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DOI: 10.14529/mmph190202
THE BARENBLATT-ZHELTOV-KOCHINA EQUATION WITH BOUNDARY NEUMANN CONDITION AND MULTIPOINT INITIAL-FINAL VALUE CONDITION
L.A. Kovaleva, E.A. Soldatova, S.A. Zagrebina
South Ural State University, Chelyabinsk, Russian Federation
E-mail: [email protected]
The article is devoted to the study of the unique solvability of the Baren-blatt-Zheltov-Kochina equation, equipped with the Neumann boundary condition and a multipoint initial-final value condition. This equation is degenerate or, in other words, it belongs to the Sobolev type equations. To study this equation, the authors used the methods of the theory of degenerate operator semigroups, created by Prof. G.A. Sviridyuk, and further developed by him and his students. We would also like to note that the equation under study is supplied with a multipoint initial-final value condition, which is not just a generalization of the Cauchy problem for the Sobolev type equations. This condition makes it possible to avoid checking the consistency of the initial data when finding a solution.
Keywords: Barenblatt-Zheltov-Kochina equation; Neumann condition; multipoint initial-final value condition; unique solvability.
Introduction
Let A and F be Banach spaces; operator L eL(A;F) (i.e. linear and continuous), and operator
Me Cl(A;F) (i.e. linear, closed and densely defined). Following [1], we introduce into consideration
L-resolvent set pL (M )={^e C : [mL -M )-1 eL(F; A)} and L-spectrum sL (M) = C \ pL (M) of
operator M . The following statements are true.
Theorem 1. [1] Let operator M be (L, p) -bounded, p e {0} u N . Then there exist such projectors
P: A ® A and Q:F ® F, that operators L e L (ker P; ker Q )n L (im P; im Q) and
MeCl(kerP; kerQ)nCl (imP; imQ).
Introduce the following condition
n
sL (M ) = u sL (M), n e N, what is more sL. (M) * 0, there exists (A)
j=0 j j
a closed contour gj c C, bounding a domain Dj 3 s^ (M), such, that Dj nsL(M) = 0, Dk nD{ =0 "j,k,l = 1,...,n, k *l.
Theorem 2. [2] Let operator M be (L, p) -bounded, p e {0} u N, and condition (A) is fulfilled.
Then there exist projectors Pj e L(A) and Qj e L (F), j = 1,..., n, having the form
Pj = 2-j(mL-m)-1 Ldm, Qj = 2-jL(mL-M)~1dU, j=1,...,n.
g g
Moreover, one more statement is true.
Corollary 1. Let conditions of theorems 1 and 2 are satisfied. Then PPj = Pj P=Pj, j = 1,...,n,
PkP = PtPk = O , k,l = 1,..., n, k * l; QQ, = Q.Q=Q,, j = 1,..., n, QkQl= QQ = O, k,l = 1,..., n, k * l.
Put
P = P - Z Pk, Q0 = Q - z Qk
k=1 k=1 due to corollary 1 operators P0 e L(A), Q0 e L(F) are projectors.
Kovaleva L.A., Soldatova E.A., The Barenblatt-Zheltov-Kochina Equation with Boundary Neumann
Zagrebina S.A. Condition and Multipoint Initial-Final Value Condition
Thus, let condition (A) is fulfilled, fix tj e M (tj < Tj+1 ), vectors Uj e A for j = 0,...,n , and consider multipoint initial-final value condition [2] (see more [3])
Pj (u(tj)-Uj ) = 0 j = n , (1)
for linear Sobolev type equation
LU = Mu + f (2)
where vector-function f e C¥ (M; F) will be defined below.
A vector-function u e C¥ (M; A), satisfying equation (2), is called a solution of equation (2). Solution u = u(t), t e M, of equation (2), satisfying conditions (1) is called the solution of multipoint initial-final value problem (1), (2).
In this paper we present the results of the Sobolev type equations theory with (L, p) -bounded operator M [1] and the unique solvability of problem (1), (2) [2]. Then the abstract results will be applied to the study of the solvability of Barenblatt-Zheltov-Kochina equation
ut -%Aut = nDu + f, (3)
defined in cylinder WxMm with boundary conditions
yu (x,t )= 0 (x), (x, t )edWx M, (4)
and with multipoint initial-final value condition of the form (1). Here WcRm is a bounded domain with the boundary of the class C¥ , and n = n(x), x e dW, is the unit normal external to domain W. Thus, the subject of the paper is divided into two parts. In the first part the information on the solvability of problem (1), (2) is given, and in the second part problem (1), (3), (4) is considered.
Note, that the Neumann conditions are a special case of the "flow balance" condition for Sobolev type equations considered on a connected oriented graph, that is in the one-dimensional case. This theory is currently being actively developed, the first studies were conducted in [4]. Sobolev type equations with Cauchy-Neumann conditions in a bounded domain were studied in [5], but studies for the case of replacing the Cauchy condition by a multipoint initial-final value condition for such a problem are considered here for the first time.
We also note, that equation (3) models the dynamics of the pressure of a fluid, filtered in a frac-tured-porous medium [6]. Here x is a real parameter characterizing the medium, v is the piezo-conductivity coefficient of the fractured rock, with x e M, v e M+. function f = f (x) plays the role of
an external load. In addition, equation (3) describes the flow of second-order liquids [7], the heat conduction process with "two temperatures" [8], the moisture transfer process in the soil [9].
1. Abstract problem
Let A and F be Banach spaces, operators L e L(A;F) (i.e. linear and continuous) and Me Cl(A;F) (i.e. linear, closed and densely defined). Suppose, in addition, operator M is (L,s)-bounded (for terminology and results see [1]), then there exist degenerate analytical groups of resolving operators
u' =—f Rlu(M)emtdm u F' =—f lL(m)emtdm, 2pi h L ' 2pi h L '
defined on spaces A and F respectively, moreover U0 ° P , F0 ° Q are projectors. Here y is a contour, bounding a domain D, containing L -spectrum <jL (M) of operator M ; RLL (M)=(mL -M) L
is a right, and L^ (M )= L (mL - M) 1 is a left L -resolvent of operator M . For degenerate analytical
m
group the concepts of kernel kerU' = ker P = ker U, for all t e M and image imU' = im P = im U for all t e M are correct. Denote by A0 = ker U', A1 = imU■, and A0 = ker F', F1 = im F', then A0 © A1 = A and F° © F1 = F . Also denote by Lk (Mk ) the restriction of operator L (M) on Fk (domM n Ak ), k = 0,1.
Вестник ЮУрГУ. Серия «Математика. Механика. Физика» 15
2019, том 11, № 2, С. 14-19
Theorem 3. [1] (Splitting theorem). Let operator M be (L,p) -bounded. Then
(i) operators Lk e L( Ak; Fk), k = 0, 1;
(ii) operators M0 e Cl(A0;F0), M1 e ^A1;F1);
(iii) there exist operators L-1 e ^(F1; A1) and M- e L(F0; A0). Put H = M0-1L0 e L( A0), S = L-lM1 e L( A1). It is true
Corollary 2. [1] Let operator M be (L, a) -bounded. Then for all me C \ D
(JUL - M )-1 = - X mkHkM -1 (I - Q) + Y,M~kSk
k=0 k=1
An operator M is called (L, p) -bounded, p e {0} u N, if Hp * O, and Hp+1 = O . Let condition (A) is satisfied. Then takes place:
Theorem 4. [2] Let operator M be (L,p)-bounded, pe {0} u N , and condition (A) is fulfilled. Then
(i) there exist degenerate analytical groups
i . _
Uj = — JRLm (M)emtdm, j = 1,n. g
(ii) UtUs} = UjU* = Us+t for all s, t e M, j = ;
(iii) UtkU\=UslUtk = O for all s, t e M, k, l = \n, k * l.
n
Put U0 = U - Yuk , t e M .
k=1
Remark 1. Units Pj ° U^ , j = 0,n, (constructed by virtue of condition (A)) of degenerate analytical groups {uj : t e m} , j = 0,n, are projectors by corollary (1). We call the operators Pj , Qj , j = 0,n, relatively spectral projectors.
Consider subspaces A1j = im Pj , Flj = im Qj , j = 0,n . By construction
A1 = © A1 j and F1 = © F1j.
j=0 j=0
Denote by L1j the restriction of operator L on A1j , j = 0,n, and by M1 j denote the restriction of operator M on domM n A1 J, j = 0,n . Since, as it easy to show, that Pje domM , if (pe domM, then the domain domM1 j = domM n A1 J is dense in A1 J, j = 0, n .
Theorem 5. [2] (Generalized spectral theorem). Let operators L e L(A; F) and M e Cl (A; F), operator M is (L, p) -bounded, p e {0} uN, and condition (A) is fulfilled. Then
(i) operators L1} e £(A1j;F1 j), M1} e L(A11;F1 j), j = 0n ;
(ii) there exist operators L-j e £(F1j; A1 J), j = 0, n.
Theorem 6. [2] Let operator M is (L, p) -bounded, p e{0}u N, moreover condition (A) is fulfilled. Then for all f e C¥ (M; F), uj e A, j = 0, n, there exists the unique solution ofproblem (1), (2), having the form
p
-1
(t) = -X HqM-1 (I - Q) f(q)(t) + X U~ jUj + X j U'r'L-iQjf (s)ds. (5)
q=0 j =0 j =0t
A = (u eW2m+2: ™ = 0 on dW\, F = Wm, m e N.
Kovaleva L.A., Soldatova E.A., The Barenblatt-Zheltov-Kochina Equation with Boundary Neumann
Zagrebina S.A. Condition and Multipoint Initial-Final Value Condition
2. Concrete interpretation
Reduce problem (3), (4) to equation (2). For this we set
du dn
All functional spaces are defined on the domain W. Let us set the operators L = I - , M = vA, moreover L, Me L(A;F) for all %e M \ {0}, ve M, and operator L is Fredholm (i.e. indL = 0).
Denote by [Xk} the set of eigenvalues of homogeneous Neumann problem for the Laplace operator A in the domain W, numbered in the order of non-increasing with allowance for their multiplicity, and by the } denote the set of orthonormalized (in the sense of the space L2 ) corresponding eigenvectors. Note, that the first eigenvalue of the homogeneous Neumann problem for the Laplace operator in domain W is zero, and the corresponding eigenfunction is constant.
Lemma 1. [5] For all %, ne M \{0} operator M is (L,0) -bounded. Note, that
>}, if c"1 ¿R}, ^ span {j : c"1 =R }.
ker L =
By theorem 1 we construct a projector
P =
I, if x~l },
I - X j j, if C"1 e{1}, «z-^l)
where (•,•> is the scalar product in L2 . Projector Q has the same form, but it is defined on the space F . The relative spectrum S (M) of operator M has the form
( vl ^
S (M) = m =vr' k e N _ I 1 -z-4 ,
Choose such the parts (M), j = 0, n, of the relative spectrum of operator M, that condition (A) is satisfied (it is clear that this can done in more then one way). Build the projectors
Pj = X (•jk >j, j=^ n.
kmesL (M)
Take Tj e M (tj < Tj+1), uj e A, j = On, f e C¥(M ;F) and for problem (3), (4) the multipoint initial-final value condition is given
X (u (tj, x) - uJ (x),j >jk = 0, j = 0, n. (6)
k:mk erf (M) By lemma 1 and theorem 6 it follows
Theorem 7. Let condition (A) is fulfilled. For all ze M, ve M \{0}, f e CM;F), u} e A, j = 0, n, equation (3) with conditions (4), (6) has the unique solution u e C¥ (M; A), which has the form
u(0 = (Q-i)f(t) + XX X eMk(t-Tj)(uJ,jk>jk + X X \emk{t-s)(f(s)jk>jkds.
j=°mkeS (M) j=°mkeoL (M)tj
In conclusion, the authors consider it their pleasant duty to express their sincere gratitude to Professor G.A. Sviridyuk for interest in the work and productive discussions.
References
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Вестник ЮУрГУ. Серия «Математика. Механика. Физика» 17
2019, том 11, № 2, С. 14-19
2. Zagrebina S.A. The Multipoint Initial-finish Problem for Hoff Linear Model. Bulletin of the South Ural State University. Series Mathematical Modelling, Programming & Computer Software, 2012, Vol. 5 (264), Issue 11, pp. 4-12. (in Russ.).
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Received April 1, 2019
Bulletin of the South Ural State University Series "Mathematics. Mechanics. Physics" _2019, vol. 11, no. 2, pp. 14-19
УДК 517.9 DOI: 10.14529/mmph190202
УРАВНЕНИЕ БАРЕ Н БЛАТТА-ЖЕЛТО В А-КО ЧИНОЙ С ГРАНИЧНЫМ УСЛОВИЕМ НЕЙМАНА И МНОГОТОЧЕЧНЫМ НАЧАЛЬНО-КОНЕЧНЫМ УСЛОВИЕМ
Л.А. Ковалева, Е.А. Солдатова, С.А. Загребина
Южно-Уральский государственный университет, г. Челябинск, Российская Федерация E-mail: [email protected]
Посвящена изучению однозначной разрешимости уравнения Баренблатта-Желтова-Кочиной, снабженного краевым условием Неймана и многоточечным начально-конечным условием. Отметим, что уравнение Баренблатта-Желтова-Кочиной моделирует динамику давления жидкости, фильтрующейся в трещинновато-пористой среде. Кроме того, оно описывает течение жидкостей второго порядка, процесс теплопроводности c «двумя температурами», процесс вла-гопереноса в почве. Данное уравнение является вырожденным или, другими словами, оно принадлежит к уравнениям соболевского типа. Для исследования изучаемого уравнения авторы воспользовались методами теории вырожденных полугрупп операторов, разработанной проф. Г.А. Свиридюком, и развитой его учениками. Отметим также, что исследуемое уравнение снабжено многоточечным начально-конечным условием, которое является не просто обобщением задачи Коши для уравнений соболевского типа. Указанное условие дает возможность избегать проверки согласования начальных данных при нахождении решения.
Ключевые слова: уравнение Баренблатта-Желтова-Кочиной; условие Неймана; многоточечное начально-конечное условие; однозначная разрешимость.
Литература
1. Свиридюк, Г. А. К общей теории полугрупп операторов / Г. А. Свиридюк // Успехи математических наук. - 1994. - Т. 49, № 4. - С. 47-74.
Kovaleva L.A., Soldatova E.A., The Barenblatt-Zheltov-Kochina Equation with Boundary Neumann
Zagrebina S.A. Condition and Multipoint Initial-Final Value Condition
2. Загребина, С.А. Многоточечная начально-конечная задача для линейной модели Хоффа / С.А. Загребина // Вестник ЮУрГУ. Серия «Математическое моделирование и программирование». - 2012. - № 5 (264). - Вып. 11. - С. 4-12.
3. Загребина, С.А. Начально-конечные задачи для неклассических моделей математической физики / С.А. Загребина // Вестник ЮУрГУ. Серия «Математическое моделирование и программирование». - 2013. - Т. 6, № 2. - С. 5-24.
4. Свиридюк, Г.А. Фазовое пространство одной неклассической модели / Г.А. Свиридюк, В.В. Шеметова // Изв. вузов. Математика. - 2005. - Т. 11. - С. 47-52.
5. Казак, В.О. Исследование фазовых пространств одного класса полулинейных уравнений соболевского типа: дис. ... канд. физ.-мат. наук / В.О. Казак. - Челябинск, 2005. - 99 с.
6. Баренблатт, Г.И. Об основных представлениях теории фильтрации в трещиноватых средах / Г.И. Баренблатт, Ю.П. Желтов, И.Н. Кочина // Прикладная математика и механика. - 1960. -Т. 24, № 5. - С. 852-864.
7. Ting, T.W. Certain non-steady flows of second-order fluids / T.W. Ting // Archive for Rational Mechanics and Analysis. - 1963. - Vol. 14. - Issue 1. - P. 1-26.
8. Chen, P.J. On a Theory of Heat Conduction Involving Two Temperatures / P.J. Chen, M.E. Gurtin // Journal of Applied Mathematics and Physics (ZAMP). - 1968. - Vol. 19. - Issue 4. - P. 614-627.
9. Hallaire, M. On a theory of moisture-transfer / M. Hallaire // Inst. Rech. Agronom. - 1964. -№ 3. - P. 60-72.
Поступила в редакцию 1 апреля 2019 г.
Вестник ЮУрГУ. Серия «Математика. Механика. Физика» 2019, том 11, № 2, С. 14-19
