CC BY
35
MSC 35R30, 35K57, 80A20
DOI: 10.14529/ mm p140403
SOME INVERSE PROBLEMS
FOR CONVECTION-DIFFUSION EQUATIONS
S. G. Pyatkov, Yugra State University, Khanty-Mansyisk, Russian Federation, s _ pyat kov@ ugrasu.ru,
E.I. Safonov, Yugra State University, Khanty-Mansyisk, Russian Federation, [email protected]
We examine the well-posedness questions for some inverse problems in the mathematical models of heat-and-mass transfer and convection-diffusion processes. The coefficients and right-hand side of the system are recovered under certain additional overdetermination conditions, which are the integrals of a solution with weights over some collection of domains. We prove an existence and uniqueness theorem, as well as stability estimates. The results are local in time. The main functional spaces used are Sobolev spaces. These results serve as the base for justifying of the convergence of numerical algorithms for inverse problems with pointwise overdetermination, which arise, in particular, in the heat-and-mass transfer problems on determining the source function or the parameters of a medium.
Keywords: parabolic system; convection-diffusion; heat-and-mass transfer; inverse problem; control problem; boundary value problem; well-posedness.
Introduction
We examine the question on recovering of the right-hand side and coefficients in a second order convection-diffusion system. Let G be a domain in Rra with boundary r of class C2 and let Q = G x (0,T). This system is of the form
ut + A(t,x,D)u = fc, (t,x) e Q, (1)
where A is a second order elliptic operator with matrix coefficients of dimension h x h. The equation (1) is supplemented with the initial and boundary conditions
u|t=o = uo, u\s = g(t,x), (2)
where S = (0,T) x r. The right-hand side and the operator A in (1) are of the form
ro
fc = £ bi(t,x)qz (t) + f, (3)
i=1
r
A(t, x, D)u = A = E qi(t)Ai(x,t) + Ar+i(x,t), Aiu = E àaDau. (4)
i=ro+1 |a|<2
The unknowns in (1), (2) are a solution u and the functions qi(t) (i = 1,2,...,r)
A
overdetermination conditions:
/ u^i(x)dx = ^i(t), i = 1, 2,...,s, r = sh, (5)
JOi
where Gi С G are some domains. Consider also the overdetermination conditions
u(xi, t) = фi(t), xi E G, i = 1, 2 ... ,s, r = sh. (6)
The problem of this type arise when describing heat and mass transfer, convection-diffusion, and filtration processes (see [1-5]). Inverse problems on recovering of the
t
where r = ^d Gi = G, are exposed in [6-12]. The linear inverse problems on recovering of the right-hand side are studied in [5,13] respectively. Similarly, both types of problems with conditions (5) and (6) are examined in [4,14] and [15,16]. A large number of physical statements and numerical methods of solving of the above-mentioned inverse problems with condition (6) is exposed in [27] and [28]. The problem on recovering of the source function (3) with the overdetermination conditions of the form (6) can be found in [29, Ch. 3], where the main attention is payed to numerical methods. In this monograph the problem of determination of the source function G(x,t), with given measurements (6) is examined. Here, the source function is replaced with its approximation of the form (3) which is calculated numerically. Note that most articles are devoted to the case of some model equations, where n =1. We can note only articles [17,18], where problems (1), (2), (6) in general statement are treated. We can also refer to monographs [2,6,14,19,20], where the reader can find statements of inverse problems for parabolic equations and systems and the corresponding existence and uniqueness theorems as well as some numerical methods for inverse problems solving.
In the present article under natural conditions on the data of the problem, we demonstrate that problem (1) - (5) is uniquely solvable and establish stability estimates for solutions. On one hand, problem (1) - (5) is of interest in its own right. On the other hand, a solution to problem (1) - (4), (6) can be approximated by solutions to the problem (1) - (5) for a suitable choice of the weights фi = ^i(x, e) depending от a parameter e > 0 (actually, we can construct an approximation of the Dirac ^-function). The convergence can be established under appropriate conditions on the data. The latter fact allows to construct numerical algorithms for solving of problem (1) - (4), (6). We should note that most of the authors for numerical solution of the inverse problems use methods based on minimization of some functional (which is not convex) (see, for instance, [29] или [27]). Algorithms relying on an approximation of solutions to the problem (1) - (4), (6) by solutions to the problem (1) - (5) allow to construct simpler methods and this fact is confirmed by numerical experiments. In the next section, we present some auxiliary statements and conditions on the data. In section 3 we state and prove our main results (Theorems 4, 5).
1. Definitions and Auxiliary Statements
Let E be a Banach space. The symbol Lp(G; E) (G is a domain in Rn) stands for the space of strongly measurable functions u(x) defined on G with values in E such that the norm ||||u(£)||e||lp(g) is finite [21]. We also use the spaces Ck(G) comprising functions having derivatives up to the order k inclusively in G continuous in G and admitting continuous extensions onto G. The notations of the Sobolev spaces W£(G; E), W£(Q; E) and so on are conventional (see [21,22]). If E = C or E = Cn then we use the notation Wps(G) or Ck(G) rather than W^(G; E) or Ck(G; E). Thus, the inclusion u E W£(G) (or u E Ck(G)) for a given vector-function u = (u\,u2,... ,uk) means that every component
of ui belongs to Wps(G) (or Ck(G)). In this case the norm of the vector-function is the sum of the norms of the coordinates. A similar convention is used for matrices as well. Given an interval J = (0,T), assign Wsr(Q) = Ws(J; LP(G)) n LP(J; Wr(G)) and, respectively, Wsr(S) = Ws(J; Lp(r)) n Lp(J; Wr(r)). In what follows the symbol Vu stands for the vector (ux1 ,ux2 ,...,uXn), i. e. the gradie nt of u in the space variables. The condition r G Ca (a > 1) is understood conventionally (see [25]).
The smoothness and consistency conditions. Fix p > n + 2. We assume that
uo(x) G W2-2/p(G), g(x,t) G Wl-1/2p'2-1/p(S), g(x, 0) = ua(x)\dG, (7)
^i(t) G C1 ([0,T]), &(0) = J uo(x)^(x)dx, i =1, 2,...,s, (8)
Gi
aa(t,x) G L„(Q) (\a\ < 2), < G C(Q) (\a\ = 2). (9)
We assume below that the boundaries of the domains {Gj} (j = 1, 2,..., s) those in (5) belongs to the class C\ We employ the following conditions on the weights {Vj (x)}:
supp Vj c Gj, Vj G Wl(Gj) (1 + 1 = 1),j = 1, 2,..., s, (10)
supp Vj c Gj, Vj G L1(G), j = 1, 2,..., s. (11)
Let Go = Us=1Gj, Qo = Go x (0,T). Under the condition (10) we require that
bj ,f G C ([0,T]; Lp (Go)) (j = 1, 2,...,ro), aa G C ([0,T],W^ (Go)) for \a\ = 2, (12)
aia G C([0,T],Lp(Go)) (i = ro + 1,ro + 2,...,r + 1) for \a\ < 1. (13)
If we replace (10) with (11) then we need the conditions
2-2 .
Vbk, Vf G Lp (Qo), Vuo G Wp p (Go), Vaia(x,t) G L^(Qo) (\a\ < 2), (14)
aa G C([0,T],L^(Go)) (\a\ < 1), bk, f G C([0,T]; L^(Go)) (k = 1, 2,...,ro), (15)
where i = ro + 1,ro + 2,..., r. In the case of the problem (l)-(5) we define the matrix B of dimension r x r whose rows with the numbers from (k — 1)h + 1 to kh, (k = 1, 2,..., s) are occupied by the h x r-matrices with columns
/ bi(x, 0)vk dx,..., br0 (x, 0)vk dx, — Ar0+i(x, 0)uoVk dx,..., — Ar (x, 0)uoVk dx.
G G G G
We require that
det B = 0 Vt G [0,T]. (16)
Assign Gs,i = {x G Gi : p(x,dGi) > 5}, Q1Si = Gs,i x (0,7), Gs = U|=iGs,i, and Qs = Gs x (0,T), QY = Gs x (0,7) (5 > 0), QY = G x (0,7). Let A = £H<2 a«Da, where aa are h x h-matrices. This expression can be also rewritten in the form A = — Y^ij=1 aijdXiXj + 'Y^n=1 aidXi + ao. We say that A is elliptic whenever
n n
Re J2(a*j (x,t)?,Zj) > 5oY, \^2, V^2,... ,C G V(x,t) G Q. (17)
i , j=1 i=1
Here S0 is some positive constant. The following theorems are valid.
Theorem 1. Assume that G is a bounded domain with boundary of class C2, the conditions (7) hold, the coefficients of the operator A = \a\<2 aaDa satisfy the condition
aa(t,x) E LX(Q) (M < 2), aa E C(Q) (M = 2) (18)
and the ellipticity condition (17) holds. If f E Lp(Q) then there exists a unique solution u E Wp,'2(Q) to the problem
ut + A(t,x, Dx)u = f, u\t=o = uo(x), u\s = g, (19)
satisfying the estimate
IMIw-^2 (q) < C \\f \\lp(q) + \\g\\Wl-i/2v,2-l/vs) + \\uo\\W2-2/V(G) , (20)
C f, uo u
2— 2
Vuo E Wp p (Go), Vaa(x,t) E Wi(Qo), M < 2, (21)
and Vf E Lp(Q0) then a solution u possesses the property Vu E Wl'2(Q&) for a 11 S > 0 and, for a fixed 6 > 0, we have the estimate
UVuH w12(qs ) + hWwt'iQ) < C\\\f hp(,Q) + \\Vf\LP(Q0) +
\g\w1-1/2p-2-1/p(S) + \uo\w2-2/p(G) + \Vuo\w2-2/p(G0)
p\Qo)
(22)
Proof. The first claim follows from Theorem 10.4 in [25]. The second claim is justified
conventionally with the use of the finite difference method and Lemma 4.6 of Ch. 2 in [26].
2 D
G C2
coefficients of A = \aa<<2 aaDa satisfy (18), the ellipticity condition (17) holds, and f E Lp(QY) (7 E (0,T]). Then there ewsts a unique solution u E Wp,'2(QY) to the problem
ut + A(t,x, Dx)u = g, u\t=o = 0, u\s = 0, (23)
satisfying the estimate
\\u\Wp'2(Q7) < C\g\Lp(QY), (24)
C7
Proof. A solution sought agrees on [0,7] with a solut ion uY to the problem (19) with the right-hand side gY = | 0' t g (0' T] E LP(Q) and the homogeneous initial and boundary
conditions. Theorem 1 yields the estimate \\uY\\w2>1(q) < c\\gY\\Lp(q) < c\g\Lp(qy) which validates (24).
□
Theorem 3. Assume that the conditions of Theorem 2 and condition (21) on aa hold. Then a solution u to problem (23) for a fixed Si > 0 satisfies the estimate
\u\Wp'2 (qY ) + \\Vu \ Wp,2(QY ) < c( \ \g \ \ Lp (QY) + \Vg \ Lp (qY )), where c is independent of 7.
Proof. The proof is in line with that of Theorem 2. _□
2014, tom 7, № 4 39
2. The Basic Results
Define the constants q0 (i = 1, 2,..., r) as solutions to the system
r ro
4>jt(0) + E q0, / ^iMo^j dx + f Ar+iu0^j dx = E q0 / bi(x, 0)<fj dx + f f<pj dx, (26)
i=ro+1 G G i=l G G
where j = 1, 2,..., s. In the following theorem we assume that
r
the operator A0 = ^^ q0Ai + Ar+1 is elliptic. (27)
i=ro+1
Theorem 4. Let the conditions (7) - (8), (9), (10), (12) - (13), (16), (27) hold. Then there exists y0 > 0 such that, for t E [0, y0]; there exists a unique solution (u,q) (q = (q1, ..,qr)) to the problem (1) - (5) such that u E W10(QYo), qi(t) E C ([0,y0]), i = 1, 2,...,r. A solution depends continuously on the data of the problem, i. e., for every two solutions (ui, ql,... ,qr) (i = 1, 2) to the problem (1) - (5) from the class
ui E Wp'2 (QY0), qj E C ([0,Yo]), (i = 1, 2, j = 1, 2,...,r),
corresponding two different collections of the data fi, -j ul0, g% (j = 1, 2,..., s, i = 1, 2) satisfying (7) - (8), (9) - (10), (12) - (13) and such that the condition (16) holds and
s
\\f%\\Lp(Q) + E \Wj Wo i([0'T|) + \\fi\\c ([0'T];Lp(Go)) + j=1
+ WuгoWw¡-2/P(G) + ^Ww^^^'^/ps) - R, (i = 1, 2 ) there exists y1 = Y1(R) such that the stability estimate
r
\\u1 - u2\wp1'2(Q71 ) + E \\q1 - q2\\c([0'Y1]) - C(\\f 1 - P\\lp(QY1) + E Uj - 1([0'Y1]) +
4 u2\ ^1,2(qyi) + E \\q1 - qj\\c([0'Y1]) - C(\\f1 - f2\\Lp(QY1) + E 11$ -j=1 j=1
^ - f 2\c ([0'T ];Lp (Go)) + \\u0 - U0WW2-2/P (G) + \\g1 - g2\\^1-1/2P 2-1/P (S~n)
is valid, where C is a constant independent of R and y1 and SY1 = dG x (0, y1). Proof. We construct a function $ as a solution to the problem
ro
$t + (^2 q0Ai + Ar+1)$ = f + Y, q0bi(x,t), $|t=0 = u0(x), $\s = g. (28)
i=ro+1 i=1
A solution to this problem exists and possesses the properties those of Theorem 1. Make the change V = u - $ and q = q1 + q0 (q1 = (q1,..., q%r), i = 0,1). In this case V is a solution to the problem
r ro r
Vt + a0v + E q1AiV = E biq1 - E q!a$, vu = 0, v\s = 0. (29)
i=ro+1 i=1 i=ro+1
Theorem 2 implies that for every f E Lp(QT) the problem
Vt + A0V = f, V|t=0 = 0, V|s = 0 (30)
is uniquely solvable and
\\VHw^Qt) < c\\g\\Lp(QT), (31)
where the constant c is independent of t E [0,T], Examine an auxiliary problem
r
Vt + A°V + J] qlAiV = g(x,t), V|t=0 = 0, VIs = 0. (32)
i=ro+1
Let q1 E = [q1 E C([0,t]) : Wq1 \c([°,r]) < V} (t < T). By Theorem 2, we have
r
V +(dt + A°)-^ qlAiV =(dt + A°)-1g(x,t). (33)
i=ro+1
The estimate (31) yields
\\(dt + A0)-1 E q1AiV\\wWqt) < c\\ it q}AiV\\lp(qt) <
i=ro+1 i=ro+1 (34)
\\q 1WC([0,T])CC1WVWw^qt) < VccA\VWw^qt).
In what follows, we assume that v < = 211- this case we infer
W'\qt) ^ 2"V '\qt)
' 1 + A0)^ J] qlAV)\\„,i,,~) < -
r p
i=ro+1
and thereby (33) is uniquely solvable and a solution V E W1'2 (Q) satisfies the estimate
\\VWw^qt) < 2\\(dt + A°)-1gWw12(QT) < 2c\\g\\LpQ), Vq1 E B^. (35)
Integrating equation (29) with the weight over G, we obtain
x V* *=11VV, d «t,«t)=* -1 *
r ro r
$it + / A°V<fiidx + E qU AjV*idx = E qU bj*idx — E qU Aj$*idx, (36)
G j=ro+1 G j=1 G j=ro+1 G
where i = 1, 2,..., s. Examine the matrix B(t) of dimension r x r whose rows with the numbers from (k — 1)h + 1 to kh, (k = 1, 2,..., s) are occupied by the h x r-matrices with the columns
J b1*k dx,..., j bro dx, — J Aro+1$*fc dx,..., — j Ar dx.
G G G G
Note that B(0) = B with the matrix B that was defined before (16). Demonstrate that the entries of B(t) are continuous in t. The continuity of the entries from the first r° columns results from (9). Consider the last r — r° columns. Si nee $ E Wp'2(Q), the embedding theorems yield
$ E C([0, T]; W^(G)), a < 2(1 — 1/p). (37)
The latter results, for example, from Theorem 1.8.3 in [21], where the Banach spaces Ao,A1 are replaced with Wp^(G) and Lp(G). In view of our conditions on p, 2(1 — 1/p) > n/p + 1 and, in particular, Wp(1-1/p)(G) C C13(G) with / < 2 — 2/p — n/p [21]. Thus,
U(Q) c cW^1 p 1(g)) C C([0,T]. C3(G)) /3 < 2 — 2 — P. (38)
W1,2(Q) C C([0,T]. Wp
In particular, changing $ on a set of a zero measure if necessary, we can assume that the functions Da$ for |a| < 1 are continuous. We have Aj$ = Aj0$ + Aj1$ j > ro, where Aj0$ = E ajaDa$ and Aj1$ = E ajaDa$■ In view of (13) and the above-pointed
|a|=2 |a|<2
properties of $, for |a| < 1 the functions fG ajaDa$vi dx are continuous in t. Consider the case of |a| = 2. We conclude that
a{(x,t)Da$<fidxdt = aa(x,t)Da$^nkdr — (aaVi)xkDa$ dx,
'G
'Gi
where Da$ = ^ Da $, nk is a coordinate of the unit outward normal to r = dG,. By
t
In view of continuity of the entries of B(t), there exi sts t0 > 0 and a const ant 8 > 0 such that dtB(t)| >8, Wt E [0,t0]. The equalities (36) can be rewritten as
B(t)q1
81 + (A°V,Vi)+ E q}(AV,Vi)
i=ro+1
8st + (AoV,Vs)+ E ql(AiV,Vs) i=ro+1
(u, v) = J uvdG.
G
Thus,
q1 = go + R(q1),
(39)
where g0 = B (t)
-1
88
1t
8
st
R(q1) = B (t)
-1
(A0V,v>1)+ E ql(AiV,V1)
i=ro+1
(A0V, Vs) + E ql(AiV^s)
i=ro+1
and V is
a solution to the problem (29), which exists whenever q1 E B^, with j < jo h t < T. Consider the vector
go = b (t)-1 '
8.
st
Let t < to. we have ||go||c([o,r]) < c max \\i)it\\c([o,t])- In view of (26), i)it(0) = 0. By
1< i<s
continuity of i8it, 3t1 < to \g0\c([0,T]) < i20, Wt0 < t1. Show that there exists t2 < t1 such that the equation (39) is uniquely solvable in the ball B^0,T2 = {q1 : Hq1 ||C([0)T2]) < jo}. Derive estimates for the operator R(q1). Let q1, q2 E B^0,T, (t < r1^d V 1,V2 are
r
solutions to problem (29) (with (j^d q2 on the right-hand side) satisfying the initial and boundary conditions Vj|t=0 = 0 Vj\s = 0 j = 1, 2). We have
ro
Vj + A0Vj + £ qjAiVj = £ qjbi — qjAiVj|t=o = 0, Vj|s = 0, (40)
j=ro+1 i=1 i=ro+1
where i =1, 2,... ,m, j = 1, 2. Estimate ^(q1) — R(q2) ||c([0,r])- We obtain
(41)
HR(q1) - RG?)||c([o,r]) < c max II(A°(V1 - V2),Pj) + E (q1(AV 1,Vi) - q2(AV2,Vi))lo([°,r]).
i=ro+1
Subtracting the equations (40) for j = 1, 2, we arrive at the equality
(V1 - V2)t + A°(V1 - V2)+ it q2(AiV1 - AV2) =
i=ro+1
ro r ro
E(q1 - q2)bi - E (q1 - q2)AV1 -£(qj - q2)A$.
i=1 i=ro+1 i=1
Note that in view of (35) the functions Vj meet the estimate
ro r
llVj IIwi2(Qr) < 2C ||E qj bi - E qj AMIlp(qt) < C1 llqj |C([o;r]) < (43)
i=1 i=ro+1
where C1 is independent of x and r. Estimate (35) and equalities (42) yield
(42)
WV1 " V^Ww^qt) < С2Ы - q2Wc([o,T]) + max WAV I^q^q1 - q2||o([o,r]) < № - q2Wo([о,т])(C2 + Сз\У 1Wwpi.2(qt)) < WO1 - q2\\c([o,T])(C2 + C3C1V0).
(44)
Estimate the right-hand side of (41) using the inclusion pj E W1(Gj). Consider \(A°v, pj)| for v = V1 - V2. We have A°v = A°v + A1v where A°v = E aaDav and A1v =
|a|=2
^ iaDav. For \a\ < 1, we infer \ fG aaDavpidx\ < c||Dav||LTC(Go) < c||v||wi,(Go). Thus,
|a|<2
\j A1v^idx\< C1HvHw^(g). (45)
Jo
Consider the expression fo A0v^idx. It contains the summands
/ aa(x,t)Dav^idxdt = aa(x,t)Da'vpndT — / (aa^i)XkDa'vdx, *J o *j r? *J Oi
д
dx
second integral is estimated as
where Dav = ^Da v, nk is the fc-th coordinate of the unit normal to Г = dGi. The
/ \aa WDa v\^\^ixk\dx + \aaxk \ • \Da v\ • \ф\dx <
J Gi J Go
1 л 1
< M\\Da v\\Lx(Go )(/ \фгХк \qdx) q + M^D*' v\\L^Go )U v\dx) q
J Gi Gi
Therefore, the second integral is estimates by CiWvWwi(G). Estimate the first integral with the use of the trace theorems (see, for instance, [21,25,26]). In view of (38) we can assume that v e C([0,T]; C 1(G)), and thus aa e C(Q°) and
\J aaDv^iUkdr\ < Mw^(G)\\a«\\LTO(Go)\\^\\Lq(ri) < c2\\v\\wi(G)-Г
We involved the fact that фi e Wj l/q(ri) due to embedding theorems. These two inequalities ensure that there exists c3 such that
\ / AoVфidx\< C3\\v\\wi(G)- (46)
G
By the embedding theorems and the interpolation inequalities (see (38), [21]), we infer
\\v\\wi,(G) < C4\\v\\W+n(G) < <b\\v\\W2-p{g ML~IgV 0(2 - 2) = 1 + n. (47)
In this case relations (45), (46), (47) imply that
max \\(A°(V1 - У2),ф3)\о([о,г|) < Ce\\v\\e 2-2 MC^l,(g))■ (48)
C(\°,T|;W, p (G)) U J P " K '
As it was noted, v e C([0,r]; Wp2~2/p(G)) after a possible modification on a set of zero measure. Next, v = f° vT(х,т)dr and thereby
\М\с([о,т^р(G)) < rq MLpdOTlLpG)), (1/P + 1/q = 1)- (49)
1-е
Therefore, \\(Av^j)\\C([0,T|) < Cr q ||v\q). In this case we have
\\(A°(V1 - V2),фj)\\с([о,т|) < C7T4е\V1 - V2\'2(qt), (50)
where C1 is independent of r. Using (44), we obtain
\\(A°(V1 - )\с([о,т|) < C8T—Hq1 - q2\с([о,г|), (51)
Consider the second summand in (41). We have
E (q1(AiV1 ,фj) - qi(AiV2, фj)) =
i=ro+1
E (q1 - q2)(AiV )+ it q2(AiV1 - AiV2^j).
i=ro+1 i=ro+1
The summands (AiV ), (Ai(V1 - V2)^j) are estimated as the expression (A°(V1 -V2),<ф3) (see (50)). We have
r -> -> 1- е
\\ E (q1 - q2)(AiV^)\\с[°>T| <\\q.1 - яЧс^r—C9\\v ) <
i=ro+1 (52)
1-е -> ->
r q C(^°)\q1 - q2\\с([°,т|).
Similarly, in view of (44), we infer
r l-e -> ->
WE q2(Ai(V1 — V2),*j )Wc[°,t ] < c1°t— \\q1 — q2Wc({°,r]). (53)
i=ro+1
Relations (51) - (53) imply that
WR(q1) — R(q2)Wc[°,r] < cut — Wq1 — q2\\c ([0,t]), Vqi E B,0,t.
l-e
Find t2 < t1 such th at c11t2 q < 1 .In this case
\R(q1) — R(q2)Wc ([° ,n]) < 2 WQ1 — Q2Wc{[° m). (54)
Since R(q2) = 0 for q2 = 0, we have
WRQ1 ^cm < 1 WQ1Wc(l°,r2]). (55)
In this case the operator R°(q 1) = g° + R(q-1) is contractive and takes the ball B^o,T2 into itself. Indeed,
\R°(q)Wc([°,-t*]) < \g°\c([° ,T2]) + \\R(q1)Wc([°n]) < y + y =
q 1
By construction, V is a solution to problem (29). Assign j° = t2. Demonstrate that V satisfies the conditions f V*jdx = ^(t). Integrate equation (29) over G with the weight
G
We have
r ro r
| / *iVdx + / A0 V*idx + E q11 Aj V*idx = E q1 f bj * dx — E q1 f Aj $*idx.
G G j=ro+1 G j=1 G j=ro+1 G
The function qj satisfies (36); subtracting the i-th equation from the previous equality, we obtain that ( f *vdx — tp-i) = 0 or JG *vdx — tpi = (f '•pivdx — tpi)It=° = 0 in view of
G t G
v
from the statement of the theorem was actually obtained in the proof and we omit the arguments.
□
Theorem 5. Let conditions (7) - (9), (11), (14) - (16), (27) hold. Then there exists y° > 0 such that on the segment [0,y°] there exists a unique solution (u,q1,..,qr) to problem (1) - (5) such that
u E W1,2(QYo), qi(t) E C ([0,y°]), i = 1, 2, ...,r.
A solution continuously depends on the data of the problem, i. e., for a fixed ^ > 0 and every two solutions (ui, q\,... ,qlr) (i = 1, 2) to the problem (l)-(5) of the class
ui E W}'2(QYo), qj E C ([0,y°]), (i =1, 2, j = 1, 2,...,r),
corresponding two different collections of the data f \ u%0, g% (j = 1, 2, . . . , s, i = 1, 2) such that conditions (7) - (9), (11), (14) - (16), (27), (16) hold and
s
\\f%\\Lp(Q) + E \Wj\\c1([0,T]) + \\P\\c([0,t];l^(gq)) + P\\lp(qq) 3=1
\\U0\\wl-2/v(G) + \\^Uo\\w2-2/v(G) + \\g%\w1-1/2p'2-1/p(S) - R> i =1, 2' there exists a constant j1 = j1 (R) such that the stability estimate
r
\\ul - u2\\wi'2(qYi) + \\^(u1 - ^Ww^qi) + E \\qj - q2\c([o,7i]) -
1 3=1
C(\\f1 - f2\\lp(qYi) + È w^} - j\ci([o,7i]) + \\f1 - f2wc([oT]MG0))+ (56) \\v(f 1 - f2)\\lp(qq) + \\u1 - u0\\w2-2/V(g) +
'МЧ0) I II "0 -rniw;-z/p(G)
_ u2o)\\W2-'2/p(Go) + W91 - 92\\w1-1/2p•2-1/p(^1 )>
holds, where S11 = dG x (0,71) and the constant C is independent of R and 71.
Proof. The proof relies on the same scheme as that of the proof of the previous theorem. As before, the claim is reduced to the solvability of equation (39).
□
The authors were supported by the Russian Foundation for Basic Research (Grant 12-01-00260a).
References
1. Alekseev G.V. Optimizatsiya v statsionamykh zadachakh teplomassoperenosa i magnitnoy gidrodinamiki [Optimization in Stationary Problems of Heat and Mass Transfer and Magnetic Hydrodynamics]. Moscow, Nauchnyi Mir, 2010.
2. Belov Ya.Ya. Inverse Problems for Parabolic Equations. Utrecht, VSP, 2002.
3. Babeshko O.M., Evdokimova O.V., Evdokimov S.M. On Taking into Account the Types of Sources and Settling Zones of Pollutants. Doklady Mathematics, 2000, vol. 61, no. 2, pp. 283-285.
4. Kalinina E. A. A Numerical Study of an Inverse Problem of Recovering the Heat Source for the Two-Dimensional Nonstationary Convection-Diffusion Equation. Far Eastern Mathematical Journal, 2004, vol. 5, no. 1, pp. 89-99. (in Russian)
5. Kriksin Yu.A., Plushchev S.N., Samarskaya E.A., Toshkin V.F. Inverse Problem of Recovering the Source Density for a Convection-Diffusion Equation. Matematicheskoe Modelirovanie [Mathematical Modelling], 1995, vol. 7, no. 11, pp. 95-108. (in Russian)
6. Iskenderov A.D., Akhundov A.Ya. Inverse Problem for a Linear System of Parabolic Equations. Doklady Mathematics, 2009, vol. 79, no. 1, pp. 73-75. DOI: 10.1134/S1064562409010219
7. Ismailov M.I., Kanca F. Inverse Problem of Finding the Time-Dependent Coefficient of Heat Equation from Integral Overdetermination Condition Data. Inverse Problems In Science and Engineering, 2012, vol. 20, no. 24, pp. 463-476.
8. Ivanchov M.I. Inverse Problem of Simulataneous Determination of Two Coefficients in a Parabolic Equation. Ukrainian Mathematical Journal, 2000, vol. 52, no. 3, pp. 379-387. DOI: 10.1007 Bl'02513132
9. Jing Li, Youjun Xu. An Inverse Coefficient Problem with Nonlinear Parabolic Equation. Journal of Applied Mathematics and Computing, 2010, vol. 34, no. 1-2, pp. 195-206. DOI: 10.1007/sl2190-009-0316-8
10. Kamynin V.L., Franchini E. An Inverse Problem for a Higher-Order Parabolic Equation. Mathematical Notes, 1998, vol. 64, no. 5, pp. 590-599. DOI: 10.1007 Bl'02316283
11. Kerimov N.B., Ismailov M.I. An Inverse Coefficient Problem for the Heat Equation in the Case of Nonlocal Boundary Conditions. Journal of Mathematical Analysis and Applications, 2012, vol. 396, issue 2, pp. 546-554. DOI: 10.1016/j.jmaa.2012.06.046
12. Kozhanov A.I. Parabolic Equations with an Unknown Coeffiecient Depending on Time. Computational Mathematics and Mathematical Physics, 2005, vol. 45, no. 12, pp. 2085-2101.
13. Vasin I.A., Kamynin V.L. On the Asymptotic Behavior of Solutions to Inverse Problems for Parabolic Equations. Siberian Mathematical Journal, 1997, vol. 38, no. 4, pp. 647-662. DOI: 10.1007 Bl'02671572
14. Prilepko A.I., Orlovsky D.G., Vasin I.A. Methods for solving inverse problems in Mathematical Physics. New-York, Marcel Dekker, Inc. 1999.
15. Tryanin A.P. Determination of Heat-Transfer Coefficients at the Inlet into a Porous Body and Inside It by Solving the Inverse Problem. Journal of Engineering Physics, 1987, vol. 52, no. 3, pp. 346-351. DOI: 10.1007 Bl'00872521
16. Dehghan M., Shakeri F. Method of Lines Solutions of the Parabolic Inverse Problem with an Overspecification at a Points. Numerical Algorithms, 2009, vol. 50, no. 4, pp. 417-437. DOI: 10.1007/sl 1075-008-9234-3
17. Pyatkov S.G., Samkov M.L. On Some Classes of Coefficient Inverse Problems for Parabolic Systems of Equations. Sib. Adv. in Math., 2012, vol. 22, no. 4, pp. 287-302. DOI: 10.3103/S1055134412040050
18. Pyatkov S.G. On Some Classes of Inverse Problems for Parabolic Equations. Journal of Inverse Ill-Posed problems, 2011, vol. 18, no. 8, pp. 917-934. DOI: 10.1515/jiip.2011.011
19. Ivanchov M. Inverse Problems for Equations of Parabolic Type Math. Studies. Monograph Series. V. 10. Lviv, WNTL Publishers, 2003.
20. Kabanikhin S.I. Obratnye i nekorektnye zadachi [Inverse and Illl-posed problems]. Novosibirsk, Sibirskoe Nauchn. Izdat., 2009. 457 p.
21. Triebel H. Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library, 18. Amsterdam, North-Holland Publishing, 1978.
22. Amann H. Compact Embeddings of Vector-Valued Sobolev and Besov Spaces. Glasnik matematicki, 2000. vol. 35, no. 55, pp. 161-177.
23. Amann H. Operator-Valued Foutier Multipliers, Vector-Valued Besov Spaces and Applications. Mathematische Nachrichten, 1997, vol. 186, no. 1, pp. 5-56. DOI: 10.1002/mana.3211860102
24. Amann H. Linear and Quasilinear Parabolic Problems. V. I. Basel, Boston, Berlin, Birkhauser Verlag, 1995. DOI: 10.1007/978-3-0348-9221-6
25. Ladyzhenskaya O.A., Solonnikov V.A., Ural'tseva N.N. Lineynye i kvazilineynye uravneniya parabolicheskogo tipa [Linear and Quasilinear Equations of the Parabolic Type]. Moscow, Nauka, 1967.
26. Ladyzhenskaya O.A., Ural'tseva N.N. Lineynye i kvazilineynye uravneniya elliptieheskogo tipa [Linear and Quasilinear Elliptic Equation of the Elliptic Type]. Moscow, Nauka, 1973.
27. Alifanov O.M., Artyukhov E.A., Nenarokom A.V. Obratnye zadaehi slozhnogo teploobmena [Inverse Problems of Complicated Heat Exchange]. Moscow, Yanus-K, 2009.
28. Alifanov O.M. Inverse Heat Transfer Problems. Springer-Verlag, Berlin Heidelberg, 1994. DOI: 10.1007/978-3-642-76436-3
29. Ozisik M.N., Orlando H.A.B. Inverse Heat Transfer. New-York, Taylor & Francis, 2000.
Received September 9, 2014
УДК 517.95 DOI: 10.14529/mmpl40403
НЕКОТОРЫЕ ОБРАТНЫЕ ЗАДАЧИ ДЛЯ СИСТЕМ УРАВНЕНИЙ КОНВЕКЦИИ-ДИФФУЗИИ
С. Г. Пятков, Е.И. Сафонов
В настоящей статье мы рассмотрим вопросы корректности некоторых обратных задач для математических моделей процессов тепломассопереноса и конвекции-диффузии. Коэффициенты и правая часть системы восстанавливаются при выполнении некоторых дополнительных условий переопределения. Эти условия есть значения интегралов решения с весами по некоторой совокупностью областей. Доказаны теоремы существования и единственности и установлены оценки устойчивости. Полученные результаты являются локальными по времени. В качестве основных функциональных пространств используются пространства Соболева. Результаты служат основой при обосновании сходимости численных алгоритмов решения обратных задач с точечными условиями переопределения, которые возникают, в частности, в задачах тепломассопереноса об определении функции источников и параметров среды.
Ключевые слова: параболические системы; конвекция-диффузия; тепломассопере-нос; обратная задача; задача управления; краевая задача; корректность.
Литература
1. Алексеев, Г.В. Оптимизация в стационарных задачах тепломассопереноса и магнитной гидродинамики / Г.В. Алексеев. - М.: Науч. мир, 2010.
2. Belov, Ya.Ya. Inverse Problems for Parabolic Equations / Ya.Ya. Belov. - Utrecht: VSP, 2002.
3. Babeshko, О.М. On taking into account the types of sources and settling zones of pollutants / O.M. Babeshko, O.V. Evdokimova, S.M. Evdokimov // Doklady Mathematics. - 2000. - V. 61, № 2. - P. 283-285.
4. Калинина, E.A. Численное исследование обратной задачи восстановления плотности источника двумерного нестационарного уравнения конвекции-диффузии / Е.А. Калинина // Дальневосточный математический журнал. - 2004. - Т. 5, № 1.
- С. 89-99.
5. Обратная задача восстановления плотности источника для уравнения конвекции-диффузии / Ю.А. Криксин, С.Н. Плющев, Е.А. Самарская, В.Ф. Тишкин // Математическое моделирование. - 1995. - Т. 7, № 11. - С. 95-108.
6. Iskenderov, A.D. Inverse Problem for a Linear System of Parabolic Equations / A.D. Iskenderov, A.Ya. Akhundov // Doklady Mathematics. - 2009. - T. 79, № 1.
- P. 73-75.
7. Ismailov, M.I. Inverse Problem of Finding the Time-Dependent Coefficient of Heat Equation from Integral Overdetermination Condition Data / M.I. Ismailov, F. Kanca // Inverse Problems In Science and Engineering. - 2012. - V. 20, № 24. - P. 463-476.
8. Ivanchov, M.I. Inverse Problem of Simulataneous Determination of Two Coefficients in a Parabolic Equation / M.I. Ivanchov // Ukrainian Mathematical Journal. - 2000.
- V. 52, № 3. - P. 379-387.
9. Jing, Li. An Inverse Coefficient Problem with Nonlinear Parabolic Equation / Jing Li, Youjun Xu // Journal of Applied Mathematics and Computing. - 2010. - V. 34, № 1-2. - P. 195-206.
10. Kamynin, V.L. An Inverse Problem for a Higher-Order Parabolic Equation / V.L. Kamynin, E. Franchini // Mathematical Notes. - 1998. - V. 64, № 5. - P. 590-599.
11. Kerimov, N.B. An Inverse Coefficient Problem for the Heat Equation in the Case of Nonlocal Boundary Conditions / N.B. Kerimov, M.I. Ismailov // Journal of Mathematical Analysis and Applications. - 2012. -V. 396, issue 2. - P. 546-554.
12. Кожанов, А.И. Параболические уравнения с неизвестным коэффициентом, зависящим от времени / А.И. Кожанов // Журнал вычислительной математики и математической физики. - 2005. - Т. 45, № 12. - С. 2168-2184.
13. Vasin, I.A. On the Asymptotic Behavior of Solutions to Inverse Problems for Parabolic Equations / I.A. Vasin, V.L. Kamynin // Siberian Mathematical Journal. - 1997. -V. 38, № 4. - P. 647-662.
14. Prilepko, A.I. Methods for solving inverse problems in Mathematical Physics / A.I. Prilepko, D.G. Orlovsky, I.A. Vasin. - New-York: Marcel Dekker, Inc. 1999.
15. Tryanin, A.P. Determination of Heat-Transfer Coefficients at the Inlet into a Porous Body and Inside It by Solving the Inverse Problem / A.P. Tryanin // Journal of Engineering Physics. - 1987. - V. 52, № 3. - P. 346-351.
16. Dehghan, M. Method of Lines Solutions of the Parabolic Inverse Problem with an Overspecification at a Points / M. Dehghan, F. Shakeri // Numerical Algorithms. -2009. - V. 50, № 4. - P. 417-437.
17. Pyatkov, S.G. On Some Classes of Coefficient Inverse Problems for Parabolic Systems of Equations / S.G. Pyatkov, M.L. Samkov // Sib. Adv. in Math. - 2012. - V. 22, № 4. - P. 287-302.
18. Pyatkov, S.G. On Some Classes of Inverse Problems for Parabolic Equations / S.G. Pyatkov // Journal of Inverse Ill-Posed problems. - 2011. - V. 18, № 8. -P. 917-934.
19. Ivanchov M. Inverse Problems for Equations of Parabolic Type Math. Studies. Monograph Series. V. 10 / M. Ivanchov. - Lviv: WNTL Publishers, 2003.
20. Кабанихин, С.И. Обратные и некоректные задачи / С.И. Кабанихин. - Новосибирск: Сибирское научное издательство, 2009. - 457 с.
21. Triebel, Н. Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library, 18 / H. Triebel. - Amsterdam: North-Holland Publishing, 1978.
22. Amann, H. Compact Embeddings of Vector-Valued Sobolev and Besov Spaces / H. Amann // Glasnik matematicki. - 2000. - V. 35, № (55). - P. 161-177.
23. Amann, H. Operator-Valued Foutier Multipliers, Vector-Valued Besov Spaces and Applications / H. Amann // Mathematische Nachrichten. - 1997. - V. 186, № 1. - P. 5-56.
24. Amann, H. Linear and Quasilinear Parabolic Problems. V. I. / H. Amann. - Basel; Boston; Berlin: Birkhauser Verlag, 1995.
25. Ладыженская, O.A. Линейные и квазилинейные уравнения параболического типа / O.A. Ладыженская, В.А. Солонников, H.H. Уральцева. - М.: Наука, 1967.
26. Ладыженская, O.A. Линейные и квазилинейные уравнения эллиптического типа / O.A. Ладыженская, H.H. Уральцева. - М.: Наука, 1973.
27. Алифанов, О.М. Обратные задачи сложного теплообмена / О.М. Алифанов, Е.А. Артюхов, A.B. Ненароком. - Москва: Янус-К, 2009.
28. Alifanov, О.М. Inverse Heat Transfer Problems / О.М. Alifanov. - Springer-Verlag. Berlin Heidelberg. 1994.
29. Ozisik, M.N. Inverse Heat Transfer / Ozisik M.N., Orlando H.A.B. - New-York: Taylor & Francis, 2000.
Работа проводилась при финансовой поддержке Российского фонда фундаментальных исследований. Грант №12-01-00260а.
Сергей Григорьевич Пятков, доктор физико-математических наук, профессор, кафедра «Высшая математика:», Югорский государственный университет (г. Ханты-Мансийск, Российская Федерация), [email protected].
Егор Иванович Сафонов, аспирант, кафедра «Высшая математика», Югорский государственный университет (г. Ханты-Мансийск, Российская Федерация), [email protected].
Поступила в редакцию 9 сентября 2014 г.
