The Barenblatt-Zheltov-Kochina equation with boundary Neumann condition and multipoint initial-final value condition

Let A and F be Banach spaces; operator L e L(A; F ) (i.e. linear and continuous), and operator M e Cl ( A ; F ) (i.e. linear, closed and densely defined). Following [1], we introduce into consideration L -resolvent set p^L ( M ) = { p _e C : ( p L - M ) ^{- 1} _eL ( F ; A ) } and L-spectrum a^L ( M ) = C\ p L ( 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 M e Cl ( ker P ; ker Q ) n Cl (im P ; im Q ).

Introduce the following condition yL (M) = u yL (M), n e N, what is more   (M) * 0, there exists              (A)

j=o a closed contour Yj c C, bounding a domainDj ^ yL (M), such, that Dj n y0(M) = 0,Dk nDi = 0 Vj,k,l = 1,...,n,k *l•

Theorem 2. [2] Let operator M be ( L , p )- bounded, p e {0} и N, and condition (A) is fulfilled. Then there exist projectors P j e L ( A ) and Q j e L ( F ) , j = 1,..., n , having the form

Pj = -1-j(pL-M) 1 Ldp, Qj = -1-JL(pL-M) 1 dp, j = 1,...,n.

2 n i ^J                        2 n i^J

Y                              Y

Moreover, one more statement is true.

Corollary 1. Let conditions of theorems 1 and 2 are satisfied. Then PP j = P j P = P j , j = 1,..., n , P k P l = P l P k = O , k , l = 1,..., n , k * l ; QQ j = Q j Q = Q j , j = 1,..., n , Q k Q / = Q / Q k = O, k , l = 1,..., n , k * l .

Put nn

P o = P — Z P k , Q o = Q — Z Q k , k = 1                   k = 1

due to corollary 1 operators P _o e L ( A ) , Q ₀ e L ( F ) are projectors.

Thus, let condition (A) is fulfilled, fix T j e R ( т j <  т j ₊ i ), vectors U j e A for j = 0,..., n , and consider multipoint initial-final value condition [2] (see more [3])

P j ( u ( t j ) - U j ) = 0, j = 0,..., n ,                                     (1)

for linear Sobolev type equation

Lu ɺ = Mu + f,                                        (2)

where vector-function f e C ” (R; F ) will be defined below.

A vector-function u e C ” (R; A ) , satisfying equation (2), is called a solution of equation (2). Solution u = u ( t ), t e R, 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 -xA ut = vAu + f,                                    (3)

defined in cylinder Qx R ^m with boundary conditions

—u ( x , t ) = 0 ( x ) , ( x , t ) edQx R,                              (4)

dn and with multipoint initial-final value condition of the form (1). Here Qc Rm is a bounded domain with the boundary of the class C” , and n = n(x), x e dQ, is the unit normal external to domain Q. 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 fractured-porous medium [6]. Here χ is a real parameter characterizing the medium, ν is the piezoconductivity coefficient of the fractured rock, with x e R , v e R ₊ . 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 M e Cl (A; F) (i.e. linear, closed and densely defined). Suppose, in addition, operator M is (L ,o)-bounded (for terminology and results see [1]), then there exist degenerate analytical groups of resolving operators w = ^ W (M) e^ dM и F = ^ \ll, (M) e^dM, 2Л1 Y                      2nl Y defined on spaces A and F respectively, moreover U0 ^P, F0 ^Q are projectors. Here у is a contour, bounding a domain D, containing L -spectrum crL (M) of operator M ; RM (M)=(mL-M) 1 L is a right, and LL (M )= L (pL - M) 1 is a left L - resolvent of operator M . For degenerate analytical group the concepts of kernel ker U. = ker P = ker Ut, for all t e R and image imU" = im P = imUt for all t e R are correct. Denote by A0 = ker U', A1 = im U', and A0 = ker F. , F1 = im F., then A0 © A1 = A and F0 © F1 = F . Also denote by Lk (Mk ) the restriction of operator L (M ) on Fk (domM n Ak ), k = 0,1.

Математика

Theorem 3. [1] (Splitting theorem). Let operator M be ( L , p ) -bounded. Then

• (i)    operators L_k e L ( A ^k ; F ^k ) , k = 0 , 1 ;

• (ii)    operators M ₀ e Cl ( A ⁰; F ⁰) , M 1 e L ( A 1 ; F ¹) ;

• (iii)    there exist operators L - ¹ e L ( F 1 ; A ¹) and M ₀ ¹ e L ( F 0 ; A ⁰) .

Put H = M ₀ ¹ L ₀ e L ( A ⁰) , S = L ^{- 1} M 1 e L ( A 1 ) . It is true

Corollary 2. [1] Let operator M be (L,o) -bounded. Then for all pe C \ D

^                   ^

( ^L - M ) ^{- 1} = - j ^H^kM 0 ¹ ( I - Q ) + E ^{u k}S^{k - 1} L - ¹ Q .

• k = 0                           k = 1

An operator M is called ( L , p ) -bounded , p e { 0 } и N , if H^p Ф O , and H^{p + 1} = O . Let condition (A) is satisfied. Then takes place:

Theorem 4. [2] Let operator M be ( L , p )-bounded, p e { 0 } u N, and condition (A) is fulfilled. Then

• (i)    there exist degenerate analytical groups

U j = — f R^L L ( M ) e ^m d u . j = 1, n .

2 n i^{J p}

Y

• (ii)    UU j = U j U = U^ for all s, t e Ж , j = 1, n;

• (iii)    U^f_kU l = U s U k = O for all s, t e Ж , k, l = 1, n, k Ф l.

Put U 0 = U t - j U , t e Ж .

k = 1

Remark 1. Units P j = U 0 , j = 0, n , (constructed by virtue of condition (A)) of degenerate analytical groups { U j : t e Ж } , j = 0, n , are projectors by corollary (1). We call the operators P j , Q j , j = 0, n , relatively spectral projectors.

Consider subspaces A1 j = im Pj , F1 j = im Qj , j = 0, n . By construction nn

A ¹ = © A ¹ j and F ¹ = © F ¹ j . j = 0                   j = 0

Denote by L 1 j the restriction of operator L on A ^{1 j} , j = 0, n , and by M 1 j denote the restriction of operator M on dom M n A ^{1 j} , j = 0, n . Since, as it easy to show, that P j ^e dom M , if ф e dom M , then the domain dom M 1 j = dom M n A ^{1 j} is dense in A ^{1 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} u N , and condition (A) is fulfilled. Then

• (i)    operators L 1 j e L ( A 1 ^j ; F 1 ^j ) , M 1 j e L ( A 1 ^j ; F 1 ^j ) , j = 0, n;

• (ii)    there exist operators L - j e L ( F 1 ^j ; A ^{1 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 ” (Ж; F), uj e A, j = 0, n, there exists the unique solution ofproblem (1), (2), having the form n„.           t-T.     nc^

• u ( t ) = - j H q M 01 (I - Q ) f ⁽ q ) ( t ) + j U j ^Ju_{] +} £J U j ^- s L j Q j f ( s ) ds .           (5)

• q = 0                              j = 0             j = 0 t j

• 2.    Concrete interpretation

Reduce problem (3), (4) to equation (2). For this we set

A = | u e W m + 2: ^ = 0 on dQ | , F = W^m , m e N.

(     2    dn            J

All functional spaces are defined on the domain Q . Let us set the operators L = I - x A , M = vJ , moreover L, M e L( A ; F ) for all xe R\ {0}, v e R, and operator L is Fredholm (i.e. ind L = 0).

Denote by { X_k } the set of eigenvalues of homogeneous Neumann problem for the Laplace operator

J in the domain Q, numbered in the order of non-increasing with allowance for their multiplicity, and by the {фк} denote the set of orthonormalized (in the sense of the space L2 ) corresponding eigenvec- tors. Note, that the first eigenvalue of the homogeneous Neumann problem for the Laplace operator in domain Q is zero, and the corresponding eigenfunction is constant.

Lemma 1. [5] For all x , ve R \{0} operator M is ( L ,0) -bounded.

Note, that ker L =

'{0}, if x-1 «{-},' Vspan\»i: x-l = -}. J

By theorem 1 we construct a projector

' I, if x ^{- l} e { X l } ,

P=in -

E < ф Ф , l :( x'= / l )

^if x 1 ^e { ^- i } ,

where <• , •) is the scalar product in L ² . Projector Q has the same form, but it is defined on the space F .

The relative spectrum o^L ( M ) of operator M has the form

^L (M) = M = —-k-, k e N . (    1 - x-k      J

Choose such the parts ^ L ( 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

P j =     E    <Ф Фк , j = ⁰ , n .

k M _k eG L ( M )

Take т j e R ( t j <  T j + 1 ), U j e A , j = 0, n, f e C “ (R ; F ) and for problem (3), (4) the multipoint initial-final value condition is given

E      < U( T j , x ) - U^j ( x}^)_k > Фк = 0, j = 0, n .                          (6)

^k : M k e ^ L ^{( M} )

By lemma 1 and theorem 6 it follows

Theorem 7. Let condition (A) is fulfilled. For all xe R , v e R \{0}, f e C ” ( R ; F ) , U j e A , j = 0, n , equation (3) with conditions (4), (6) has the unique solution u e C ” (R; A ), which has the form

u ( t ) = ( Q - 1) f ( t ) + ]T     E     e “ ^{k (} t ^{-T j}) < u^ k Ф .' "E    E ' -" ⁽ t ^- s ) < f ( s)Фk >Ф k dS'

^j = ⁰ M k ^e^ j ^{( M} )                     ^j = ⁰ M k ^e^ j ^{( M) T} j

In conclusion, the authors consider it their pleasant duty to express their sincere gratitude to Professor G.А. Sviridyuk for interest in the work and productive discussions.