Inverse problems of recovering the boundary data with integral over determination conditions
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In the present article we examine an inverse problem of recovering unknown functions being part of the Dirichlet boundary condition together solving an initial boundary problem for a parabolic second order equation. Such problems on recovering the boundary data arise in various tasks of mathematical physics: control of heat exchange prosesses and design of thermal protection systems, diagnostics and identification of heat transfer in supersonic heterogeneous flows, identification and modeling of heat transfer in heat-shielding materials and coatings, modeling of properties and heat regimes of reusable heat protection of spacecrafts, study of composite materials, etc. As the overdetrermination conditions we take the integrals of a solution over the spatial domain with weights. The problem is reduced to an operator equation of the Volterra-type. The existence and uniqueness theorem for solutions to this inverse problem is established in Sobolev spaces. A solution is regular, i. e., all generalized derivatives occuring into the equation exists and are summable to some power. The proof relies on the fixed point theorem and bootstrap arguments. Stability estimates for solutions are also given. The solvability conditions are close to necessary conditions.
Inverse problem, parabolic equation, boundary and initial condition, sobolev space, existence and uniqueness theorem, solvability
Короткий адрес: https://sciup.org/147158975
IDR: 147158975 | DOI: 10.14529/mmph180204
Текст научной статьи Inverse problems of recovering the boundary data with integral over determination conditions
We consider the parabolic equation n∂ n
Lu = ut - ∑ aij (t, x)ux + ∑ai (t, x)ux + a0 (t, x)u = f , i,j=1∂xi j i=1 i where x ∈ G ⊂ Rn is a bounded domain with boundary Γ of the class C2 (see the definition, for instance, in [1, Sect. 1]), t∈ (0,T) . Put Q = (0,T)×G and S = (0,T)×Γ . The equation (1) is furnished with the following initial and boundary conditions:
u | S = g , u | t =0= u 0( x ).
Put ∂ ∂ Nu = ∑ i n , j =1 aij ( t , x ) uxj ( t , x ) ν i , where ν = ( ν 1, ν 2
,...,vn ) is the outward unit normal to 5 . The
inverse problem is to find a g = ∑ i m =1 qi ( t ) Φ i ( t , x ) , where overdetermiantion conditions
solution u to the problem (1)–(2) and a function g of the form the vector <7 = (qv q2, ^, qm) is unknown, with the use of the
G ( x , t )фк ( x ) dx = Y k ( t ), к = 1,2, ^ , m.
Inverse problems of recovering boundary regimes, in particular, the convective heat exchange problems are conventional (see, for instance, [2–11]). They arise in different problems of mathematical physics such as the problems of control of heat exchange prosesses and design of thermal protection systems, diagnostics and identification of heat transfer in supersonic heterogeneous flows, identification and modeling of heat transfer in heat-shielding materials and coverings, modeling of properties and heat regimes of reusable heat protection of spacecrafts, the study of composite materials, etc. Mathematical models describing these prosesses and the corresponding inverse problems in both one-dimensional and multidimensional cases are described, for example, in [2]. The essential attention here is paid to numerical methods of solving inverse problems in question and some uniqueness theorems together stability estimates for solutions. We refer also to the monograph [3] mainly devoted to numerical
Математика
methods of determining a solution, where in the one-dimensional case different inverse problems for parabolic equations and problems of recovering the boundary regimes as well are studied. The overdetrermination conditions are the values of a solution at some points lying inside the spatial domain. These problems are studied in different settings in dependence on the type of the ovedetermination conditions. It is often the case when these problems are ill-posed in the Hadamard sense. In particular, it is true in the case of the overdetermination conditions in the form of values of a solution at separate points or on some surfaces lying in the spatial domain (see [2]). At the present article we examine the problems with overdetermination conditions in the form of some integrals with weights of a solution over a spatial domain. Note that these conditions arise in applications and they are often used in literature. Inverse problems of recovering the right-hand side or coefficients of an equation with integral ovedetermination conditions are studied in the articles [12–18], the monographs [19, 20], and some other articles. In particular, the existence and uniqueness theorem of a generalized solution to the problem (1)–(3) (from the class u ∈ W 20,1( Q ) ) in the case of m = 1 and the Neumann boundary condition was obtained in [9] and a similar result for a heat-and-mass transfer system including the Navier–Stokes system and a parabolic equation for the concentration of an admixture was obtained in [10]. The article [11] is devoted to a regular solvability ( u ∈ W 21,2( Q ) ) in the case of m = 1 and the Robin boundary conditions. The case of the Dirichlet boundary condition happens to be more complicated than the case of the Neumann (Robin) boundary conditions and was not studied before. The present article is devoted to this case. Under some conditions on the data we prove well-poseness of this problem. The article in some sense is an extension of [21], where the Robin boundary conditions are treated. Some our auxiliary statements are taken from this article.
Preliminaries
Let E be a Banach space. Denote by Lp ( G ; E ) ( G is a domain in R n ) the space of strongly measurable functions defined on G with values in E and the finite norm || || u ( x ) || E || L ( g ) [1]. We also employ the Holder spaces (see the definition for instance, in [22]) C α , β ( Q ), C α , β ( S ) , Ck ( G ) and the Sobolev spaces Wps ( G ; E ) , Wps ( Q ; E ) (see [21, 23]). If E = R or E = R n then the latter space is denoted by Wps ( Q ) . Given an interval J = (0, T ) , put Wps , r ( Q ) = Wps ( J ; Lp ( G )) ∩ Lp ( J ; Wpr ( G ) . Respectively, Wps , r ( S ) = Wps ( J ; Lp ( Γ )) ∩ Lp ( J ; Wpr ( Γ )) . All spaces and the coeffciecients of the equation (1) are assumed to be real. Let ( u , v ) = ∫ Gu ( x ) v ( x ) dx , Q γ = (0, γ ) × G and S γ = (0, γ ) × Γ .
We endow the space Wps (0, τ ; E ) ( s ∈ (0,1) , E is a banach space, with the norm
|| q ( t ) || =( | q || p m + < q > p _ ) 1/ p , < q > p- =[ 1 q ( t 1)— q ( t 2 ) 1 E dtxdt2 . If E = R then we
W p s (0, τ ; E ) L p (0, τ ; E ) s , τ s , τ 0 0 | t 1 - t 2 | 1 + sp 1 2
obtain the conventional Sobolev space Wps (0, τ ) . For s ∈ (1/ p ,1] , we put
Wɶps(0,τ) = {q∈Wps(0,τ) : q(0) = 0} . This class is a Banach space with the norm || • II „ . We can
W p s (0, τ )
equivalence results from
define also the equivalent norm
| q ( t ) | ps = | q | p p (0.)+ < q > p r. The
Wp s (0, τ ) t s L p (0, τ ) s , τ
Lemma 1 of the subsection 3.2.6 [1]. Similarly, we can define the spaces W ɶ p s (0, τ ; Lp ( G )) , W ɶ p s ,2 s ( Q τ )
comprising , the functions v ( t , x ) in Wps (0, τ ; Lp ( G )) and Wps ,2 s ( Q τ ) , respectively, such that v (0, x ) = 0 .
The new norms || • |L, , | • |L,7, T are defined naturally with the use of the above norm
W p s (0, τ ; L p ( G )) W p s ,2 s ( Qτ )
W p s (0, τ ) .
Lemma 1. Let s ∈ (1/ p ,1) and p ∈ (1, ∞ ) . Then the following statements are valid.
-
1) Let q ( t ) ∈ Wps (0, τ ) ( τ ∈ (0, T ]) . Then q ∈ C ([0, τ ]) after a possible change on a set of zero measure . If q (0) = 0 and q ɶ is an extension by zero of q for t ≤ 0 then
II q II - c II q II , (4)
Wps ( - T + τ , τ ) 1 Wps (0, τ )
where the constant c 1 is independent of τ ∈ (0, T ] and q .
-
2) The product q ⋅ v of functions in Wps (0, τ ) ( τ ∈ (0, T ]) belongs to Wps (0, τ ) and if q ∈ Wps (0, τ ) and v ∈ Wps (0, τ ) then qv ∈ Wps (0, τ ) . Moreover, the following estimate holds:
-
1 qv Wp(0 T ) - c 2 1 q ^ p (0 T ) (< v > s T + 1 v 1 '(0 т ) ) (5)
where the constant c 2 is independent of q , v, and τ ∈ (0, T ] .
-
3) If a function v is strictly bounded from zero on [0, τ ] , i. e. δ 0 = inf t ∈ [0, τ ] | v ( t ) |> 0 then the ratio
q / v of functions in Wps (0, τ ) ( τ ∈ (0, T ]) belongs to Wps (0, τ ) again and
|| q / v || 5 - c 3 || q || s || v || s , (6)
Wps (0, τ ) 3 Wps (0, τ ) Wps (0, τ )
where the constant c 3 is independent of q but it depends on δ 0 and tends to ∞ as δ 0 → 0 .
-
4) Assume that q ( t ) ∈ W ɶ p s (0, τ ) ( τ ∈ (0, T ]) , v ( t ) ∈ Wps (0, T ) , and Φ ( t , x ) ∈ Wps ,2 s ( S ) . Then qv ∈ W ɶ p s (0, τ ) , q Φ ∈ W ɶ p s ,2 s ( S τ ) , and
|| qv || - c || q || || v || , (7)
W ɶ p s (0, τ ) W ɶ p s (0, τ ) W p s (0, T )
H q Ф 11-525 Г- c 4 || q || || Ф || 25 , (8)
W ɶ p s ,2 s ( Sτ ) 4 W ɶ p s (0, τ ) W p s ,2 s ( S )
where the constant c is independent of τ ∈ (0, T ].
The proof can be found in [21].
We describe now the conditions on the data used below. Fix a number s = 1 - 1/2 p and assume that p >3/2 .
The conditions on the coeficients aij∈C([0,T];W∞1(G))∩Lq(G;Wps(0,T)),∇xaij∈Lq(G;Wps(0,T)),(9)
where i , j = 1,2, … , n , 1/ p + 1/ q = 1 .
ai∈Lq(G;Wps(0,T)),(i=0,1,…,n),∇xai∈Lq(G;Wps(0,T))(i=1,2,…,n).(10)
Suppose also that there exists a constant δ 0 > 0 such that
∑n aijξiξj ≥δ0|ξ|2, ∀(t,x)∈Q,∀ξ∈Rn.(11)
-
i , j =1
The conditions on the data of the problem f∈Lp(Q), u0(x)∈Wp2-2/p(G),(12)
g∈Wps,2s(S),g(0,x)=u0|Γ,(13)
ϕk∈W∞2(G),Φk∈Wps,2s(S),ψk∈Wps+1(0,T),(f,ϕk)∈Wps(0,T),k=1,2,…,m.(14)
As a consequence of Theorem 9.1 in [22, Ch. 4] (see also Theorem 10.4 in [22, Ch. 7]) we have the following theorem.
Theorem 1. Assume that G is a bounded domain with boundary of the class C 2 and the conditions (9)–(13) hold . Then there exists a unique solution u to the problem (1)–(2) such that u ∈ Wp 1,2( Q ). A solution meets the estimate
^ u I WP ,2 ( Q ) - C (ll f ^ Lp (Q ) + ^ u 0 I Wp 2 - 2/ p ( G ) + ^ g I Wp ,2 s ( s ) ).
As a consequece of Theorem 1 we have that
Математика
Theorem 2. Assume that G be a bounded domain with boundary of the class C 2 and the conditions (9)–(13) hold, where f ≡ 0 and u 0 ≡ 0 . Let γ ∈ (0, T ] . Then there exists a unique solution u to the problem (1)–(2) such that u ∈ Wp 1,2 ( Q γ ). A solution meets the estimate
II u " 1 7 V
W 1 p ,2( Q γ )
< c " g " Wp,2 s ( s y )
, where the constant c is independent of γ ∈ (0, T ] and g .
| g ( t , x ), t e ( - T + y , Y )
Proof. Extend the function g by zero for t < 0 and put g = . Obviously,
[ g (2 y - t , x ), t e [ Y , T + y ]
g e Wp,2s (S). By Theorem 1, we can construct a solution to the problem (1)-(2), where u0 = 0, f = 0, and g = g such that u e W1,2(Q). Theorem 1 yields the estimate " u " 12 < c " g " 2 . Estimate
W p ( Q ) W p ( S )
the right-hand side. Lemma 1 implies that
" g " Wp ,2 s ( S ) < " g " Wp ,2 s (( - T + Y , T + Y ) хГ ) < c ( " g " Wp ,2 s (( - T +YY ) хГ )) + " g " Wp ,2 s (( y , T + Y ) хГ )) ) < c 1 " g " Wp ,2 s ( SY ) .
We employ the additivity of the Sobolev space with respect to the partition of a domain (see Remark 3 of Subsect. 4.4.1 in [1]) and the definition of the corresponding norm.
Basic results
In addition to the above conditions we require that
|det B ( t ) | ≥ δ 0 >0 ∀ t ∈ [0, T ],
where B (t) is the matrix with entries b„ = [ ^L(x) ф(t, x) d г, ^ \ =0 (k = 1,., m ); ij Γ ∂N j k Γ u0(x)ϕk(x)dx=ψk(0),ψkt(0)=(u0,L*0ϕk)- u0(x)∂ϕi(x)dΓ+(f,ϕk)|t=0, G Γ ∂N where k = 1,., m
and L *0 is
a formally adjoint to the operator
L 0 ,
L o ® = Z
∂ aω -
1, j =1 d x i 4 x j Z
n
n aω i=1 i xi
-
a 0 ω ;
-
(A) the functions Ф 1 (0, x ),..., Ф m (0, x ) are linearly independent on Г and u 0( x ) |г belongs to the span of these functions.
We can note that (16) is a necessary solvability condition of the inverse problem.
Theorem 3. Assume that G is a bounded domain with boundary of the class C 2 and the conditions (9)-(12), (14)-(16), and (A) hold . Then there exists a unique solution ( u , q ) ( q = ( q 1 , . , qm )) to the problem (1)-(3) such that u e W p ,2( Q ), <7 e W p (0, T ) . A solution satisfies the estimate
m
" u " Wp ,2( Q ) + " q " Wp (0, T ) < C ( I f " L p ( Q ) + " u 0 " W p-2/ p ( G ) + i =1 ( I ^ ‘ " Wp + s (0, T ) + " ( f , ^ ) " Wp (0, T ) )).
Proof. Let u e W p ,2( Q ) be a solution to the problem (1)-(3), where g = Z m= q li Ф i . The conditions (15) and (A) imply that there exists a unique collection of constants qi (0) such that u 0 1Г = Z m =i q i (0) Ф i (0, x ). Put Z m =i q i (0) Ф i ( t , x ) = g 0 ( t , x ) and denote by ve Wp>’2(Q ) a solution to the problem (see Theorem 1)
Lv = f , v | S = g 0( t , x ), v | t =0= u 0( x ). (17)
Let q e W ps (0, T ). In view of our conditions Ф j - e W p ,2 s ( S ). Lemma 1 yields qi ( t ) Φ i ( t , x ) ∈ Wps ,2 s ( S ) and thus g ∈ Wps ,2 s ( S ) . Make the change of variables u = v + ω . The function ω ∈ Wp 1,2( Q ) is a solution to the problem
L to =0, B to l s = g - g 0 = g; , to\ t =o =0. (18)
The condition (3) transforms into the form
\ G toV k ( x ) dx = / k — j G v ( t , x )Фк ( x ) dx = / k , k = 1,2, ^ , m.
By (16), / 7 k (0) = 0 and we have at least that / 7t ( t ) e W p (0,T ). Below we demonstrate that / 7 k ( t ) e Wp+5 (0, T ) and / к '(0) = 0 . Multiply the equation in (18) by ф к ( x ) and integrate the result over G . We obtain that ( ω t , ϕ k ) = ( L 0 ω , ϕ k ). Using (18), (19), and integrating by parts, we infer
/ к '( t ) = ( to , L ^ k ) - Z 7 i ( t ) | Ф i d^ i (7)dd Г , к = 1, ^ , m , < i ( t ) = < ( t ) — < (0). i =1 Γ ∂ N
The last inequality can be written in either of the forms
m
Z
or
7 _,_.!
Список литературы Inverse problems of recovering the boundary data with integral over determination conditions
- Трибель, Х. Теория интерполяции. Функциональные пространства. Дифференциальные операторы/Х. Трибель. -М.: Мир, 1980. -664 с.
- Алифанов, О.М. Обратные задачи в исследовании сложного теплообмена/О.М. Алифанов, Е.А. Артюхин, А.В. Ненарокомов. -Москва: Янус-К, 2009. -299 с.
- Ozisik, M.N. Inverse heat transfer/M.N. Ozisik, H.A.B. Orlando. -New-York: Taylor & Francis, 2000. -352 p.
- Костин, А.Б. О некоторых задачах восстановления граничного условия для параболического уравнения. I/А.Б. Костин, А.И. Прилепко//Дифференц. уравнения. -1996. -Т. 32, № 1. -С. 107-116.
- Борухов, В.Т. Применение неклассических краевых задач для восстановления граничных режимов процессов переноса/В.Т. Борухов, В.И. Корзюк//Вестник Белорусского университета. -1998. -Сер. 1, № 3. -C. 54-57.
- Трянин, А.П. Определение коэффициентов теплообмена на входе в пористое тело и внутри него из решения обратной задачи/А.П. Трянин//Инженерно-физический журнал. -1987. -Т. 52, № 3. -С. 469-475.
- Борухов, В.Т. Сведение одного класса обратных задач теплопроводности к прямым начально-краевым задачам/В.Т. Борухов, П.Н. Вабищевич, В.И. Корзюк//Инженерно-физический журнал. -2000. -Т. 73, № 4. -C. 742-747.
- Короткий, А.И. Реконструкция граничных режимов в обратной задаче тепловой конвекции несжимаемой жидкости/А.И. Короткий, Д.А. Ковтунов//Тр. ИММ ДВО АН. -2006. -Т. 12, № 2. -C. 88-97.
- Абылкаиров, У.У. Обратная задача интегрального наблюдения для общего параболического уравнения/У.У. Абылкаиров//Математический журнал. -2003. -Т. 3, № 4(10). -С. 5-12.
- Абылкаиров, У.У. Обратная задача для системы тепловой конвекции/У.У. Абылкаиров, А.А. Абиев, С.Е. Айтжанов//Тезисы докладов Молодежной международной научной школы-конференции «Теория и численные методы решения обратных и некорректных задач». -Новосибирск, ИМ СО РАН, 2009. -C. 10-11.
- Кожанов, А.И. Линейные обратные задачи для некоторых классов нелинейных нестационарных уравнений/А.И. Кожанов//Сибирские электронные математические известия. -2015. -Т. 12. -C. 264-275.
- Iskenderov, A.D. Inverse problem for a linear system of parabolic equations/A.D. Iskenderov, A.Ya. Akhundov//Doklady Mathematics. -2009. -Vol. 79, no. 1. -P. 73-75.
- 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. -Vol. 20, № 24. -P. 463-476.
- Li, J. An inverse coefficient problem with nonlinear parabolic equation/J. Li, Y. Xu//J. Appl. Math. Comput. -2010. -Vol. 34. -P. 195-206.
- Kerimov, N.B. An inverse coefficient problem for the heat equation in the case of nonlocal boundary conditions/N.B. Kerimov, M.I. Ismailov//J. of Mathematical Analysis and Applications. -2012. -Vol. 396. -Issue 2. -P. 546-554.
- Kozhanov, A.I. Parabolic equations with unknown time-dependent coefficients/A.I. Kozhanov//Comput. Math. and Math. Phys. -2017. -Vol. 57, № 6. -P. 956-966.
- Пятков С.Г. Об определении функции источника в математических моделях конвекции-диффузии/С.Г. Пятков, А.Е. Сафонов//Математические заметки СВФУ. -2014. -Т. 21, № 2. -C. 117-130.
- 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.
- 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. -744 p.
- Ivanchov, M. Inverse problems for equations of parabolic type/M. Ivancho//Mathematical Studies Monograph Series 10. -Lviv: VNTL Publishers, 2003. -238 p.
- Вержбицкий, М.А. O некоторых обратных задачах об определении граничных режимов/М.А. Вержбицкий, С.Г. Пятков//Матем. Заметки СВФУ. -2016. -Т. 23, № 2. -С. 3-16.
- Ladyženskaja, O.A. Linear and quasilinear equations of parabolic type/O.A. Ladyženskaja, V.A. Solonnikov, N.N. Uralceva//Translations of Mathematical Monographs. -Vol. 23. -American Mathematical Society, Providence, R.I., 1968. -648 p.
- Denk, R. Optimal Lp-Lq-estimates for parabolic boundary value problems with inhomogeneous data/R. Denk, M. Hieber, J. Prüss//Math. Z. -2007. -Vol. 257, no. 1. -P. 193-224.