The Barenblatt-Zheltov-Kochina model on the segment with Wentzell boundary conditions Текст научной статьи по специальности «Математика»

КРАТКИЕ СООБЩЕНИЯ

MSC 35G15 DOI: 10.14529/mmp190211

THE BARENBLATT-ZHELTOV-KOCHINA MODEL ON THE SEGMENT WITH WENTZELL BOUNDARY CONDITIONS

N.S. Goncharov, South Ural State University, Chelyabinsk, Russian Federation, [email protected]

In terms of the theory of relative p-bounded operators, we study the Barenblatt-Zheltov-Kochina model, which describes dynamics of pressure of a filtered fluid in a fractured-porous medium with general Wentzell boundary conditions. In particular, we consider spectrum of one-dimensional Laplace operator on the segment [0,1] with general Wentzell boundary conditions. We examine the relative spectrum in one-dimensional Barenblatt-Zheltov-Kochina equation, and construct the resolving group in the Cauchy-Wentzell problem with general Wentzell boundary conditions. In the paper, these problems are solved under the assumption that the initial space is a contraction of the space L2(0,1).

Keywords: Barenblatt-Zheltov-Kochina model; relatively p-bounded operator; phase space; C0-contraction semigroups; Wentzell boundary conditions.

Dedicated to the 60-th birthday of outstanding mathematician Jacek Banasiak.

Introduction

Let us consider the Cauchy-Wentzell problem

u(x, 0) = v0(x), x £ [0,1],

uxx( 0, t) + ao ux(0, t) + aiu(0, t) = 0, (1)

Uxx (1, t) + A)Ux(1, t) + ftu(1,t) = 0

for the Barenblatt-Zheltov-Kochina equation on the segment [0,1]

Aut(x,t) - utxx(x, t) = auxx(x,t) + f (x, t), (x,t) £ [0,1] x R+, (2)

which describes dynamics of pressure of a filtered fluid in a fractured-porous medium. Here a and A are the material parameters characterizing the environment, the parameter a £ R+, the function f = f (x,t) plays the role of external loading.

For the first time Wentzell boundary condition were considered in [1] in order to find diffusive processes for Markov processes homogeneous in time on the segment. Independently, these conditions were investigated in [2]. More general case were studied later in [3]. Namely, the domain belongs to n-dimensional Euclidean space which is a circle or a sphere, and semigroup is C0-contracting and invariant under rotations.

Further the results of [3] were developed and generalized in papers [4-7]. In particular, the classification of general Wentzell boundary conditions for a fourth-order differential operator in the one-dimensional case was established in [4], the role of Wentzell boundary conditions in linear and nonlinear analysis was shown in [5], Wentzell boundary conditions

for the Sturm-Liouville operator were studied in [6], the Laplace operator with general Wentzell boundary condition in Sobolev space was considered in [7]. These papers formed the basis of the new scientific direction, which endures a blossoming time in the present.

The purpose of this work is to research resolvability of problem (1) - (2) with Wentzell boundary conditions. The article contains two sections except introduction, conclusion, and references. The relative spectrum of the Laplace operator with Wentzell boundary condition is found in the first section. The main results on resolvability of the Cauchy-Wentzell problem in the Barenblatt-Zheltova-Kochina model are given in the second section.

1. Relative Spectrum of the Laplace Operator with Wentzell Boundary Condition

Let us consider the differential operator

Au(x) = u''(x), x e [0,1] (3)

with general Wentzell boundary conditions

Au(0) + ao u'(0) + aiu(0) = 0, (4)

Au(1) + fto u'(1) + ft u(1) = 0. (5)

By formulas (3) - (5) we define the linear operator A : dom A C F ^ F- Here F is a space

L2 [0,1],dx

Ф nds

(0,1)

with the norm

{0,1}/ 1

u||| = |u(x)|2dx + no|u(0)|2 + ni |u(1)|,

where dx is a Lebesque measure on the segment (0,1), ds is a point measure at the boundary, rjo = rji = j^, where a\ < 0 < ft, are positive weights. The

full construction of the space F is given in [8]. We consider also the linear manifold dom A = {u £ C2[0,1] : conditions (4), (5) are fulfilled} as the domain of the operator A.

Lemma 1. Let the operator A be defined by formulas (3)-(5). Then

(i) dom A = {u £ C2[0,1] : conditions (4), (5) are fulfilled} is a Banach space with regard to the norm ||u||C2[0 i];

(ii) dom A is densely embedded in F;

(iii) A £ L(dom A; F).

Let us give an idea of the proof. Statement (i) is obviously, since dom A forms a subspace closed in C2[0,1]. Statement (ii) obviuosly follows from the fact that the operator of embedding G : C2[0,1] ^ F is compact. Statement (iii) is obviously.

We consider the spectral problem for the operator A with general Wentzell boundary conditions. Prove the following theorem.

Theorem 1. Suppose that the operator A satisfies the conditions of Lemma 1. Then A has a real, discrete, finite multiplicity spectrum with the unique limit point at infinity.

Proof. It follows from [8] that the operator A on F is essentially self-adjoint. This means that the spectrum of the operator A is real. Let us define the spectrum of the operator A and find its resolvent. We have (AI - A)u = f (x), x £ [0,1], for f £ C2[0,1].

Consider the case of A < 0. Solve the differential equation with general Wentzell boundary conditions by classical methods, and obtain the resolvent of the following form:

x

u{x) = (AI-A)"1/ = Rxf = CTlcos(VZXx) + C^sm(V^Xx) + J -^=sm(y/^X(x-s))ds.

0

Write coefficients C\ = —^ and C2 = —¡^ for the resolvent, where

B B

В = (A + ai) ^A sin(V—A) + /Зол/-A cos(V-A) + Pi sin(V—A)^ — а0л/—Â^A cos(-\/—A) — -/Зол/-A sin(V—A) + Pi cos (л/— A)^ ,

Л = /(0) ^A sin (л/—A) + Po л/—A cos ( л/—A) + Pi sin(vQ)j -

1 __1 __1 __\

-/Щ sin(v^A(l - s))ds 00 J f(s) cos(v/rA(l - s))ds f Ш sin(v^A(l - s))ds ,

0 0 0 J

/ 1 __1

Ai = (A + ai) /(1) -f^à sin(v^A(l - s))ds -Pof f(s) со8(л/=Л(1 - s))ds-

0 0

f(s)

-fafjj&siniy/zA(1 - s))ds^-f( 0)^Acos(v^)-/3ov/ZAsin(v/rA) +/3i cos(v/zA)^.

The resolvent operator Ra is the sum of a two-dimensional operator (a linear combination of sine and cosine) and an integral operator of Hilbert-Schmidt type. A two-dimensional operator is finite-dimensional, and hence compact, since the coefficients Ci and C2 depend continuously on / in the metric of Hence, the operator R\ = (AI — A)~l is compact in F as the sum of finite-dimensional and compact operators. By Hilbert's theorem, Ra has a discrete, finite multiplicity spectrum with the unique limit point at zero.

Let us show that the operator A has a discrete, finite multiplicity spectrum with the unique limit point at infinity. Fix an arbitrary eigenvalue Ao of the operator Ra and express the eigenvalues of the operator A through the eigenvalues of the resolvent Ra. We obtain Ra/ = A0f, where / is the eigenvector of the resolvent. By acting with the operator (AI — A) on both parts of the equality and dividing by A0 (A0 = 0), we get the expression

Af = (\I - i-) /,

A0

which shows how the eigenvalues of the original and resolvent operators are related. Due to the behavior of the spectrum of the operator Ra, we proved that for A < 0 the operator A has a discrete, finite multiplicity spectrum with the unique limit point at infinity.

Similarly, consideration of the case A > 0 by the Sturm-Liouville method shows that the set of eigenvalues is finite or empty depending on the conditions on the coefficients in (4), (5).

Consider the case of A = 0. Find sufficient conditions for the set of eigenvalues of the operator A. Note that if the coefficients in (4), (5) satisfy the equality

«0^1 = a 1(^0 + A),

then A = 0 belongs to the set of eigenvalues of the operator A. The theorem is proved. □

The Barenblatt-Zheltov-Kochina equation

Aut(x, t) — utxx(x, t) = auxx(x, t) + f (x, t), (x, t) G [0,1] x R+ can be considered as a non-homogeneous Sobolev type equation Lut = Mu + f, where the operators L = A — A G L(domA; F), M = aA G L(domA; F), the function f = f (x,t) G C2([0,1] x R+;F). In order to solve the Cauchy-Wentzell problem (6), (7), we find the L-spectrum operator of M. Since the L-resolvent of the operator M takes the form

uA 1 1

^ + a

- A

(uL — M )-1 = (u(A — A) — aA)-1 = (u + a = 0} = (u + a)-1 with u + a = 0, then u belongs to relative spectrum aL(M) if

_ aa(A) fJ'~ X-a(AY

Therefore, according to Theorem 1, with u + a = 0, we have a discrete, finite L-spectrum aL(M) of the operator M with the limit point —a at infinity.

Consider the case of u + a = 0. With A = 0 we have aL(M) = (—a}. With A = 0 we have aL(M) = (0}, if a = 0, and aL(M) = (0}, if a = 0. We described the L-spectrum, of the operator M, getting the following corollary of Theorem 1.

Corollary 1. The L-spectrum of the operator M in the Barenblatt-Zheltov-Kochina equation with Wentzell boundary conditions is discrete, finite multiplicity, with the limit point —a at infinity.

2. The Cauchy-Wentzell Problem in the Barenblatt-Zheltov-Kochina Model

Let us consider the Cauchy-Wentzel problem in the previously introduced space F on the segment [0,1]

u(x, 0) = v0(x), x G [0,1],

uxx(0, t) + aoux(0, t) + ai u(0, t) = 0, (6)

uxx(1, t) + ftux(1, t) + ftu(1,t) = 0

for the Barenblatt-Zheltov-Kochina equation

Aut(x,t) — utxx(x,t) = auxx(x,t) + f (x, t), (x,t) G [0,1] x R+. (7)

By Corollary 1, the operator M is (L, a)-bounded, therefore, the following theorem holds.

Theorem 2. Suppose that the linear operator A satisfies the conditions of Lemma 1, and f G F is a fixed vector. Then

(i) if A G °"(A), then for any v0 G domA and f G F there exists the unique solution u G C2(R; domA) to problem (6)-(7), which has the form,

°° aA °° / aA \ ^ f ^

u(x, t) = e1^4 < Vo, <Pk >3- <Pk(x) + ( eJ^t - 1 )-—-^k(x);

k=1 k=1 ^ ' a k

(ii) if A G a(A), then for any f G F and v0 G P/ = < u G domA : a A < u, >f=

— < f, >f, Ak = A j there exists the unique solution u G C2(R; Pf) to problem (6), (7), which has the form,

1

^ ✓ / i •«k ft: •«k ft: ( ^ )

a A

± x ^ » ^ 3lXk t

u(x,t) =--- < f,Vk >3 ^fc(^) + eA_Afc < Vo(%),<-Pk >3 +

' a Ak

Proof. The proof of this theorem depends on the kernel of the operator L and consists in applying either the classical theorem for a non-homogeneous differential operator equation, or Sviridyuk's theorem. According to Theorem 1, the Laplace operator has a real, discrete, finite multiplicity spectrum having the limit point at —to, and (Ak : k G N} are eigenvalues of the Laplace operator, which are numbered in non-increasing order taking into account the multiplicity, and correspond to eigenfunctions (<^k : k G N}. Then, according to the completeness of the eigenfunctions, for v G F we have

R (A)v = - A) v

\ - < v, <~pk >3 <~Pk h ;

and, therefore,

Ri(M)v = (ßL - M)-' = №

fc=l

Termwise integration is admissible, since the series uniform convergences by the norm of the space domA. Therefore, substituting L-resolvent (8) of the operator M and applying the residue theorem, we obtain corresponding expressions (i), (ii). □

Conclusion

We constructed the resolution group in the Cauchy-Wentzell problem. To this end, we used the Sviridyuk's theory, and the space, the structure of which is specified in [8]. Further, we plan to continue the results of the paper by applying the Wentzell boundary conditions in directions related to [10, 11].

Acknowledgements. The work was supported by Act 211 Government of the Russian Federation, contract no. 02.A03.21.0011.

References

1. Wentzell A.D. Semigroups of Operators Corresponding to a Generalized Differential Operator of Second Order. Doklady Academii Nauk SSSR, 1956, vol. 111, pp. 269-272. (in Russian)

2. Feller W. Generalized Second Order Differential Operators and Their Lateral Conditions. Illinois Journal of Mathematics, 1957, vol. 1, no. 4, pp. 459-504. DOI: 10.1215/ijm/1255380673

3. Wentzell A.D. On Boundary Conditions for Multidimensional Diffusion Processes. Theory of Probability and its Applications, 1959, vol. 4, pp. 164-177. DOI: 10.1137/1104014

4. Favini A., Goldstein G.R., Goldstein J.A., Romanelli S. Classification of General Wentzell Boundary Conditions for Fourth Order Operators in One Space Dimension. Journal of Mathematical Analysis and Applications, 2007, vol. 333, no. 1, pp. 219-235. DOI: 10.1016/j.jmaa.2006.11.058

5. Coclite G.M., Favini A., Gal C.G., Goldstein G.R., Goldstein J.A. Obrecht E., Romanelli S. The Role of Wentzell Boundary Conditions in Linear and Nonlinear Analysis. Advances in Nonlinear Analysis: Theory, Methods and Applications, 2009, vol. 3, pp. 279-292.

6. Gal C.G. Sturm-Liouville Operator with General Boundary Conditions. Electronic Journal of Differential Equations, 2005, vol. 2005, no. 120, pp. 1-17.

7. Favini A., Goldstein G.R., Goldstein J.A. The Laplacian with Generalized Wentzell Boundary Conditions. Progress in Nonlinear Differential Equations and Their Applications, 2003, vol. 55, pp. 169-180. DOI: 10.1007/978-3-0348-8085-5_13

8. Favini A., Goldstein G.R., Goldstein J.A., Romanelli S. The Heat Equation with Generalized Wentzell Boundary Condition. Journal of Evolution Equations, 2002, vol. 2, pp. 1-19. DOI: 10.1007/s00028-002-8077-y

9. Sviridyuk G.A., Manakova N.A. The Barenblatt-Zheltov-Kochina Model with Additive White Noise in Quasi-Sobolev Spaces. Journal of Computational and Engineering Mathematics, 2016, vol. 3, no 1, pp. 61-67. DOI: 10.14529/jcem160107

10. Banasiak J. Mathematical Properties of Inelastic Scattering Models in Linear Kinetic Theory. Mathematical Models and Methods in Applied Sciences, 2000, vol. 10, no 2, pp. 163-186. DOI: 10.1142/S0218202500000112

11. Banasiak J., Lachowicz M., Moszynski M. Chaotic Behavior of Semigroups Related to the Process of Gene Amplification-Deamplification with Cell Proliferation. Mathematical Biosciences, 2007, vol. 206, no 2, pp. 200-215. DOI: 10.1016/j.mbs.2005.08.004

Received February 3, 2019

УДК 517.9 Б01: 10.14529/шшр190211

МОДЕЛЬ БАРЕНБЛАТТА - ЖЕЛТОВА - КОЧИНОЙ В ОБЛАСТИ С ГРАНИЧНЫМИ УСЛОВИЯМИ ВЕНТЦЕЛЯ

Н. С. Гончаров, Южно-Уральский государственный университет, г. Челябинск, Российская Федерация

В терминах теории относительно р-ограниченных операторов исследуется модель Баренблатта - Желтова - Кочиной, описывающая динамику давления фильтрующейся жидкости в трещинновато-пористой среде с общими граничными условиями Вент-целя. В частности, рассматривается спектр одномерного оператора Лапласа на отрезке [0,1] с общими граничными условиями Вентцеля; ставится вопрос об относительном спектре в одномерном уравнении Баренблатта - Желтова - Кочиной и построении разрешающей группы в задаче Коши - Вентцеля с общими граничными условиями Вентцеля. В работе решены указанные задачи в предположении, что исходное пространство, в котором действует оператор Лапласа на отрезке, есть сужение пространства Ь2(0,1).

Ключевые слова: модель Баренблатта - Желтова - Кочиной; относительно р-ограниченный оператор; фазовое пространство; С0-сжимающие полугруппы; краевые условия Вентцеля.

Литература

1. Вентцель, А.Д. Полугруппы операторов, соответствующие обобщенному дифференциальному оператору второго порядка / А.Д. Вентцель // ДАН СССР. - 1956. - Т. 111. -С. 269-272.

2. Feller, W. Generalized Second Order Differential Operators and Their Lateral Conditions / W. Feller // Illinois Journal of Mathematics. - 1957. - V. 1, № 4. - P. 459-504.

3. Wentzell, A.D. On Boundary Conditions for Multidimensional Diffusion Processes / A.D. Wentzell // Theory of Probability and Its Applications. - 1959. - V. 4. - P. 164-177.

4. Favini, A. Classification of General Wentzell Boundary Conditions for Fourth Order Operators in One Space Dimension / A. Favini, G.R. Goldstein, J.A. Goldstein, S. Romanelli // Journal of Mathematical Analysis and Applications. - 2007. - V. 333, № 1. - P. 219-235.

5. Coclite G.M., Favini A., Gal C.G., Goldstein G.R., Goldstein J.A. Obrecht E., Romanelli S. The Role of Wentzell Boundary Conditions in Linear and Nonlinear Analysis. Advances in Nonlinear Analysis: Theory, Methods and Applications, 2009, vol. 3, pp. 279-292.

6. Gal C.G. Sturm-Liouville Operator with General Boundary Conditions / Ciprian G. Gal // Electronic Journal of Differential Equations. - 2005. - V. 2005, № 120. - P. 1-17.

7. Favini, A. The Laplacian with Generalized Wentzell Boundary Conditions / A. Favini, G.R. Goldstein, J.A. Goldstein, Enrico Obrecht, S. Romanelli // Progress in Nonlinear Differential Equations and Their Applications. - 2003. - V. 55. - P. 169-180.

8. Favini, A. The Heat Equation with Generalized Wentzell Boundary Condition / A. Favini, G.R. Goldstein, J.A. Goldstein, S. Romanelli // Journal of Evolution Equations. - 2002. -V. 2. - P. 1-19.

9. Sviridyuk, G.A. The Barenblatt-Zheltov-Kochina Model with Additive White Noise in Quasi-Sobolev Spaces / G.A. Sviridyuk, N.A. Manakova // Journal of Computational and Engineering Mathematics. - 2016. - V. 3, № 1. - P. 61-67.

10. Banasiak, J. Mathematical Properties of Inelastic Scattering Models in Linear Kinetic Theory / J. Banasiak // Mathematical Models and Methods in Applied Sciences. - 2000. - V. 10, № 2. - P. 163-186.

11. Banasiak J., Lachowicz M., Moszynski M. Chaotic Behavior of Semigroups Related to the Process of Gene Amplification-Deamplification with Cell Proliferation / J. Banasiak // Mathematical Biosciences. - 2007. - V. 206, № 2. - P. 200-205.

Никита Сергеевич Гончаров, магистрант, Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), [email protected].

Поступила в редакцию 3 февраля 2019 г.