst,
is not satisfied. Later we are going to use theorem 2 in such situation, therefore we need to make some explanations. Let U0. = [U1;U^]a, a E [0,1] be some interpolation space constructed by the operator S. Supplement the condition "operator F E Crx(U^; F), vector-function f E CF)" in theorem 2 with the condition "operator H E C™(Um;U1), h E CU1)", and replace (10) with estimate
/ \\U1\l(Ui;U^)dt< ™ Vt E R+. (14)
Jo
Then theorem 2 is true. See discussion of these problems in [17, Chapter 9 ].
Remark 8. Let the conditions of theorem 2 be satisfied (possibly taking into account the remark 7). Construct the plane A = {u E UM : (I — Pr)(I — P)u = u^} and the set M = {u eUm : Pr((I — P)u + G(u) + g(t)) = 0}.
By hypothesis, their intersection A fl M = 0, so it contains at least a point u0. Moreover, there exists a C^-diffeomorphism I + ¿, mapping neighborhood Oui to some neighborhood Ouo C A f M. Consequently, not only point u0 can be taken as the initial value, it may be an arbitrary point of neighborhood Oui. This means that Oui is the part of extended, phase space B1 of equation (4).
Now let Uk and Fk be Banach spaces, operators Ak E C(Uk, Fk), and operators Bk : dom Bk ^ F be linear and closed with domains dom Bk dense in Uk, k = 1, 2. Construct spaces U = U1 x U2, F = F1 x F2 and operators L = A1 ® A2, M = B1 ® B2. By construction operator L E L(U; F), and operator M : dom M F is linear, closed and densely defined, in U dom M = dom B1 x dom B2.
Theorem 3. [18] Let operators Bk be strongly (Ak,pk)-seetoriaI, k = 1, 2; and p1 > p2. Then operator M is strongly (L,p1)-sectorial.
2. The Concrete Interpretation
Consider problem (2) for system (1) given by
(1 - AV2)vt = vV2v - (v • V)v - p - g^O + f,
0 = V(V- v), (15)
Ot = ^V2O - v •VO + v • y.
Here p = Vp, since in many hydrodynamic problems knowledge of the pressure gradient is preferable [19]. This change in the continuity equation was made for the first time by G.A. Sviridyuk in [20]. We are interested in a local unique solvability of problem (15), (2), equivalent to the original problem (1), (2). It's convenient to consider this problem in the frame of the semilinear Sobolev type equations theory, briefly described in section 2.
In order to reduce problem (15), (2) to (3), (4) we introduce, following [20], the spaces H2, H2, and Hn .Her e H2 and Ha are subspaces of solenoidal functions in spaces
o
(W|(Q))n fi (W2(Q))n and (L2(Q))n respectively, and H^ and Hn are their orthogonal (in sense of (L2(Q))n) complements. By symbol £ denote the orthogonal projection on Ha,
o
where its restriction to space (W22(Q))n f (W2(Q))n is denoted by the same symbol. Let n = I - £.
By formula A = V2En : H2 © H ^ Ha © Hn, (En is the identity matrix of order n,) determines the continuous linear operator with the discrete finite spectrum a (A) C R, condensed only at -<x>. Formula B : v ^ V(V • v) determines the continuous linear surjective operator B : H2 © H ^ Hn with the kernel ker B = H2.
Using natural isomorphism of the direct sum and Cartesian product of Banach spaces, introduce spaces U1 = H2 x H^ x Hp and F1 = Ha x Hn x Hp, where Hp = Hn. Construct the operators
i £(I - AA) £(I - AA) 0\ i v£A v£A O
A1 = I n(l - AA) n(l - AA) O I , B1 = I vnA vnA -I \ O O O J \ O B O
Remark 9. Denote by Aa the restriction of £A on H2. By the Solonnikov-Vorovich-Yudovich theorem the spectrum o(Aa) is real, discrete, with finitely-multiplicities one and condenses only at -to.
Theorem 4. (i) The operat ors A1,B1 e L(U1; FF1). If A-1 £ a (A), then opera tor A1 is bi-splitting, ker A1 = {0} x {0} x Hp, im A1 = Ha x Hn x {0}.
(ii) If A-1 £ a(A) U a(Aa), then operator B1 is an (A1,1)-bounded one.
Remark 10. The proof of theorem 4 is given in [9]. For the first time the concept of relatively bounded operator was introduced in [21]. The case of relatively sectorial operator was considered in [7,22,23].
Set U2 = F2 = L2(Q) and by formula B2 = ^V2 : dom B2 ^ F2 determine the closed
o
linear and densely defined operator B2, dom B2 = W22(Q) f W2(Q). If operator A2 is equal I, B2
B2A2
Let U = U1 x U2, F = F1 x F2. The vector u of space U has the form u = col (ua,un, up, ue), where col (ua,un,up) E U1, and ue E U2. The vector f E F has the same form. Operators L and M are defined by formulas L = A1 ® A2 and M = B1 ® B2. Operator L E L(U; F), and operator M : dom M — F is linear, closed and densely defined, dom M = U1 x domB2. From theorem 4 and [10] it follows that operator B1 is strongly (A1; 1)-sectorial. Therefore, by theorems 3 and 5 the following theorem is true.
Theorem 6. Let X-1 E °(A), then operator M is a strongly (L, 1)-sectorial one.
F.
to represent it in the form F = F1 ® F2, where F1 = F1(ua,un, ue) = col (—E(((Ha + un) • V)(ua + un) — ((ua + un) • V)(ua + un) + g^ue + f), — n(((uCT + uw) • V)(ua + un) — ((ua + un) • V)(ua + Un) + giue + f), 0), and F2 = F2(ua,un) = (ua + un) • (7 — Vue). Formally, we find the Frechet derivative F' of operator F at point u,
/ Sa(uCT ,un) £a(uCT ,un) O —g^Y \
„1 _ na(uCT ,un) na(uCT ,un) O —gnY
Fu = O O O O ,
\ (7 — Vue) • (*) (y — ^ue) • (*) O —(ua + un) • (*) /
where a(ua,un) = —((*) • V)(ua + un) — ((ua + un) • V)(*), and the character * should be changed the corresponding coordinate of vector v in case of finding a vector F'v.
Um = U1 x dom B2 B1
u E UM Fu' E
L(UM; F). Similarly we establish that the second Frechet derivative F' of operat or F is a continuous bilinear operator from UM x UM to F, and F'' = O. So the following theorem is valid.
Theorem 7. The operator F E CUM; F).
The vector-function f = f1 ® f2, where f1 = col(Ef1, nf1,0), f2 = 0. We assume that f EC ~(R +; F).
Thus, the reduction of problem (15), (2) to (3), (4) is finished. Further we identify problems (15), (2) and (3), (4).
Now let's check the conditions of theorems 1 and 2. By theorem 6 and the corresponding results of [10] there exists the analytic semigroup {U1 : t E R+} of the resolving operators for equation (4) which is in this case naturally presented in the form Uf = V1 ® W1, where Vt(Wt) is the restriction of operator U* on U1 (U2). Since the operator B2 is sectorial, then W* = exp(tB2), which leads to W0 = {0} and W1 = U2.
Consider the semigroup {V* : t E R+}. By theorems 4, 6 and monograph [10] this semigroup can be extended to the group {V* : t E R}. Its kernel V0 = U°° ®U° where U0° = {0} x {0} x Hp(= ker A1 (by theorem 4 ), and U°01 = EA-1A-1[H.2n] x H2n x {0}. Here A\ = I — XA, A\n is the restriction of operator nA-^o Hn. It is known that if X-1 E a(A) U a(Aa), then operator A\n : Hn — H^ is topological linear isomorphism (see,
U11 V1 . U1
the direct sum of subspaces: U1 = U0 © U^ © U\.
Construct the operator R = B-qA10 E £(U10 ©U^), where A10 (B10) is the restriction of operator A1 (B1) on U^ © U^. operator B-0 exists the theorem 6 and the
corresponding results of [10]).
Obviously, ker R = U°°, and it is shown [20] that im R = U°°. Hence, the operator R is bi-splitting. Let PR be a projector in the space U°° ©U®1 on U00 along U01. By construction of space Um project or Pr e C(U°M), where U°M = Um n (U?° © U? 1 )(= U?° © U? 1).
Lemma 1. Let X- 1 e a(A) U a(Aa). Then the operator R is bi-splitting, and PR e L(Um)■
Introduce the projectors
/ O O O\ ( O P\2 O\
P° = ( O O O I , Pi = I O n O I , \O O n) \O O O)
where P/ 2 = SA- 1A—1. From [20] since the kernel W° = {0}, projector I — P = (P° + P1) ® O. Applying projector I — P to the equation (4) in this transcription we obtain
n(vA(ua + un) — ((ua + Un) •V)(ua + Un) — up — gjug + f (t)) = 0,
(16)
Bun = 0.
Hence, by theorem 1 and properties of operator B we obtain the necessary condition for quasi-stationarity of the semi-trajectory un = 0. In other words, all solutions of the problem (if they exist) need to lie in the plane B = {u e UM : un = 0}.
But since nup = up, from the first equation in (16) we obtain relation (8) in a transcription
up = n(vAua — (ua • V)ua — g^ue + f (t)). (17)
Obviously, P° = PR, so the second equation in (16) is relation (9) with respect to our situation. So, we have
Lemma 2. Under the conditions of lemma 1, any solution of (3), (4) belong to the set A = {u e Um : un = 0, up = n(vAua — (ua • V)ua — g^ue) + f (t)}.
Remark 11. Relation (17) immediately implies the condition (iii) of theorem 2 for every point u° e Um(= U°° x {0}). Therefore, by remark 8 the set A1 (a simple Banach Crx-manifold diffeomorphic to the subspace U\ x U2) is the candidate for extended phase space of problem (15), (2).
o
We proceed to verify conditions (10) and (14). Construct the space Ua = U1 x W^Q). This space is obviously the interpolation space for the pair \U,UM}a, with a = 1/2. As noted above, the semigroup {Uf : t e R+} extends to a group {V1 : t e R} on U\, where V1 is the restriction of the o perator on U\ .Sine e Um = UM nU\ (by construction) and B1
{U* : t e R+} we have
/t nt
wwcUi
;UM)dt — C°nStWB1WL(Wi; Fi) / WVtWcundt < ~ e R+. (18)
Further, by Sobolev inequality [17, Chapter 9] semigroup {Wf : t G R+} satisfies
Let U\ = Ua nU1, where U1 = U\ x U2. Then from (18) and (19) imply the following
Lemma 3. Under the conditions of lemma 1, relation (10) is fulfilled.
Finally, for checking the requirement (14), we should find the operator H and the vector-function h. For this purpose we construct the projector Q : F ^ F1. According to [20] Q = (I - Qo - Qi) 0 I, where
Q13 = ZAA-1A-1B-1, Qi3 = nAA-1A-lB-1, Q03 = -Qi3, and operator Bn is a
restriction of operator B on H (by Banach theorem about the inverse operator the operator Bn : H ^ Hn is a toplinear isomorphism). So, the operator H = H1 ® H2, where H1 = A—1 (I — Q0 — Q1)F1, and H2 = F2 (A11 is the restriction of A on U\).
The inclusion H E C™ (UM; U^), where U\ = Ua HU1 is shown in the same way inclusion F E Crx(UM; F). Vector-function h(t) is defined as h1(t) ® h2(t), where h1 = A— (I — Q0 — Q1)f1 ^d h2 = 0.
The vector-function f = f1 ® f2 has ^te ^^^^^^e ^^^^^toess by const ruction. So h E
Thus, all conditions of theorem 2 are satisfied. Therefore we have
Theorem 8. Let A-1 E a(A) U a(Aa). Then for any u0 such th at (u0, 0) E and some T E R+ there exists a unique solution u = (ua, 0,up,ug) to problem (1), (2), which is a quasi-stationary semi-trajectory, and (u(t),t) E A*, for all t E (0,T).
Other non-stationary models of the incompressible viscoelastic fluids are considered, in [9,25-30]. Linearized models of different orders were studied in [31-36].
The non-autonomous case is described in detail in [37]. The Taylor problem for the generalized model of the non-zero order is studied in [38]. Different models of non-zero order in the autonomous case are studied in [39]. Investigation of magnetohydrodynamic models using the semigroup approach was initiated in [40,41].
The work is supported by The Ministry of education and science of Russian Federation (State task no. 1.857.20U/K).
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(19)
Qo =
Q1 =
C ~(R; U1).
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Received September 15, 2014
УДК 517.9 DOI: 10.14529/mmpl40401
ОБ ОДНОМ КЛАССЕ УРАВНЕНИЙ СОБОЛЕВСКОГО ТИПА
Т.Г. Сукачева, А.О. Кондюков
Статья содержит обзор работ Т.Г. Сукачевой и ее учеников в области исследования моделей несжимаемых вязкоупругих жидкостей Кельвина - Фойгта в рамках теории полулинейных уравнений соболевского типа. Основное внимание уделяется нестационарному случаю ввиду его большей общности. Идея исследования демонстрируется на примере нестационарной задачи термоконвекции для модели Осколкова нулевого порядка. Вначале изучается абстрактная задача Коши для полулинейного неавтономного уравнения соболевского типа, а затем соответствующая начально-краевая задача рассматривается как конкретная интерпретация этой задачи. Доказана теорема существования единственного решения указанной задачи, являющегося квазистационарной полутраекторией, и получено описание ее расширенного фазового пространства. Подобным образом могут быть исследованы и другие задачи гидродинамики, например, линеаризованные модели Осколкова, задача Тейлора, а также некоторые модели, описывающие движение несжимаемых вязкоупругих жидкостей Кельвина - Фойгта в магнитном поле Земли.
Ключевые слова: уравнения соболевского типа; несжимаемая вязкоупругая жидкости; относительно р-секториальные операторы; расширенное фазовое пространство.
Литература
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Работа поддержана Министерством образования и науки Российской Федерации в рамках выполнения государственного задания № 1.857.2014/К.
Тамара Геннадьевна Сукачева, доктор физико-математических наук, профессор, кафедра «Алгебра и геометрия:», Новгородский государственный университет имени Ярослава Мудрого (г. Великий Новгород, Российская Федерация), tamara.sukacheva@novsu.ru.
Алексей Олегович Кондюков, аспирант, кафедра «Алгебра и геометрия», Новгородский государственный университет имени Ярослава Мудрого (г. Великий Новгород, Российская Федерация), k.a.o_leksey999@mail.ru.
Поступила в редакцию 15 сентября 2014 г.
