ON EVOLUTIONARY INVERSE PROBLEMS FOR MATHEMATICAL MODELS OF HEAT AND MASS TRANSFER Текст научной статьи по специальности «Математика»

MSC 35R30, 35Q35, 65M60, 65M32 DOI: 10.14529/mmp210101

ON EVOLUTIONARY INVERSE PROBLEMS FOR MATHEMATICAL MODELS OF HEAT AND MASS TRANSFER

S. G. Pyatkov, Yugra State University, Khanty-Mansiisk, Russian Federation, s_pyatkov@ugrasu.ru

This article is a survey. The results on well-posedness of inverse problems for mathematical models of heat and mass transfer are presented. The unknowns are the coefficients of a system or the right-hand side (the source function). The overdetermination conditions are values of a solution of some manifolds or integrals of a solution with weight over the spatial domain. Two classes of mathematical models are considered. The former includes the Navier-Stokes system, the parabolic equations for the temperature of a fluid, and the parabolic system for concentrations of admixtures. The right-hand side of the system for concentrations is unknown and characterizes the volumetric density of sources of admixtures in a fluid. The unknown functions depend on time and some part of spacial variables and occur in the right-hand side of the parabolic system for concentrations. The latter class is just a parabolic system of equations, where the unknowns occur in the right-hand side and the system as coefficients. The well-posedness questions for these problems are examined, in particular, existence and uniqueness theorems as well as stability estimates for solutions are exposed.

Keywords: inverse problem; heat and mass transfer; filtration; diffusion; well-posedness.

Introduction

First, we consider the system

ut - vAu + (u, V)u + Vp = f + C + 6, div u = 0, (1)

8t - div (A,V6) + (u, V)6 = fg, (2)

n n

Ct + (u, V)C - Lu = fc, Lu = ^ aijCxix, + ^ aiCXi + aoC, (3)

i,j=1 i=l

where v = const > 0, (x,t) £ Q = G x (0,T), G C Rn, T < to, u, 6, p, C are the velocity vector, the temperature of a fluid, the pressure, the concentrations of admixtures (inorganic or organic) in a fluid, and fc is the volumetric density of sources of admixtures, respectively. The system (1) - (3) describes the propagation of admixtures in a fluid. In particular, it includes the classical Oberbeck-Boussinesq model (see, for instance, [1-3]). Here aij, ai, a0 are matrices of dimension h x h, with h the number of admixtures, fie is a matrix of dimension n x h, is a vector of length n, A© > 0 is a scalar function. The description of these class of models can be found, for instance in [4], where even more general models can be found derived on the base of thermodynamics of noninvertible processes. The functions fg and f are the densities of the heat sources and external forces. The coefficient Ag stands for the thermal diffusivity. In the Oberbeck-Boussinesq model,

the vector-functions [c and [e are the mass transfer coefficient and the heat-transfer coefficient multiplied by the free fall acceleration. For generality, we assume below that [c and [d are vector-functions of the variables (x,t).

For simplicity, we assume the domain G to be bounded though the main results are valid for a wide class of unbounded domains as well. The system (1) - (3) is furnished with the initial and boundary conditions

u|t=o = uo, uls = gi(t,x), r = dG, S = r x (0,T), (4)

6|i=o = 8o, Bi8|s = g2(t, x), (5)

C |t=o = Co, B2C |s = gs(t,x), (6)

n

where Biu = u or Biu = + cri(x,t)u, with 7h,<7i some functions, and

i=1 1

n

B2u = u or B2u = ^72i(x,t)-j^- + a2(x,t)u, with 72i,<T2 some matrices of dimension

i=1 1

h x h.

We consider an inverse problem of defining a solution to the system (1) - (3) and the right-hand side fc in (3) using the data of additional measurements on cross-sections of G. Let x" = (xk+1, xk+2,..., xn), k = 0,1,..., n — 1. If k > 1 then we put x' = (x1, x2,..., xk). The right-hand side is of the form

r

fc = fo(x,t) + J] fi(x, t)qi(x',t), (x, t) G Q, (7)

i=1

where fi (i = 0,1,... ,m) are given vector-functions. The functions qi(x',t) (qi(t)) in this representation are unknown and the overdetermination conditions for recovering these functions are of the form

C|si = ^(t,x), Si = (0,T) x ri, i = 1, 2,..., s, (8)

where {ri} is a collection of smooth k-dimensional surfaces lying in Go. For k = 0, the surfaces ri are just points lying in G. One more overdetermination condition is of the form

J(C, <fi(x))dx = ^i(t), i =1, 2,..., r, (9)

G

where the brackets denote the inner product in Rh, ^i(x) is a vector with h components, and ^i(t) are given functions. We do not know the articles where the inverse problems (1) -(8) or (1) - (7), (9) for the complete system are studied except for the author articles [5-7]. We can refer to [8], where a series of results devoted to optimal control problems for the systems occurring in the class (1) - (3) in the stationary case can be found. Optimal control problems for some simpler models are studied in [9-11]. The description of numerical methods of solving directs problems for Oberbeck-Boussinesq model is exposed in [3].

Many results connected with solvability of inverse problems for the Navier-Stokes system and the linearized Navier-Stokes system are presented in [12], where the main results are connected with the ovedetermination conditions of the form (9).

The inverse problems (1) - (8) and (1) - (7), (9) as well as the problems (3), (6), (8) and (3), (6), (9) for parabolic equations and systems arise when describing heat and mass transfer, filtration, diffusion, and some other physical processes [13,14]. We can note that, in a real situation, even the simplest one-dimensional models used in monitoring and warning systems for river basins include several parabolic equations relative to concentrations. For parabolic equations and systems, the problems of the above type are studied in many articles and we can refer to the book [15], where these problems are discussed in the case of parabolic equations of the second order and k = n — 1. The overdetermination conditions here are values of a solution on sections of a spatial domain and the coefficients are independent of some spatial variables, the latter allows to apply the Fourier transform and to simplify the problem. The inverse problems with additional data on planes (sections of a spatial domain) are considered also in [16,17] and some other articles. More general inverse problems with data on surfaces of arbitrary dimension are studied in [18-21]. The most known overdetermination conditions used in these problems are the values of a solution at some collection of interior points of G. Thus, additional conditions are the data of measurements (for example, the concentration of the transferred substance) at certain points in the domain. The data are employed to determine both the sources (for example, sources of pollution in water or air) and environmental parameters. The unknowns qi(x',t) depend in this case only on t. Thus, the right-hand side in (3) is

r

representable as fc = f0(x,t) + fi(x,t)qi(t). The inverse problem is to find a solution

i=l

to the system (3) and the functions qi (t), i = 1, 2,...,r, that appear in the right-hand side (3) or in the equation itself from the data (6) and (8), where Si = {xi} are points. In the heat and mass transfer and filtration problems, the right-hand side fc characterizes the distribution of sources (sinks) and their intensities. In the case of point sources, i.e. fi = 5(x — xi), where 5 is the Dirac delta function, qi is the intensity of the i-th source in the heat and mass transfer problems, and in filtration problems, for example, in oil production qi, is the flow rate i-th well, in this case u is the pressure [22]. In various practical problems distributed and point sources as well are both employed. First, we describe some results devoted to problems with spatially distributed sources. A large number of results was obtained in the case a linear second order equation.

We can refer to the article [23], where a theorem on the existence and uniqueness of solutions to problem (3), (6), (8) on determining the source in Holder spaces in the case h = 1 and r = 1 is obtained. Similar results in the case of the problems of determining the source function and coefficients were obtained in the monograph [24] but in a one-dimensional situation (n = 1). In [25] the problem of determining the lowest order coefficient in a parabolic equation was considered, and in [26], the lowest order coefficient and the right-hand side of the form q(t)f (t,x) are determined. In both cases, the well-posedness of the corresponding inverse problems is proven. There are many articles devoted to model equations and systems mainly in the one-dimensional situation (see, for example, [27,28]). The first most essential results for quasilinear equations of the form (3) were obtained in [29], where conditions for a nonlinear function depending on u, Vu were derived that guarantee the global solvability (in time) of the problem (3), (6), (8) in the Holder spaces for case r = 1. The authors of [30] obtain a similar result already in the case of a parabolic system and in the Sobolev spaces. The problem of local or global well-posedness of linear and quasilinear problem of the form (3), (6), (8) in the Sobolev

spaces was further considered in the articles [31-33]. In the general setting quasilinear inverse problems are considered in the book [12], where the relevant bibliography can be found. The authors consider a nonlinear nonautonomous first order operator-differential equation. The operator in the main part is a generator of an analytic semigroup. The overdetermination conditions are a collection of functionals defined on a given Banach space. The inverse parabolic problems with the overdetemination conditions (8), with k = 0, and (9) are thus included in this statement. The problem was studied in the spaces of functions continuously differentiable with respect to t. However, the constraints imposed on the nonlinearity are rather strong and can be essentially weakened. Weaker assumptions on the nonlinearity are used in [34], where the domain of the operator A(t) in the main part can depend on time and the main results are stated in the Sobolev spaces. The article [35] contains the results on solvability of a linear inverse problem of recovering the function f (t) in the operator-differential equation ut + Au = f (t)z with the overdermination conditions Ф(и) = ^(t) (Ф is a functional). A huge amount of articles is devoted to numerical solving the problems of the form (3), (6), (8). We can refer, for example, to the articles [36-38]. There is a large number of monographs devoted to numerical methods for solving inverse problems. Almost all inverse parabolic problems and a large number of numerical methods are considered in the monograph [14] in the case n =1. The monographs [39,40] are devoted to a more general situation; number of interesting statements and problems (including those of convective heat transfer) are considered in [41,42].

Describe some results in the case of point sources. As already noted, these problems are not well-posed in the classes of finite smoothness, and there are practically no existence and uniqueness theorems for solutions [43]. There is a huge number of articles devoted to the numerical solution of the problem of determining point sources, however, as a rule, these articles do not contain any theoretical justifications and very often both non-existence of solutions and their non-uniqueness can take place in the corresponding problems for certain values of parameters. The articles [45-47] can serve as examples. Let us single out the articles, where there is some theoretical justification of algorithms for finding solutions [48-53]. Note that in this case we need to determine the number of sources, their locations and intensities. The most interesting idea of constructing point sources is presented in [51]. It was subsequently used in [52]. Note that the problems of determining point sources are nonlinear, in contrast to the case of distributed sources.

The well-posedness questions for parabolic equations and systems with the overdetermination conditions (9) (including numerical methods) are treated in many articles. The first article probably is that by A.I. Prilepko [56] with coauthors, where the question of recovering the right-hand side f = q(t)g(x, t) + f0(x,t) (the unknown is the function q(t)) in a parabolic equation was examined in Holder spaces. Next, we should refer to the well-known article [57] (see also [58]), where general nonlinear parabolic problems were considered in the one-dimensional case also in Holder spaces. In particular, it is established under certain condition (at most linear growth of the nonlinearity in u,ux at infinity if the main part of the equation is linear) that the solvability of the inverse problem with the overdetermination conditions of the form (9) is global in time. Next, we can refer to [59-67], where inverse problems of recovering coefficients depending on time in the case of the r =1 in a linear parabolic equation. There are examples of simultaneous determination of the right-hand side and the coefficient (see, for instance, [68], where

n =1). The inverse coefficient problems for the parabolic system of the second order having a special structure was studied in [59] in the Holder spaces. Here the overdetermination conditions (9) and (8) for the parabolic system (3) are used simultaneously. Examples of recovering the right-hand sides in a second order parabolic equation are presented in [69-72]. The multidimensional inverse problems of recovering the right-hand side and coefficients simultaneously are studied also in [44,73,74] in the Sobolev spaces.

Among the monographs devoted to inverse problem for parabolic equation and systems we note the monographs [12,14,15,42,75-81], where the sufficient bibliography can be found.

Describe the contents of the article. The next section contains some conventional definitions. Section 2 contains the results on solvability of the problems (1) - (8) and (1) - (7), (9). Section 3 is devoted to solvability of the parabolic problems (3), (6), (8) and (3), (6), (9), where the operator L is replaced with a higher order elliptic operator. The notations of function spaces are conventional (see, for example, [54,55]).

1. Definitions and Notations

Let E be a Banach space. By Lp(G; E) (G is a domain in Rn) we mean the space of strongly measurable functions defined on G with values in E endowed with the norm II\\u(x) I|e||lp(g) [54]. We employ also the Holder spaces Ca(G) (see the definition in [54]). The Sobolev space notations are conventional, i.e. W*(G; E), WpS(Q; E), etc. (see the definitions in [54,55]) designate the Sobolev spaces of functions with values in E .If E = C (E = R) or E = Cn (E = Rn) then the latter space is denoted by Wps(Q). Similarly, we use the notations Wp(G) or Ca(G) rather than Wp(G; E) or Ca(G; E). Thus, the membership u £ Wp(G) (or u £ Ca(G)) for a given vector-function u = (u\,u2, ■ ■ ■ ,Uk) means that every of its component Ui belongs to Wp(G) (or Ca(G)). The norm of the vector is just the sum of the norms of the coordinates. The same meaning has the membership a £ Wp(G) for a matrix-function a. Given an interval J = (0,T), put Wp'r(Q) = Wp(J; Lp(G)) f LP(J; Wp(G)). Respectively, W*'r(S) = W;(J;LP(T)) n Lp(J; W^(T)). Similarly, we can define the Holder space Cr's(Q).

Next, we present some auxiliary statements. For simplicity, we assume that G is a bounded domain, though many of the results are valid in the case of unbounded domains as well. Let the symbol (xi) stand for the ball centered at xi of radius 5. As conventionally, we denote by Lp,a(G) the closure of solenoidal C0°-vector-functions in the norm of Lp(G) and put W^a(G) = Wps(G) f Lp(G) and WP^/2(Q) = Wps,s/2(Q) f

o

Lp(0,T; Lp,a(G)) (s > 0). The symbol W s(G) designates the closure of C^(G) in the norm of the space Wqs(G) and W^G) = {p £ Lq1oc(G) : Vp £ Lq (G)}. We identify functions which differ by a constant and endow this space with the norm HpIIw^g) = l|Vp||Lq(g). It is a Banach space.

Consider the parabolic problem

ut + Lu = f, Bju\s = gj, u(0,x)= u0(x), (10)

where Lu = \a\<2m aa(t,x)Dau, Bju = \a\<mj bja(t,x)Dau. Introduce the operators L0u = \a\=2m aa(t, x)Dau and B0ju = \a=m. bja(t,x)Dau. We say that the problem (10) satisfies the (PL) condition (see [82, p. 198] and [83, Ch. 7]) if

(PL) a) there exists a constant > 0 such that any root p of the polynomial

det (Lo(t,x,îf) + pE) =0

(E is the identity matrix) satisfies the inequality

Re p <-#i|£|2m V£ G Rn V(x,t) G Q;

b) for every point (to,xo) G S, £ G Rn such that (£, v(xo)) = 0 (v(x) is the outward unit normal to r at x), and all h G Ch, A such that Re A > 0 and |£| + |A| = 0, the system

(AE+(-1)mAo(xo,to,£+iv(xo)dy))v(z) = 0, y> 0, Bo(xo,to,£+iv(xo)dy)v(0) = h, (11)

has a unique solution decreasing at infinity of the class C([0, œ)).

The algebraic conditions ensuring the condition (PL) (the parabolicity condition and the Lopatinskii condition) can be found in [83], for example.

2. Inverse Problems (1) - (8) and (1) - (7), (9)

First, we describe the conditions on the data of the problems (1) - (8) and (1) - (7),

(9).

(A) The case of k > 0. There exists a domain Q C Rk with boundary of class C2 such that G C Q x Rn-k,

r = {x G Rn : x'' = /(x') = (^k+1(x'), ^fc+2(x'),..., <(x')), x' G Q},

Lpl(x') G C2(Q) and there exists a constant 8 > 0 such that

USi = {(x', <^(x') + n) : x' G Q, n G Rn-k, |n| < 8} C G

for i =1, 2,..., s, and Usi if = 0, for i = j, i, j = 1, 2,..., s.

The case of k = 0. In this case the sets ri are points and we assume that these points are interior points of G.

In what follows, we use the following notations: Qo = Q x (0,T), So = dQ x (0,T), Qo = Q x (0,t), ST = r x (0,t), Gs = UiUsi, and Qs = Gs x (0,T), QS = Gs x (0,t), QT = G x (0, t).

The condition (A) have been used in all articles devoted to the problems in question. As is easily seen, it ensures uniqueness of solutions. The condition (A) is fulfilled if G = Q x Rn-fc, with Q a bounded or unbounded domain of class C2.

First, we present our conditions for the data. Let f (x', x'', t) ((x', t) G Rfc+1, x'' G Rn-k, j = 1, 2,..., r) be the zero extension of the function fj from Q to Rn+1, i. e., fj = 0 on Rn+1 \Q. In view of the condition (A), we can assume that the functions Vj (j = 1, 2,..., s) in (8) depends on the variables x', t only, i. e., Vj = Vj(x', t) ((x', t) G Qo). We also assume that the parameter 8 > 0 used below is that of this condition. First, we consider the case of the Dirichlet boundary conditions in Theorems 1, 2 below.

The agreement and smoothness conditions. Let q > n + 2 and there exist vector-functions $1, $3 and function $2 such that

$i(t,x) G W2'1(Q): $1|t=o = uo, $2|t=o = ©o, $3|t=o = Co, $i|s = gi, (12)

а1уф! = 0, Фз|sj = ^, /с,/в,/ е Lq(Q), / е (Qo;Lq(Rn-s)), (13) Vx»Фз е Wq^Q*), Vx-/с е Lq(Q*), /,Vx»/ е L^(Q*), (14)

where j = 1, 2,..., r and i =1, 2, 3. As a consequence of these conditions and embedding theorems, we can conclude that

uo, Co, во е Bq2-2/q(G), gi е Wq2-1/q'1-1/(2q)(S), i = 1, 2, 3,

Vx''Co е Bq2-2/q(G*), Vx-g3 е Wq2-1/q'1-1/(2q)(S*), ^(t,X) е Wq^Qo),

where j = 1, 2 ...,r and S* = (dG* П Г) x (0,T). If these smoothness conditions and the corresponding agreement conditions (see trace theorems, for instance, in [54]) hold then we can construct the corresponding functions Фг. For example, if g1 = 0, div uo = 0, q > 3/2, and the above smoothness condition for uo holds then the agreement condition on the vector-function uo ensuring the existence of Ф1 is the condition uo|r = 0.

Let B(X,t) be the matrix whose rows with the numbers from (j — 1)h +1 to jh, j = 1, 2,..., s are occupied by the vectors

[A(X, (X), t), /2(X, (X), t),..., /r(X, (X), t)].

We require that there exist a constant > 0 such that

| det B(X,t)| > a.e. in Qo. (15)

We also assume that

(В) Ae(x,t) > 5 > 0 V(x,t) G Q, Ae(x,t) G W^(Q), аг] G C(Q), and Ух„аг] G L^Qe) for all i, j = 1, 2,..., n; вс, a, ao, вв е Lq(Q), Vx»аг, Vx»ao е Lq(Q*), i = 1, 2,..., n. The proofs of Theorems 1, 2 below can be found in [5-7].

Theorem 1. Assume that Г е C2, q > n + 2, the problem (3), (6) satisfies the (PL) condition, and the conditions (A), (B), (12) - (15) hold. Then there exists a number to е (0,T] such that there exists a unique solution (u,p, в, C, q1,..., qr) to the problem (1) -(8) from the class

u е Wq2'1(QT0), p е Lq (0,To; W^G)), qj е Lq (Qo0), j = 1,2,..., r,

в, C е Wq2,1(Qr0), Vx''C е Wq^QSO) W2 < 5.

Let collections (иг,рг, вг, Cl, q1,..., q^), i = 1, 2 be solutions to the problem (1) - (8) from the above class corresponding two different collections of the data /г, /в, /o, , ulo, g^, eo, and Co, j = 1,... s, n =1, 2, 3, i = 1, 2 satisfying (12) - (14) with some functions Ф*, i = 1, 2, j = 1, 2, 3 and

3 / ■

м 11Ф Hw^Q) + llVx'' ^llwf1^) + II Лк (Q) + j=1v

+ H Ув H Lq (Q) + H/0 HLq (Q) + H Vx'' Уо HLq (Qi ^ ^ R , i = 1 2.

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Fix 52 <5. Then there exists a constant t0 > 0 such that the following stability estimate holds:

||ul - u2|Wq2,1(QTc) + H©1 - ©2|Wq2,1(QT0) + |V(p' - p2 )|Lq (Qto ) +

r

+ llC 1 - ^IIw^qo) + |lVx''(C1 - C2)llwti(Q?) + E Hal - a2HLqqoo) <

2 j=i

3

< c(£ 11$) - $2Wwl'\QT0) + IIVx''($3 - ^llwf1^) + ll/a - /o Ik(QT0) + j=1

+ W/1 - / 2||Lq (QT0 ) + W/) - fe 11 Lq (QT0 ) + WVx'' (/01 - /02)WLq (QT0 )) >

where the constant c depends on the quantities Ra and 82.

Proceed with the linearized statement. We examine the system

n

ut - vAu + Vp Bjuxj + Bau + / + pcC + pe8, div u = 0, (16)

j=1

nn

8t - A* A8 + £ bj8X. + bo8 = / + J] buj, (17)

j=i j=i

n n n

Ct - Lu = /c + cj uj, Lu = aij Cxixj + a Cxi + aaC, (18)

j=1 i,j=1 i=1

where Bj ,Ba are n x n matrices. We assume that

(C) bj(x,t),ba,bj,cj,cj,ca,Bj,Ba G Lq(Q), Vx''cj(x,t) G Lq(Qs),

where j = 1, 2,... ,n and ô is the parameter that of the condition (A).

Theorem 2. Assume that r G C2, q > n + 2, the problem (3), (6) satisfies the (PL) condition, and the conditions (A), (B), (C), (12) - (15) hold. Then there exists a unique solution (u,p, 8, C,q1,..., qr) to the problem (16) - (18), (4) - (8) from the class

u G Wq2,1(Q), p G Lq (0,T ; W)(G)), q3 G Lq (Qo), j = 1, 2,...,r,

8, C G Wq2,1(Q), Vx'' C G Wq2,1(QÔ2 ) VÔ2 < ô. Fix ô2 < ô. A solution meets the estimate

r

WuWWq2,1(Q) + W8WWq2,1(Q) + WVPWLq (Q) + WC WWq2,1(Q) + E W qj W Lq (Q0) +

j=1

3

+ WVx''C WW2'1(QS2 ) < c(E W$i WWq2'1(Q) + WVx'' $3WWq2'1(Q5 ) +

(19)

i=1

+ W/0WLq (Q) + WVx'' /0 W Lq (QS ) + W/\W Lq (Q) + W/e \W Lq (Q)).

Next, we consider the integral overdetermination conditions (9). Actually, these results are new. We describe them without proofs. The proofs can be found in the forthcoming paper in Itogi Nauki i Tekhniki (2020, vol. 187).

In this case, we have that qj = qj(t), i. e., the functions qj depend only on t. Our conditions on the data can be written in the following form

uo G Wq2-2/q(G), divuo = 0, Uo|r = 0, /,/,/c G Lq(Q), q > n + 2, (20)

Co(x) G Wq2-2/q (G), gs(x,t) G Wqs2'2s2 (S), B2(x, 0,D)Co|r = g3(x, 0), (21) ©o(x) G Wq2-2/q(G), g2(x,t) G Wqsi'2si (S), Bi(x, 0,D)0o|r = g2(x, 0), (22)

where sj = 1 — 1/2q if Bju = u (i = 1, 2) and sj = 1/2 — 1/2q otherwise and we take gi(x,t) = 0 in (4)

^i(t) G W(0,T), ^(0) = J(Co(x), ^j(x)) dx, i = 1, 2,..., r, (23)

Gi

a,i(t, x) G Lg(Q) (i = 0,1,... ,n), ctij G G([0, T]; Geo(G)), i,3 = l,...,n,

Ъз, о- G c^-vap+eo.i-i/p+eo^ j = 1,..., n, г = 1,2, ^ ^

where eo > 0 is a positive constant

/i(x, t) G L^(0, T; Lq(G)), i = 1, 2,..., r. (25)

Let {Gj} be a collection of subdomains of G with boundaries of the class C1. We assume that

щ G LP{G), supp^- С Gj С G, щ G ^(Gi), J + J = 1, j = 1, 2,..., r, (26) for some e1 > 0.

Define the entries bjj(t) of the matrix B by the equalities bjj = /(/•, (x)) dx and

G

suppose that there exist constants , > 0 such that

| det B|> io > 0, a.a. on (0,T), (27)

Ae(x,i) > 8г > 0, {x,t) G Q),\e G C(Q); G Lq{Q). (28)

Introduce the set BR of vectors U7 = (uo, Co, 0o, g2, g3, /, /o, /, ..., ), satisfying (20) - (23) and such that

llu°llw,2-2/9 (G) + IIc°IIw2-2/« (G) + ll0olw2-2/« (G) + lg2lff;i'2si (S) + 1Ы1^2'282 (S) +

s

+ |/ |Lq (Q) + |/e |Lq (Q) + |/o|Lq (Q) + E H^i |Wi(o,T) <

j=1

Theorem 3. Assume that Г G C2, the problems (3), (6) and (2), (5) satisfy the (PL) condition, q > n + 2, and the condition (20) - (28) hold. Then there exists a number to G (0, T] such that there exists a unique solution (u,p, 0, C, q1,..., qr) to the problem (1) - (7), (9) from the class

u G Wg'1(QT°), p E Lq{0,ro-,Wlq{G)), q3ebq(QT0°), j = 1,2,... ,r.

Вестник !Ю"УрГ"У. Серия «Математическое моделирование 13

и программирование» (Вестник ЮУрГУ ММП). 2021. Т. 14, № 1. С. 5-25

Fix R0 > 0. Then there exist constants t0 = t0(R0) and c = c(R0) such that for every two solutions u%, ©i ,Ci,qi, qi = (qi1,qi2 ,...,qir), i = 1, 2 relating to the collections U1,U2 £ Bro, Ui = (u0, C0, ©0, g2, g3, fi, f0, fe, "01,..., ^r), i = 1, 2, the following estimate holds:

l|u1 - u2|Wq2,1(QT0) + ||©1 - ©2|Wq2'1(QT0) + ||C 1 - C2|Wq2,1(QT0 ) + r (

+ E Hq1j - q2j 11 Lq (0,To) < C |u0 - u0HW2-2/q (G) + ||f 1 - f2|Lq (QT0) + j=1 q

+ Hfe1 - fe2|Lq(Qto) + ||f0 - f02|Lq(Qto) + ||C0 - C02HW2-2/q(G) + H©0 - ©0HWq2-2/q(G) +

+||g21 - giHwqsi'2si (s to ) + ||g1 - glK-^ {sn ) + =^ - )).

Consider a linearized statement. We consider the system (16) - (18), where

B0,b0,aj,Bj,bj, £ Lq(Q), j = 1, 2,... ,n. (29)

Theorem 4. Assume that r £ C2, p > n + 2, the problems (3), (6) and (2), (5) satisfy the (PL) condition, and the conditions (20) - (29) hold. Then there exists a unique solution (u,p, ©, C,q1,..., qr) to the problem (16) - (18), (4) - (7), (9) from the class

u £ Wq2,1(Q), p £ Lq (0,T; W1(G)), q3 £ Lq (Q), j = 1, 2,...,r.

A solution satisfies the estimate

Hu|Wq2'1(Q) + H©|Wq2'1(Q) + ||C HWq2'1(Q) + E Hqj H|Lq (0,T) < c( 11 u0 11 w2 - 2/q (G) + ||f HLq (Q) + + Hfe 11 Lq(Q) + 11 f0 H Lq(Q) + |C0HWq2-2/q(G) + H ©0 H Wq2-2/q(G) + + Hg2|Wq1-1/2q,2-1/q(S) + ||g3HWql0'2s0(S) + E H^i|Wq1(0,T^ .

3. Inverse Problems for Parabolic Systems

In this section, we examine parabolic equations and systems of the form

r

ut + A(t, x, D)u = bi(t, x)qi(t, x') + f, (t,x) £ Q, x =(x',x''), (30)

i=1

where x' = (x1 ,x2,..., xk), x" = (xk+1, xk+2,..., xn), bi, i = 1, 2,... ,r, and f are given vector-functions and A is a matrix elliptic operator of order 2m with matrix coefficients of dimension h x h representable as

ro

A(t,x,D)=J2 qi(t,x')Ai(t,x,Dx) + Aro+1(t,x,Dx), (31)

i=r+1

Ai = Y, aia(t, x)Da, i = r + 1,...,r0 + 1, r0 = sh, D = (dx1, dx2 ,...,dxn). (32)

\a\<2m

The unknowns in (30) are a solution u and functions qi(t,x'), i = 1,2,..., r0 occurring in the right-hand side (30) and the operator A as well; in the latter case

they are coefficients. The equation (30) is complemented with the initial and boundary conditions

= ua, Bju|S = bj/3(t,x)D^u|S = Qj(t,x), (33)

u\

t=0

where mj < 2m, j = 1, 2,... ,m. Let r G C2m. The overdetermination conditions have the

form

u|s = (t,x), i = 1, 2,...,s.

(34)

The results exposed in this section can be found in author's articles [18-21,44]. The condition on the data are written as follows.

3$(t,x) G Wp2m'1(Q), p > n + 2m : $|t=0 = u0(x), Bj$|g = gi, I = 1,...,m, dXi$ e W2m>l(Qs), S]=^(t,x') e C([0,T]-,C2m(tt)), e C(Q~0),

f G LP(Q), dXifGLp(Qs), f\s, eC(Q~0), i>k + l, j = l,...,s. (35)

As a consequence of the conditions (35) and the embedding theorems, we have

uo(x) G Wp2m-2m/p(G), g3 G Wp2mfc''fcj(S), BjUo|r = g(0,x), where kj = (2m — mj — 1 /p)/(2m) and j = 1, 2,... ,m,

(36)

dxiQj G Wp2mfcj'fcj (Ss), OxMx) G Wp2m-2m/p (Gs),j = 1, 2,... ,m, i = k + 1,...,n, (37)

where Ss = (dGs H r) x (0,T). The conditions on the coefficients of the operators A,Bj are more or less conventional. For simplicity, we will use the conditions that are not sharp. We assume that

(t,x) G L00(Q) (M<2m), aiaeC{Q ) (\a\ = 2m), г = r + 1,..., r0 + 1, (38)

big G C

i2m—rrii, 1 —

'Jjß t О ^-^(S), dXibjß G

(Ss),

< mj, j = 1,... ,m, i > k, (39)

dXiaja(t,x) £ LX(QS), \a\ < 2m, j = r + 1,...,r0 + 1,i>k, (40)

bi(t,x) £ Lp(Q), dxibi £ Lp(Qs), (l = 1,...,r, i> k). (41)

We look for the functions qi in the class of continuous functions. Hence, we require that

(42)

for all l = 1,... ,r, j = 1, 2,..., s, and |a| < 2m.

Now we introduce the matrix B(t,x') of dimension sh x sh whose rows with the numbers from (j - 1)h + 1 to jh are occupied by the column vectors

( — bi(t,x), —b2(t,x),..., —br (t,x),Ar+1&(t,x),..., As^(t,x))

x"=ipj (x')

It can be shown with the use of the embedding theorems and the conditions (35), (36), (42) that the elements of this matrix are continuous on G. We require also that there exists a constant 50 > 0 such that

det B(t,x')\ > öo Vx' G Ü, t G [0,T].

(43)

<mj

a

a

Consider the system of equations

B(t,X)q 0 = g,

q 0 = ,

(44)

where g is the column vector whose coordinates with the numbers from (j — 1)h + 1 to jh are the vector

/(t,xV(X)) - t,XV(X)) - (t,XV(X)).

(45)

Under the condition (43) the system (44) has a unique solution q 0 = (q°,...,q°h)

(B(i,x')) 1g(t,x'). The above conditions for the data ensure that q° G C(<5o)- Consider the operator

A°(t,X,D) = J] q°(t,X)Ai(t,x,Dx) + ),

i=r + 1

and the problem

ut

+ A°(t,x,Dx)u = g ((t,x) G Q)

ul

t=0

u°(x),

Bj uls

gj •

(46)

Fix i G {1, 2,..., s} and make the change of the variables y' = X, y'' = x'' — ^(X), t = t in the domain Q^j, with ^ < i. After this change, the operators A and Bj are transformed into some operators Al(t,y,Dy) and Bj(t,y,Dy). Denote by Ay, and Bjy, the parts of the operators A1 and Bj not containing the derivatives with respect to the variables yk+1, yk+2,..., yn and by Ay,, and Bjy„ the remainders. Similar sense has the notations Ax, Bjx', Ax,,, Bjx», and A0x,, A0x,,. Describe the connections between the derivatives with respect to the new and old variables. We have

r=fc+1

+ ^ iïxj(X)dx r=fc+1

j < k, dXj. j < k dVj

Vj'

j > k, j > k.

Thus, we infer

Ay,(t,y,Dy') = Ax'(t,y',y'' + ^(y'), Dy), Bjy,(t,y,Dy) = BjV(t,y',y'' + <^(y'), Dy).

As is easily seen, the operators Ax, and preserve their form. Consider the auxiliary problems

W + A°v' (t,y', 0, Dy'= 0, (t,y') G Q° (0,y' ) = 0, j = 1, 2,...,s,

Byj = 0, j = 1, 2,..., s, i = 1, 2,...,

m.

(47)

(48)

(49)

Theorem 5. Assume that the condition (A), where dQ G C2m and the conditions (35), (38) - (43) are fulfilled and the problem (46) satisfies the condition (PL). If Bjy' (t, y', 0, Dv) = 0 for all i = 1, 2,..., m and j = 1, 2,..., s then there exists a number t° G (0, T] such that there exists a unique solution (u, q1,..., ) to the problem (30), (33), (34) of the class

u G W%m'\Qro) : Vx»« G %2m,1(Qg) W2 < 5, q3 G ), j = 1, 2,. . . , sh.

r

j

Proceed with the linear case. Assume that all coefficients of the operator A are known functions, i.e., r = sh, A = aa(t,x)Dau, and all unknown functions q enter

|a|<2m

the right-hand side of (30). All conditions for the data have the same form. In particular, we assume in the next theorem that the problem ut + Au = f, u(0,x) = u0, Bju|S = gj, j = 1,..., m satisfies the confition (PL). In our case the rows of the matrix B(t, x') of dimension sh x sh with the numbers from (j — 1)h +1 to jh are occupied by the column vectors ( — bi(i,x',^j(x')), — b2(t,x',^j(x')),..., —br(t,x',^j(x')). The conditions (35), (41) can be rewritten as follows:

ЗФ(£,x) G Wp2m'1(Q) (p> n + 2m): Ф|4=0 = uo(x), = gi, l = 1,...,m,

Ф G Wp2m,1(Qi), f G Lp(Q), 5хгf G Lp(Qi), i > k + 1, (50)

bi(t, x) G L^(Q), dxibi G L^(Qi), l = 1,..., r, i > k + 1. (51)

Let Ф0 be the class of vector-functions ф = (ф^ф2, ...,ф5) G Wpm,1(Q0) whose coordinates meet (47), (48) and there exists a function Ф satisfying (50), with u0 = 0, gj =

0 (j = 1,..., m), such that Bix (t, x', (x'), Dx') —j | = Bix' (t, x', (x'), Dx') Ф | , where

1 = 1, 2,..., m, j = 1,..., s. We say that the equalities (34) are fulfilled in a generalized sense whenever there exists a vector-function ф = (ф1, ф2,..., фs) G Ф0 such that

u|S = ф,(t, x')+ —^(t,x'), (t, x') G Q0, i = 1, 2,..., s. (52)

I

The fulfillment of the equality (34) in a generalized sense means that it is fulfilled in the quotient space (Wpm'1(Q0))s/Ф0, where Wpm,1(Q0) is a space of vector-functions ф G Wp2m>1(Q0) of length h.

Theorem 6. Assume that the condition (A), where дП G C2m, and the conditions (38) —

— (40), (43), (50), (51) are fulfilled. Fix 81 < 8. Then the following statements are valid.

1. There exists a constant c > 0 such that a solution (u, q1,..., qr) to the problem (30)

- (34) from the class

u G W2m,1(Q): Vx»u G Wp2m,1(Q*2) V82 <8, qj G Lp(Q0), j = 1, 2,. . . , sh meets the estimate r

||u| Wpm,1(Q) + ||Vx'' ) + Y1 ||qj IlLp (Qo) -

j=1

- с(|Ф|Жр2т-1(д) + ||Vx'' ^W2™'1 (Q4) + ||f ||Lp(Q) + ||Vx''f ¡Lp (QS)). (53)

2. There exists a unique solution (u,q1,..., qr) to the problem (30) - (34), where (34) is understood in the generalized sense, from the class

u G W?™'1 (Q), Vx» u G Wp2m'1(Qi1) V81 <8, qj G Lp (Q0), j = 1, 2,...,r.

3. Solutions (u, q1,..., qr) to the problem (30) - (34), with u0 = 0, f = 0, gj = 0 and ф = (ф1 ,ф2,... ,—s) G Ф0, from the class

u G W2™'1 (Q) : Vx''u G Wp2m'1(Qi1) V81 < 8, do not exist whenever ф = 0.

4. If B3iy„ (t,y', 0, Dy) = 0 for all i = 1, 2,...,m and j = 1, 2,...,s then ={0} and there exists a unique solution (u,q\,... ,qs) to the problem (30) - (34), where the equality (34) is understood in the usual sense from the class

u £ Wp2m,1(Q): Vx- u £ Wp2m,1(QSl) Wi <S, qj £ LpQ), j = 1, 2,...,r.

Remark 1. We note that the function qi are sought in the space Lp(Q0) in the previous theorem. However, the results are valid if a solution is sought in the class indicated in Theorem 5. The conditions on the data and the coefficients in this case coincide with those of Theorem 5 (see [19]).

Remark 2. Note that the above theorems are valid in the case of the pointwise overdetermination as well, i.e., k = 0. The condition (A) in this case is reduced to the following conditions: the points {xi}S=1 are interior points of G. Moreover, in this case we can replace the conditions (35) with the more natural conditions (36), (37) and the consistency condition u0\Sl = ipi(0,x'), l = 1,..., s.

Remark 3. If the condition (43) fails then very often the problem becomes ill-posed in the Hadamard sense. In this case the problem becomes unsolvable if the data have finite smoothness. The corresponding example can be found in [7, Example 3]. The condition of additional smoothness of the data in some neighborhood Gs about the set, where the overdetermination data are imposed (see conditions (35), (37), (41), (40), etc.) also cannot be omitted. For example, in the condition (35) we require that Vx» f £ Lp(Qs)). In the case of the pointwise overdetermination this condition can be written as f £ Lp(0,T; Wp1(Gs)) (Gs is a neighborhood of the set {xi} of the overdetermination points). We can replace this condition with the condition f £ Lp(0,T; Wp(Gs)) with s > n/p (see [31]). But if s < n/p we can construct ill-posedness examples again. In the general case of k > 0 additional smoothness in the variables x" can be characterized by the number s > (n — k)/p.

Next, we present an analog of Theorem 3 in the case of a higher order parabolic system. The results are published in [44,73,74]. We consider the problem (30), (33), where the operator A admits the representation (31). We slightly refine some of the statements in these articles. We assume that

iPi G Lq{G), supp^ CG.CG, dGl C C1, ipi G WhGi), i=l,...,r0, - + - = 1, (54)

pq

f £ Lp (Q), p > n + 2m, bi (x,t) £ L^(0,T; Lp(G)), i = 1, 2,...,r, (55)

^i(t) £ Wp1[0,T], ^(0) = J(u0(x),Vi(x))dx, i =1, 2,...,r-0, (56)

Gi

__171 j _

aia G C(Q), M = 2m, alot G Lp(Q), M < 2m, bjP G (S), (57)

where j = 1,... ,m, \ < mj. Let G0 = ur=1 Gi and assume that

bj,f £ C([0,T]; Lp(G0)) (j = 1, 2,... ,r), ata £ C([0,T], Wp1^)) for \a\ = 2m, (58) aia £ C([0,T],Lp(G0)), i = r + 1,r + 2,...,r0 + 1 for \a\ < 2m. (59)

18 Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming

& Computer Software (Bulletin SUSU MMCS), 2021, vol. 14, no. 1, pp. 5-25

Define the matrix B of dimension r0 x r0 with the rows

(6i(0,x),<£fc) dx,..., / (6r(0,x),^fc) dx,

G

G

- /(Ar+i(0,x)uo,^fc) dx,..., - / (Ar0 (0,x)uo,^fc) dx.

G G

We require that

det B = 0.

Determine the constants q°, i = 1, 2,..., r° as solutions to the system

(60)

ro

«0)+ Ê q°/) dx + /(Aro+iUo,^j) dx =

i=r+1 G G r

= E q0 / (bi(0,x), ) dx + /(/ ) dx,

i=1 G G

(61)

r0

where j = 1,2,...,ro, and construct the operator A0 = E q^A + Ar+1. Our

overdetermination conditions take the form

i=r + 1

(u, ^j(x)) dx = ^j(i), i = 1, 2,..., r0.

(62)

G

Theorem 7. Assume that the conditions (36), (54) - (60) hold and the problem (46) satisfies the condition (PL) with the above-defined operator A°. Then there exists a number t° < T such that on the segment [0,t°] there exists a unique solution (u,q1,..,qr0) to the problem (30), (33), (62) such that

u G Wp2m'1(QT0), q*(t) G C([0, t°]), i = 1, 2,r°.

In the linear case, we can weaken our conditions on the coefficients. In this case all coefficients of the operator A = A° = aaDa are known functions and the unknowns

|a|<2m

q enter the right-hand side of (30). We require that

G Lq(G), supp c Gj c G, 1/p + 1/q = 1, ^j G Wq£0(Gj), e° > 0,

aa G C(Q), a« G £«*,(0, T, G£°(G,)), M = 2m, j = 1,..., r,

mj

т-гщ, 1-тг±

G Lp(Q) (|a| < 2m), j G C2

The matrix B of dimension r x r has the rows

(S),

< mj, j = 1,..., m.

(63)

(b1 (0,x),^k ) dx,..., / (br (0,x),<£k ) dx, k =1,...,r.

G

G

The claim of the previous theorem can be reformulated as follows.

а

а

Theorem 8. Assume that the conditions (36), (55), (56), (60), (63) hold and the problem, (46) satisfies the condition (PL) with the above-defined operator A0. Then there exists a unique solution (u, q1,.., qr0) to the problem (30), (33), (62) such that

u £ Wp2™'1 (Q), qt(t) £ Lp(0,T), i = 1, 2,..., r.

A solution satisfies the estimate

lull Wj2m,1(Q)

+ Y1 il qi (i)Hc ([O,T ]) <

i=1

< c

+ £ ii*.

j=1

jIIWp2mfcj'fcj (S) +

'(S )

Uo|

W,

2m-2m / p

(G)

j=i

i|W1(0,T )

Remark 4. The results of the above Theorems 1-8 remain valid in the case of unbounded domains G for which the solvability theorems of the direct problems are valid (the conditions on the coefficients slightly differ from the above-presented, see those in [83, Theorem 9.1], the Theorem 5.7 for G = Rn, Theorem 7.11 for G = R+ in [82]. Note also that the results in [5-7] employ more general condition rather than the condition (A).

Remark 5. First, we note that the conditions on the lower order coefficients in Theorems 4, 5 can be weakened. It suffices to require that aia G Lp(Q) or aa G Lp(Q) rather than aia G Lœ(Q) or aa G Lœ(Q). Second, we can note that stability estimates for solutions similar to those in Theorems 1, 2 are also valid in all remaining theorems.

m

s

L

p

Acknowledgment. The author was supported by the Russian foundation for basic research (Grant 18-01-00620a).

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Received August 19, 2020

УДК 517.956 Б01: 10.14529/шшр210101

ОБ ЭВОЛЮЦИОННЫХ ОБРАТНЫХ ЗАДАЧАХ

ДЛЯ МАТЕМАТИЧЕСКИХ МОДЕЛЕЙ ТЕПЛОМАССОПЕРЕНОСА

С.Г. Пятков, Югорский государственный университет, г. Ханты-Мансийск, Российская Федерация

Представлены результаты о корректности обратных задач для математических моделей тепломассопереноса. Неизвестными являются правая часть в уравнении (функция источников) и коэффициенты уравнения. Условия переопределения -значения решения на некоторых многообразиях или в отдельных точках. Рассматриваются два класса математических моделей. Первая включает систему уравнений Навье - Стокса, дополненную параболическим уравнением для температуры и параболической системой для концентраций примесей. Правая часть неизвестна и характеризует объемную плотность источников в жидкости. Неизвестные функции зависят от времени и части пространственных переменных и входят в правую часть уравнения. Второй класс систем - параболическая система уравнений уравнений для концентраций переносимых веществ, где неизвестные входят как в правую часть так и саму систему в качестве коэффициентов. Показана корректность этих задач, в частности полученные теоремы существования, единственности и оценки устойчивости для решений. Далее, мы опишем некоторые алгоритмы решения обратных задач о восстановлении точечных источников по точечным данным переопределения, основанные на асимптотике решений функций Грина соответствующих эллиптических задач.

Ключевые слова: обратная задача; тепломассообмен; фильтрация; диффузия; корректность.

Сергей Григорьевич Пятков, доктор физико-математических наук, профессор, Высшая цифровая школа, Югорский государственный университет (г. Ханты-Мансийск, Российская Федерация), s_pyatkov@ugrasu.ru.

Поступила в редакцию 19 августа 2020 г.