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1 Math-Net.Ru A Russian mathematica porta D. Zaora, On properties of root eements in the probem on sma motions of viscous reaxing fuid, Zh. Mat. Fiz. Ana. Geom., 217, Voume 13, Number 4, DOI: Use of the a-russian mathematica porta Math-Net.Ru impies that you have read and agreed to these terms of use Downoad detais: IP: January 18, 219, 3:12:21

2 Journa of Mathematica Physics, Anaysis, Geometry 217, vo. 13, No. 4, pp doi: On Properties of Root Eements in the Probem on Sma Motions of Viscous Reaxing Fuid D. Zaora Voronezh State University 1 University Sq., Voronezh, 3946, Russia E-mai: dmitry.zr@gmai.com Received October 2, 215, revised May 11, 216 In the present wor, the properties of root eements of the probem on sma motions of a viscous reaxing fuid competey fiing a bounded domain are studied. A mutipe p-basis property of specia system of eements is proven for the case where the system is in weightessness. The soution of the evoution probem is expanded with respect to the corresponding system. Key words: viscous fuid, compressibe fuid, basis. Mathematica Subject Cassification 21: 45K5, 58C4, 76R Introduction This paper is connected with [1 and deas with the probem on sma motions of a viscous reaxing fuid competey fiing a bounded domain. We study root eements of the corresponding spectra probem. A mutipe p-basis property of specia system of eements is proven for the case where the system is in weightessness. The soution of an evoution probem is expanded with respect to the corresponding system. This wor was supported by the grant of the Russian Foundation for Basic Research project no , Voronezh State University. c D. Zaora, 217

3 On Properties of Root Eements... Sma Motions of Viscous Reaxing Fuid 2. Statement of the Probem 2.1. Statement of the boundary-vaue probem. Consider a container competey fied by a viscous inhomogeneous fuid. The fuid is said to fi a bounded region Ω R 3. We introduce a system of coordinates Ox 1 x 2 x 3 which is toughy connected with the container so that the origin of coordinates is in the region Ω. In what foows, we suppose the hydrodynamica system to be in weightessness. The probem on sma motions of a rotating viscous reaxing fuid is described by the foowing initia-boundary vaue probem see [1, 2: ut, x t ρt, x t ρ 1 µ ut, x + η µ div ut, x t + a ρ 1/2 ρt, x e t s a ρ 1/2 ρs, x ds = ft, x, + a ρ 1/2 div ut, x = in Ω, ut, x = on S := Ω, u, x = u x, ρ, x = ρ x, 1 where ρ = const is the given stationary density of the fuid, a is the given sound veocity, ft, x is a wea fied of externa forces, ut, x is the fied of the veocities in the fuid, a 1 ρ 1/2 ρt, x is the dynamic density of the fuid, µ and η are the dynamic and the second viscosity of the fuid. The numbers b 1 are used as the times of reaxation in the system < < +1 = 1, m 1, and > are some structura constants Operator formuation of the probem. To pass to the operator formuation of the probem, we introduce basic spaces and a number of operators see [2. Let us introduce a vector Hibert space L 2 Ω, ρ with the scaar product and the norm u, v L2 Ω,ρ := ρ ux vx dω, u 2 L2 Ω,ρ = ρ ux 2 dω. Ω We introduce a scaar Hibert space L 2 Ω of the square summabe functions in the region Ω, and aso its subspace L 2,Ω := { f L 2 Ω f, 1 L2 Ω = }. Lemma 1. The foowing statements hod. 1. Let the boundary S of the region Ω beong to the cass C 2. Consider a boundary-vaue probem µ u + η µ div u = w in Ω, u = on S. ρ 1 Ω Journa of Mathematica Physics, Anaysis, Geometry, 217, Vo. 13, No. 4 43

4 D. Zaora For every fied w L 2 Ω, ρ, the generaized soution of the probem given by the formua u = A 1 w exists and it is unique. The operator A is sefadjoint and positive definite in L 2 Ω, ρ. The { operator A 1 beongs to the cass S p p > 3/2. In addition, DA 1/2 = u W } 2 1Ω u = on S. 2. Let us define an operator B u := a ρ 1/2 div u, DB := DA 1/2. The adjoint operator B ρ = a ρ 1/2 ρ, DB = W2 1Ω L 2,Ω, B BA 1 L L 2 Ω, ρ. 3. The operator Q := BA 1/2 is bounded: Q L L 2 Ω, ρ, L 2,Ω. The operator Q + := A 1/2 B admits extension to the bounded operator Q : Q + = Q. Using the operators introduced above, we can write probem 1 as a system of two equations with initia-vaue conditions in a Hibert space H := L 2 Ω, ρ L 2,Ω : d u dt + 2ω is u + A u B ρ + t dρ dt + B u =, u; ρτ = u ; ρ τ. The upper index τ denotes the transposition. e t s B ρ ρs ds = ft, Definition 1. A strong soution to probem 2 is said to be a strong soution to initia-vaue probem 1. A function ζt := ut; ρt τ is a strong soution to probem 2 if ζt DA DB for a t R +, A ut; B ρt τ CR +, H, ut; ρt τ C 1 R +, H, and ζt satisfies 2 for a t R + := [, Reduction to the first-order differentia equation. Let us suppose that probem 2 has a strong soution ut, ρt and ρ DB. From Lemma 1, it foows that ut and ρt satisfy the system d u dt + A1/2 {A 1/2 u Q [ t Q 1 e t s ρ ρ dρs ds dρ dt + QA1/2 u =, u; ρ τ = u ; ρ τ. ds ρ } = ft + e t ρ B ρ, In what foows, we suppose the physica parameters satisfy the condition ρ ϕ := 1 > Journa of Mathematica Physics, Anaysis, Geometry, 217, Vo. 13, No. 4

5 On Properties of Root Eements... Sma Motions of Viscous Reaxing Fuid This condition supposes that the times of reaxation b 1 and the structura constants = 1, m are sufficienty sma. Let us define the foowing objects connected with the function ρt: t [ ϕ 1/2 ρt =: rt, e b ρ 1/2 t s dρs ds ds =: r t = 1, m. 5 f t := ft + The functions rt and r t = 1, m are continuousy differentiabe. From 3, it foows that these functions satisfy the system { d u [ } dt + A1/2 A 1/2 u ϕ 1/2 Q ρ 1/2 r Q r = f b t, [ dr dt + ϕ1/2 QA 1/2 dr u =, dt + ρ 1/2 QA 1/2 u + r = = 1, m, 6 e t ρ B ρ. Let us write system 6 as a first-order differentia equation in a Hibert space H := H H H := L 2 Ω, ρ, H := m+1 L 2,Ω: where dξ dt + Aξ = Ft, ξ = ξ, 7 ξ := u; w τ, w := r; r 1 ;... ; r m τ, ξ := u ; w τ, 8 w := ϕ 1/2 ρ ; ;... ; τ, Ft := f t; ;... ; τ. 9 The operator A satisfies the foowing formuae: A = diag A 1/2 I Q, I diag A 1/2, I Q G, 1 I A = QA 1/2 diag A, G+ QQ I A 1/2 Q, 11 I I { } DA = ξ = u; w τ H u + A 1/2 Q w DA, 12 where I and I are the identity operators in H and H, respectivey, [ Q := ϕ 1/2 ρ 1/2 [ 1 ρ 1/2 τ 1 Q, Q,..., Q, b 1 b m G := diag, b 1 I,..., b m I. From [1, it foows that A is a maxima sectoria and accretive operator. From this and 7 one can obtain the theorem on strong sovabiity of probem 1. Journa of Mathematica Physics, Anaysis, Geometry, 217, Vo. 13, No. 4 45

6 D. Zaora Theorem 1 see [1. Let the fied ft, x satisfy the Höder condition τ R + K = Kτ >, τ, 1 : ft fs K t L2 Ω,ρ s for a s, t τ. Then for any u DA and ρ DB there exists a unique strong soution to initia-boundary vaue probem The basic spectra probems and the theorem on recacuation of root eements. We consider the spectra probem to the evoution probem 7. Assuming Ft, a dependence on time for the unnown function be of the form ξt = exp λtξ, where λ is a spectra parameter and ξ is an ampitude eement, we have Aξ = λξ, ξ DA H. 13 Let ξ = u; w τ DA. Changing the sought eement in probem 13, diaga 1/2, I ξ = ζ =: v; w τ, using factorization 1, we have the foowing spectra probem: I λa 1 Q Aλζ := v =, ζ H = H H Q G λ w. 14 Let λ / {, b 1,..., b m } = σg. From 14 it foows that Lλ v := [ I λa 1 + Q G λ 1 Q v =, v H. 15 From [1, we obtain the theorem on the spectrum of the operator A. Theorem 2 see [1, Theorems 2, 4. The foowing statements hod. 1. σ ess A = Λ E Λ L, where Λ E := Λ L := { { λ C 1 λ C 1 } ρ + 3η = λ4µ λ 3a 2, ρ } ρ + 3η = λ7µ λ 3a 2. ρ The set C\σ ess A consists of reguar points and isoated eigenvaues of finite mutipicity of the operator A. 2. λ = is not an eigenvaue of the operator A. If A 1/2 < b 1/2 q, then λ = b q is not an eigenvaue of the operator A. Otherwise the point λ = b q can be an eigenvaue of finite mutipicity of the operator A. 46 Journa of Mathematica Physics, Anaysis, Geometry, 217, Vo. 13, No. 4

7 On Properties of Root Eements... Sma Motions of Viscous Reaxing Fuid 3. If λ is a non-rea eigenvaue of the operator A, then [ γ 1 := 2 A 1/2 2 1 < Re λ < b m + Q 2 + Q b m + Q 2 1/2 =: γ2, λ 2 < b m + 2 Q Q b m + Q 2 1/2 2bm + Q 2. The spectrum of the operator A is rea if the foowing condition hods: 2 A 1/2 2 b m + Q 2 + Q b m + Q 2 1/ Let us introduce the foowing definition. Definition 2 see [3, p. 61. Let λ and v be an eigenvaue and an eigenvector of the operator penci Lλ, i.e., Lλ v =. The eements v 1, v 2,..., v n 1 are said to be adjoint to v if j =! 1 L λ v j = j = 1, 2,..., n 1. The number n is caed the ength of the chain v, v 1,..., v n 1 of root eements. From [4, we obtain the foowing theorem on connection between root eements of the operator A and the operator penci Lλ. Theorem 3. Let {ξ = u ; w τ } n 1 = denote a chain of root eements of the operator A and this chain corresponds to λ λ, b 1,..., b m. Then { v } n 1 = := { A 1/2 } n 1 u is a chain of root eements of the operator penci Lλ = and this chain corresponds to λ. Let { v } n 1 = denote a chain of root eements of the operator penci Lλ and this chain corresponds to λ. Then { ξ = A 1/2 v ; w τ } n 1 =, where w := = G λ +1 Q v, is a chain of root eements of the operator A. 3. A Mutipe p-basicity of Specia System of Eements 3.1. The theorem on competeness and p-basicity of root eements of the operator A. The foowing considerations are based on the theory of spaces with indefinite metrics see [5, 6. Therefore we assume that H = H + H, where H + := H = L 2 Ω, ρ, H := H. We reca here some concepts and facts of this theory. Define J := diagi, I and introduce in H an indefinite scaar product by the formua [ξ 1, ξ 2 := Jζ 1, ζ 2 H = v 1, v 2 H+ w 1, w 2 H. Denote by P + and P orthogona projectors of the space H onto the subspace H + and H, respectivey: P + H = H +, P H = H. A subspace L + is caed nonnegative if [ξ, ξ for a ξ L +, and maxima nonnegative L + M + if it is not a principe part of any other nonnegative subspace. In the same way, we can define nonpositive subspace L. Journa of Mathematica Physics, Anaysis, Geometry, 217, Vo. 13, No. 4 47

8 D. Zaora From [5, p. 7, it foows that L + M + if there exists a restriction K + : H + H K + 1 such that L + = {ξ = ξ + + K + ξ + : ξ + H + }. This restriction is caed an anguar operator of the subspace L +. A positive subspace L + is said to be uniformy positive if it is a Hibert space with respect to the scaar product generated by the indefinite metric. We say that the subspace L + beongs to the cass h + if it can be represented as the J-orthogona sum of the uniformy positive subspace and the finitedimensiona neutra subspace. In particuar, L + h + if K + S see [5, p. 84. Let L ± M ±. If L + and L are J-orthogona, then we say that L + and L form a dua pair {L +, L }. We can write {L +, L } h, if L ± h ±. We say that a continuous J-sefadjoint operator B beongs to Heton s cass B H if there exists at east one dua pair {L +, L } h of invariant with respect to B-subspaces and every B-invariant dua pair beongs to the cass h. Theorem 4. A 1 H. Proof. The Schur Frobenius factorization 11 of the operator A and Theorem 2 ρa impy A 1 I A = 1/2 Q diag A 1, G+ QQ 1 I I QA 1/2 I A = 1 A 1/2 Q G+ QQ 1 QA 1/2 A 1/2 Q G+ QQ 1 G+ QQ 1 QA 1/2 G+ QQ The operator A 1 is J-sefadjoint and bounded. The compactness of the operator A 1/2 impies the compactness of the operator P + A 1 P. From this and [5, p. 287, it foows that the operator A 1 has a dua invariant pair {L + A 1, L A 1 }. Let K + denote the anguar operator of invariant nonnegative subspace L + A 1. Then K + : H + H, K + 1, and L + A 1 = { u; w τ H + H u; w τ = u; K + u τ, u H + }. Let u 1 ; w 1 τ = u 1 ; K + u 1 τ L + A 1. Then A 1 u 1 ; K + u 1 τ = u 2 ; K + u 2 τ. From this and 17, we deduce the equation for anguar operator K + : G+ QQ 1 K + = G+ QQ 1 QA 1/2 + K + A 1 A 1/2 Q G+ QQ 1 QA 1/2 From A 1/2 S, it foows that K + S H +. K + A 1/2 Q G+ QQ 1 K Journa of Mathematica Physics, Anaysis, Geometry, 217, Vo. 13, No. 4

9 On Properties of Root Eements... Sma Motions of Viscous Reaxing Fuid Remar 1. Theorem 4 is vaid for the case where the system is in the gravitationa fied. In particuar, Theorem 4 impies that the nonrea spectrum of the operator A consists of at most finite number of eigenvaues with regard for their agebraic mutipicity see [5, p. 245, Coroary A system {ξ } =1 is said to be a Riesz basis in a space H if ξ = Tζ, where T, T 1 LH, and {ζ } =1 is an orthonorma basis in the space H. If T = I + K, where K S p H, then the system {ξ } =1 is caed a p-basis in the space H. A basis in the J-space H is said to be amost J-orthonorma if it can be represented as a unity of a finite number of eements and a set of J-orthonorma eements, and the sets are J-orthogona to each other. Let L λ A denote a root subspace of the operator A which corresponds to the eigenvaue λ λ σ p A. Let us aso introduce the foowing notations: FA := sp{l λ A λ σ p A}, F A := sp{kera λ λ σ p A}. We write λ sa R if KerA λ is degenerate, i.e., there exists ξ KerA λ such that [ξ, ξ = for a ξ KerA λ. Theorem of T.Ya. Azizov see [5, p. 271, Theorem 2.12 impies the foowing statement. Theorem 5. The foowing statements hod. 1. codim FA codim F A <. 2. FA = H sp{l λ A λ σ ess A γ 1, γ 2 } is a non-degenerate subspace. The numbers γ 1 and γ 2 are defined in Theorem F A = H L λ A = KerA λ as λ λ and sa =. If γ 2 γ 1 then F A = H. 4. If F A = H FA = H, then the eigeneements root eements of the operator A form an amost J-orthonorma p-basis p > 3 in H. If γ 2 γ 1, then the eigeneements of the operator A form a J-orthonorma basis in H. Proof. Theorem 4 impies A 1 H. From 18 and A 1 S p p > 3/2 it foows that K + S p p > 3. From Theorem 2 we concude that the spectrum of the operator A 1 has at most finite number of accumuation points. Therefore the operator A 1 satisfies a assumptions of T.Ya. Azizov s theorem. 1. From FA = FA 1, F A = F A 1 we obtain the first statement. 2. FA 1 = H sp{l λ 1A 1 λ 1 sa 1 } is a non-degenerate subspace. To prove FA 1 = H, we shoud chec that L λ 1A 1 is a nondegenerate subspace ony for those λ 1 sa 1 which are accumuation points Journa of Mathematica Physics, Anaysis, Geometry, 217, Vo. 13, No. 4 49

10 D. Zaora of the operator A 1 see [5, p. 271, Remar 3.8. From L λ 1A 1 = L λ A we have the foowing concusion. To prove FA 1 = H, we shoud chec that whether L λ A is a non-degenerate subspace for λ σ ess A sa. Now we shoud determine where the set sa is positioned. Let λ = λ σ p A, λ / σg, and KerA λ be degenerated. By Theorem 3, it is equivaent to the foowing. There exists v KerLλ such that ξ = A 1/2 v ; G λ 1 Q v τ is J-orthogona to a eements ξ = A 1/2 v; G λ 1 Q v τ, where v KerLλ, i.e., [ξ, ξ =. From the above it foows that L λ v, v H =. In particuar, we have two equations: Lλ v, v H =, L λ v, v H =. From this it foows that λ is a mutipe root of the equation 1 λp 1 λ q q λ =, 19 p := A 1/2 v 2 v 2, q := Q v 2 v 2, q := ρ q = 1, m. Let us write 19 in the form = λ λ 2 p q m λ m λ m+1 [1 + p m q b λ = p 1 m λ m+2 1 m λ m q + + p b i b j +. 2 i<j Equation 19 has m rea roots λ λ 1,, = 1, m, b := and a rea doube root λ. Then [ m = p λ λλ λ 2 = p 1 m λ m m λ m+1 p 2λ + λ 1 m λ m p λ 2 + 2λ m λ + i<j λ i λ j Let us compare the coefficients of λ m+1 in 2, 21, and the coefficients of λ m in 2, 21. We have λ 2 + 2λ m 2λ + λ + i<j λ = 1 m p +, 22 λ i λ j = q p + 1 p + b i b j. 23 i<j 41 Journa of Mathematica Physics, Anaysis, Geometry, 217, Vo. 13, No. 4

11 On Properties of Root Eements... Sma Motions of Viscous Reaxing Fuid From 22, it foows that 2λ = p 1 + m λ > p 1 A 1/2 2. Hence λ > [ 2 A 1/2 2 1 = γ1. In what foows, we use the ideas from [6, p Define δ := 2 1 m λ, ω := 2p 1. Then λ = ω + δ see 22. Extract m λ from 22 and use this expression in 23. It foows that 2δ λ i<j b i b j λ i λ j = ω 2 + 2ωδ + q δ From λ 1,, = 1, m b :=, one can obtain the foowing inequaity see [6, p. 38, Formua 5.24: m b i b j λ i λ j < b j λ j λ i = 2δ λ. 25 i<j j=1 From 25, it foows that the right-hand part in 24 is positive. Consequenty, ω < δ + q + 2δq + q 2 1/2. Therefore, λ < 2δ + q + 2δq + q 2 1/2 b m + Q 2 + Q [ b m + Q 2 1/2 = γ2. Hence, λ γ 1, γ 2. Theorem 2 impies / σ p A. Hence, / sa. Let b q / γ 1, γ 2. Then from inequaities b q b m < γ 2 we obtain b q γ 1. Let us suppose that b q γ 1 and KerA b q is degenerated. This means that there exists ξ KerA b q such that [ξ, ξ = for a ξ KerA b q. In particuar, [ξ, ξ =. Let ξ KerA b q. Then 1 impies [ [ A 1/2 A 1/2 u ϕ 1/2 Q ρ 1/2 r Q r b q u =, ϕ 1/2 QA 1/2 u b q r =, [ ρ 1/2 QA 1/2 u + b q r =, = 1, m, q, i=1 [ ρ 1/2 q QA 1/2 u =. b q Mutipying the first equation by u and using the other equations, we rearrange it to the form v 2 H = b q A 1/2 v 2 H, where v := A 1/2 u. Hence b q A 1/2 2 > γ 1, which contradicts b q γ 1. From the above, it foows that sa γ 1, γ 2, and the second statement of the theorem is proven. 3. The first parts of the statements 3 and 4 foow directy from [5, p. 271, Theorem If γ 2 γ 1, then sa = and the operator A has no non-rea Journa of Mathematica Physics, Anaysis, Geometry, 217, Vo. 13, No

12 D. Zaora eigenvaues. Consequenty, F A = H. Hence the eigeneements of the operator A form a J-orthonorma basis in H Representation of soution of evoution equation. Let us assume that γ 2 γ 1. Then from Theorem 5 it foows that the eigeneements of the operator A form a J-orthonorma basis in H. This basis is aso a p-basis p > 3 in H. By Theorem 3, this basis after being divided into positive and negative eements can be represented in the form τ } {ξ ± := A 1/2 v ± ; G λ± 1 Q v ±, =1 ξ ± L ±A 1, [ξ +, ξ+ j = δ j, [ξ, ξ j = δ j, [ξ +, ξ j =. 26 Let us represent the soution ξt of probem 7 in the foowing form: ξt = c + tξ+ + c j tξ j, c+ = [ξ, ξ +, c j = [ξ, ξj. 27 =1 j=1 From 7, 26, 27, the formuae for ξ and Ft, we find that ξt = e λ+ t [ ξ, ξ + t + e λ+ t s [ Fs, ξ + ds ξ + [ ξ, ξ ± =1 j=1 = u, A 1/2 v ± e λ j t [ ξ, ξ j + t e λ j t s [ Fs, ξ j ϕ ρ, Q v ±, N, H L 2,Ω λ ± [Ft, ξ ± = ft, A 1/2 v ± H m + ρ e t ρ, Q v ±, N. L 2,Ω ds ξj, 28 From 28, we obtain the representation of the soution ut, ρt of probem 2 with respect to a system of eigenvectors { v ± }+ =1 of the operator penci Lλ this system is connected with J-orthonorma basis 26 and is normaized in a specia way: ut [ = T + A ρt t 1/2 v + λ + =1 1 Q v + T A t 1/2 v λ 1 Q v, t T ± t := e λ± t s fs, A 1/2 v ± ds + H e λ± t u, A 1/2 v ± H [ m ρ λ ± + e bt e λ± t λ ± λ ± b e λ± t ρ λ ±, Q v ±. L 2,Ω The author thans Professor N.D. Kopachevsy for attention to this wor and the anonymous referee for usefu suggestions. 412 Journa of Mathematica Physics, Anaysis, Geometry, 217, Vo. 13, No. 4

13 On Properties of Root Eements... Sma Motions of Viscous Reaxing Fuid References [1 D. Zaora, On the Spectrum of Rotating Viscous Reaxing Fuid, Zh. Mat. Fiz. Ana. Geom , No. 4, [2 D. Zaora, A Symmetric Mode of Viscous Reaxing Fuid. An Evoution Probem, Zh. Mat. Fiz. Ana. Geom , No. 2, [3 A.S. Marcus, Introduction to Spectra Theory of Poinomia Operator Pencis, Shtiinca, Kishenev, 1986 Russian. [4 D.A. Zaora, Operator Approach to Iushin s Mode of Viscoeastic Body of Paraboic Type, Sovrem. Mat. Fundam. Naprav , Russian; Eng. trans.: J. Math. Sci. N.Y , No. 2, [5 T.Ya. Azizov and I.S. Iohvidov, Basic Operator Theory in Spaces with Indefinite Metrics, Naua, Moscow, 1986 Russian. [6 N.D. Kopachevsy and S.G. Krein, Operator Approach to Linear Probems of Hydrodynamics. Vo. 2: Nonsef-Adjoint Probems for Viscous Fuids, Birhäuser Verag, Base Boston Berin, 23. Journa of Mathematica Physics, Anaysis, Geometry, 217, Vo. 13, No