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On the solvability of the initial-value problem for the motion equations of nonlinear viscoelastic medium in the whole space
Nonlinear Analysis 58 (2004) 631 – 656
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On the solvability of the initial-value problem for the motion equations of nonlinear viscoelastic medium in the whole space D.A. Vorotnikov, V.G. Zvyagin∗ Research Institute of Mathematics, Voronezh State University, Universitetskaya Pl. 1, 394006 Voronezh, Russia Received 18 March 2004; accepted 1 May 2004
Abstract The paper deals with the initial-value problem for the system of motion equations of a wide class of incompressible nonlinear viscoelastic mediums, i.e. mediums possessing both nonlinear viscous and viscoelastic properties, in the whole space R2 or R3 . The existence and uniqueness of global in time solutions of this problem for small data are obtained. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Nonlinear viscoelastic medium; Frame-indi6erent derivative; Initial-value problem; Nonlinear evolution equation in a Banach space
1. Introduction It is known [6] that the motion of an incompressible medium with constant density = const is determined by the system of di6erential equations in the form of Cauchy n @u @u = Div T + f0 ; (t; x) ∈ [0; T] × Rn ; ui (1.1) + @t @xi i=1
div u = 0; (t; x) ∈ [0; T ] × Rn :
(1.2)
Here u is the velocity vector, f0 is the body force, T is the stress tensor (all of them depend on a point x of the space Rn ; n = 2; 3 and on a moment of time t). ∗
Corresponding author. Fax: +7-0732-208755. E-mail address: [email protected] (V.G. Zvyagin).
0362-546X/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.05.012
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The divergence div is taken with respect to the variable x. The divergence Div of a n @ tensor is the vector with the coordinates (Div )j = i=1 @xiji . Without loss of generality we can consider the density to be equal to 1. The type of a medium is determined by the choice of the constitutivelaw of the @u @ui relation between and the strain velocity tensor E(u)=(Eij (u)); Eij (u)= 12 @x + @xji . j For instance, one class of media is connected with the Stokes conjecture that the deviator of the stress tensor in every point is completely determined by the strain velocity tensor in the same point at the same moment of time. It is the conception of linear- and nonlinear-viscous Huid [9]. Prandtl and Eiring models [17] are examples of models of nonlinear-viscous Huids. However this conception is not satisfactory for all incompressible mediums. In particular, it is not suitable for media “with memory”: concrete, various polymers and solutions of polymers, the earth’s crust, etc. To take into account the e6ects of memory one may introduce time derivatives into the constitutive law. In the case of using this method the models of Maxwell, Je6reys [5,12], Oldroyd [10,11], Larson, Giesekus, Phan-Thien and Tanner [4,5], Spriggs [3] and a lot of other ones have appeared. The mathematical investigation of a part of these models was carried out in [4,14]. The models of nonlinear viscoelastic mediums, i.e. the media possessing both nonlinear viscosity and viscoelasticity, were investigated in [1,2]. In this work we study the motion equations for a much wider class of nonlinear viscoelastic mediums. This class contains a large number of existing models of linear and nonlinear-viscous, viscoelastic and nonlinear viscoelastic Huids and mediums, including all the models listed above. Namely, we research nonlinear viscoelastic media with the following combined constitutive law: T = s + p :
(1.3)
The tensor s is a nonlinear-viscous constituent of the stress tensor: s = −pI + (E(u));
(1.4)
where p is the pressure function, I is the identity matrix. The tensor p may consist of several viscoelastic (Maxwell) constituents with various relaxation times r k; (1.5) p = k=1
k + k
Da k + k ( k ; E) = 2k E: Dt
(1.6)
Here and k are arbitrary functions with values in the space of symmetric matrixes. The expression DDta A is the objective (frame-indi6erent) Oldroyd derivative of a tensor [3,5]. For a function A(x; t) with values in the set of matrices n × n it is deOned by the formula: n Da A @A @A ui + AW − WA − a(EA + AE) (1.7) = + Dt @t @xi i=1
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@uj @ui is the vorticity tensor, a is some and in this expression W = (Wij ); Wij = 12 @x − @xi j number, k ¿ 0 are the relaxation times, k ¿ 0 are the viscosities, k = 1; : : : ; r. It follows from the principle of material frame-indi6erence (see [3,16, Section 1 of chapter VI]), that without loss of generality of the model the functions and k ; k = 1; : : : ; r can be represented as (E) = ’1 E + ’2 E2 ;
(1.8)
k (; E) = 0k I + 1k E + 2k E2 + 3k + 4k 2 + 5k (E + E) + 6k (E2 + E2 ) + 7k (E2 + 2 E) + 8k (E2 2 + 2 E2 ); where ’1 ; ’2 and
jk 2
(1.9)
are scalar functions
’i = ’i (Tr (E ); det E);
i = 1; 2;
jk = jk (Tr E2 ; Tr E3 ; Tr(); Tr(2 ); Tr(3 ); Tr(E); Tr(2 E); Tr(E2 ); Tr(2 E2 )); k = 1; : : : ; r;
j = 0; : : : ; 8:
In this paper, we prove the global existence and uniqueness for small data of the solutions of the initial-value problem for the motion equations for the described above class of nonlinear viscoelastic mediums in the whole space Rn ; n = 2; 3. Let us mention some results concerning certain special cases of the considered problem. GuillopPe and Saut [4] proved the local in time existence and global existence for small data of the Dirichlet initial-boundary value problem for the system of motion equations of a viscoelastic Huid in a bounded domain in the case ’1 ≡ const; ’2 ≡ 0, i ≡ 0; i = 1; : : : ; r, and also for several speciOc functions i . Talhouk [14] generalized their local existence result (for r = 1; 1 ≡ 0) onto the case of unbounded domains. The motion equations of a nonlinear viscoelastic Huid were investigated by Agranovich and Sobolevskii [1,2] provided that the time derivative (1.7) was replaced by the partial derivative @t@ , what essentially narrows the class of mediums satisfying the model (see [11]). The plan of the paper is the following. In the second section the basic notations are introduced, the statement of the problem is carried out and the basic result is formulated. In the third section the operator treatment of the considered problem is realized. In the fourth section an auxiliary problem depending on a parameter is introduced and investigated. The existence of solutions of this problem and a uniform a priori estimate are proved. In the Ofth section the passage to the limit as the parameter tends to zero is carried out and the solution of the original problem is obtained. In the appendix some technical lemmas are proved. 2. Notations and the main result 2.1. Basic notations We will use the following notations:
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Denote by Rn×n the space of matrices of the order n × n with the following scalar product: for A = (Aij ); B = (Bij ) (A; B)Rn×n =
n
Aij Bij :
i; j=1
Denote by RSn×n its subspace of symmetric matrices. Denote by Rn×n×n the space of ordered collections of n matrices of the order n × n with the following scalar product: for A = (A1 ; : : : ; An ); B = (B1 ; : : : ; Bn ) (A; B)Rn×n×n =
n
(Ai ; Bi )Rn×n :
i=1
@p @p The symbol grad p will stand for the gradient @x of a function ; : : : ; @x n 1 n p : R → R. The symbol ∇u will stand for the Jacobi matrix of a vector function u : Rn → Rn . The symbol ∇ will denote the ordered collection of the Jacobi matrices of the columns of a matrix function : Rn → Rn×n . The symbol the " will denote n @2 @ n @1 Laplacian i=1 @x2 . The symbol D will denote the derivative @x 1 ; : : : ; @x nn where i 1 = ( 1 ; : : : ; n ) is a multi-index. The symbol ∇$ will stand for the Frechet derivative of functions or matrix functions of one or two matrix arguments ’ : RSn×n → R, : RSn×n × RSn×n → RSn×n , etc. The partial derivative of a function with the matrix argument $ = ($ij ) with respect to an element $ij will be denoted as @$@ij . Let us remind that if an operator B : D(B) → E with a dense in a Banach space E range of deOnition D(B) satisOes condition (4.4), it generates the analytical semigroup e−Bt ; t ¿ 0 then one can deOne its fractional powers [8,13]: ∞ 1 B = s− −1 e−Bt ds( ¡ 0); &(− ) 0 where & is the Euler function B = (B− )−1 ( ¿ 0);
B0 = I:
We shall use the function spaces of Sobolev type HVm = {u ∈ H m (Rn ; Rn ); div u = 0} and HMm = H m (Rn ; RSn×n ). The scalar product and the Euclidean norm in both spaces will be denoted as (·; ·)m ; · m . Let us remind that the scalar product in these spaces can be deOned by the equality (u; v)m = ((I − ")m=2 u; (I − ")m=2 v)L2 : In the case of natural m there is also another formula (u; v)m = (D u; D v)L2 : | |6m
The symbol P will stand for the Leray projection, Pu = u − "−1 ∇(div u). It is an orthogonal projection from H m (Rn ; Rn ) onto HVm .
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The symbols C([0; T ]; X ); L2 (0; T ; X ), etc. will denote the Banach spaces of continuous, quadratically integrable or other functions on the interval [0; T ] with values in some Banach space X . The symbol K will stand for various positive constants. 2.2. Statement of the initial-value problem and the main result Let us provide the statement of the initial-value problem for system (1.1)–(1.9) describing the motion of a nonlinear viscoelastic medium. For this purpose we have to add two conditions to (1.1)–(1.9). Generally, the pressure p may be determined up to an arbitrary scalar function of time. For deOniteness the following condition is imposed: p(t; x) d x ≡ 0; (2.1) 3
where 3 is some Oxed bounded domain in Rn . The initial conditions have the following form: u(0; x) = a(x);
k (0; x) = k0 (x);
x ∈ Rn ;
k = 1; : : : ; r:
(2.2)
Let us turn now to the formulation of the existence and uniqueness theorem for a solution of problem (1.1)–(1.9), (2.1)–(2.2). Let 0 = ’1 (0;0) 2 . Assume that 0 ¿ 0. This is a natural condition since the physical meaning of 0 is a viscosity parameter. Let us also assume that ’i and jk are C 4 - and C 3 -smooth functions, respectively, and 0k (4) = 1k (4) = 3k (4) = 0;
@ 0k (4) =0 @ Tr()
(4 stands for the point (0; 0; 0; 0; 0; 0; 0; 0; 0)). This assumption is also natural in the considered model since the functions k in it correspond to “nonlinear” e6ects, i.e. e6ects of the second order and higher. Therefore the coeQcients 1k and 3k at the Orst order terms E and should be “of the Orst order”, i.e. they should vanish in the point 4, and the coeQcient 0k at the zero-order term I should be of the second order, i.e. it should vanish in 4 and its partial derivative with respect to the “linear” argument Tr() should vanish in 4. Remarks. 1. Instead of the C 4 —smoothness of ’1 and ’2 it is suQcient for these functions to be di6erentiable in zero provided the function : RSn×n → RSn×n is C 4 -smooth or has locally Lipschitz third derivatives. 2. All these assumptions take place for the mentioned above models of Oldroyd, Larson, Giesekus, Phan-Thien and Tanner, Spriggs, Prandtl, Eiring and their combinations. Now we can formulate the main result.
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Theorem 2.1. Given, a ∈ HV3 ; k0 ∈ HM3 ; k = 1; : : : ; r; f0 ∈ L1 (0; T ; H 3 (Rn ; Rn )) ∩ L2 (0; T ; H 2 (Rn ; Rn )), there exists a constant K0 ¿ 0, independent on T, such that provided r
a 3 +
k0 3 + f0 L1 (0;T ;H 3 ) ¡ K0 (2.3) k=1
problem (1.1)–(1.9), (2.1)–(2.2) has a solution in the class u ∈ L2 (0; T ; HV4 ) ∩ C([0; T ]; HV3 ) ∩ W21 (0; T ; HV2 )
(2.4)
3 T ∈ L2 (0; T; HM; loc );
(2.5)
3 (Rn ; R)): p ∈ L2 (0; T ; Hloc
(2.6)
Furthermore, we have the following information about the constituents of the stress tensor: s + pI ∈ L2 (0; T ; HM3 ) ∩ C([0; T ]; HM2 ) ∩ W21 (0; T ; HM1 ) p ; k ∈ L∞ (0; T ; HM3 ) ∩ C([0; T ]; HM2 ) ∩ C 1 ([0; T ]; HM1 ); This solution is unique in class (2.4)–(2.8). If f0 ∈ L1 (0; +∞; H 3 (Rn ; Rn )) ∩ L2 (0; +∞; H 2 (Rn ; Rn )) and r
a 3 +
k0 3 + f0 L1 (0;+∞;H 3 ) ¡ K0
(2.7) k = 1; : : : ; r:
(2.8)
(2.9)
k=1
then problem (1.1)–(1.9), (2.1)–(2.2) has a unique solution in class (2.4)–(2.8) for every T ¿ 0. 3. Operator treatment of the problem In this section problem (1.1)–(1.9), (2.1)–(2.2), describing the motion of a nonlinear viscoelastic medium, will be rewritten as a Cauchy problem in a Banach space. Below for simplicity we shall consider r = 1 (concerning r ¿ 1, see Section 5.3). Let for brevity = 1 , = 1 , = 1 =1 . Let also 5(E) = (E) − 20 E:
(3.1)
Denote by g the function 1 (; E(u)) : (3.2) Note that Div pI = grad p and under condition (1.2) 2 Div E(u) = Su. We have from (1.1)–(1.7), (3.1)–(3.2) n @u @u ui + grad p − 0 Su − Div (5(E) + ) = f0 ; (3.3) + @t @xi i=1 n @ @ + + ui + g(; ∇u) = 2E: (3.4) @t @xi g(; ∇u) = W − W − a(E + E) +
i=1
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Let 5 = (5ij ). Let us introduce the following notations: (A(u)v) j =
n @5ij @Ekl (v) (E(u)) ; @$kl @xi
j = 1; : : : ; n;
(3.5)
i; k;l=1
(u)v + v − 0 Sv; A(u)v = −P A F1 (u; v) = −P
n i=1
ui
@v ; @xi
(3.6) (3.7)
n @ F(u; ) = − ui ; @xi
(3.8)
F(u) = F1 (u; u) + u;
(3.9)
G(u; ) = −g(; ∇u);
(3.10)
N1 () = P(Div );
(3.11)
N2 (u) = 2E(u);
(3.12)
B0 = I − ";
(3.13)
A0 = A(a);
(3.14)
f = Pf0 :
(3.15)
i=1
Consider the problem du + A(u)u = F(u) + N1 () + f; dt d + = F(u; ) + N2 (u) + G(u; ); dt u(0) = a;
(0) = 0 :
(3.16) (3.17) (3.18)
The statement of Theorem 2.1 will be deduced from the following result. Theorem 3.1. Given a ∈ HV3 ; 0 ∈ HM3 ; f ∈ L1 (0; T ; HV3 ) ∩ L2 (0; T ; HV2 ), there exists a constant K1 ¿ 0, independent on T, such that provided
a 3 + 0 3 + f L1 (0;T ;HV3 ) ¡ K1
(3.19)
problem (3.16)–(3.18) has a unique solution in the class u ∈ L2 (0; T ; HV4 ) ∩ C([0; T ]; HV3 ) ∩ W21 (0; T ; HV2 );
(3.20)
∈ L∞ (0; T ; HM3 ) ∩ C([0; T ]; HM2 ) ∩ C 1 ([0; T ]; HM1 ):
(3.21)
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4. Auxiliary problem 4.1. Solvability of an auxiliary problem Before proving Theorems 2.1 and 3.1, we investigate the solvability of an auxiliary problem. Introduce a family of operators A; () = − ;S; ; ¿ 0 (4.1) and consider the following equation: d (4.2) + A; = F(u; ) + N2 (u) + G(u; ): dt Theorem 4.1. Given a ∈ HV4 ; 0 ∈ HM4 ; f ∈ L1 (0; T ; HV3 ) ∩ C 1 ([0; T ]; HV2 ), there exists a constant K2 ¿ 0, independent on T and ;, such that provided
a 3 + 0 3 + f L1 (0;T ;HV3 ) ¡ K2 there is a unique solution of problem (3.16), (4.2), (3.18) in the class u ∈ C 1 ([0; T ]; HV2 ) ∩ C([0; T ]; HV4 ); ∈ C 1 ([0; T ]; HM2 ) ∩ C([0; T ]; HM4 ):
(4.3)
In order to prove this theorem we need some lemmas. 4.2. On an equation in a Banach space Lemma 4.1. Let ; ; R; M be some numbers, 0 6 ¡ 6 1. Let B : D(B) → E be a linear operator in a Banach space E; D(B) = E, M ; ∈ C; Re ¿ 0: (4.4)
(B + I )−1 6 1 + || For every v0 such that v0 ∈ D(B );
B v0 ¡ R;
(4.5)
let a linear operator A(v0 ) = A0 be de>ned on D(A0 ) = D(B) and for all v ∈ D(B): K3 A0 v 6 Bv 6 K4 A0 v ;
(4.6)
where K3 ; K4 do not depend on v; v0 . Let also M
(A0 + I )−1 6 ; Re ¿ 0: 1 + ||
(4.7)
For every v ∈ D(B ) such that B v ¡ R let a linear operator A(v)(·) be de>ned on D(B) and
(A(v) − A(w))A0−1 6 K5 B (v − w) ;
(4.8)
where K5 does not depend on v; w ∈ D(B ) such that B v ; B w ¡ R.
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Let f1 : [0; T ] × D(B ) → E and
f1 (t; v) − f1 (s; w) 6 K6 ( B (v − w) + |t − s|);
(4.9)
where K6 does not depend on t; s and v; w ∈ D(B ) such that B v ; B w ¡ R. Then for all v0 ∈ D(B) satisfying (4.5), there exists a solution v of the problem dv (4.10) + A(v)v = f1 (t; v); dt (4.11) v(0) = v0 in the class v ∈ C 1 ([0; t0 ]; E);
Bv ∈ C([0; t0 ]; E);
(4.12)
where t0 = t0 (v0 ; f1 ) ¿ 0 is some number. If for every such solution it is a priori known that it is bounded in the following sense:
B v(t) ¡ K7 ;
t ∈ [0; t0 ];
(4.13)
where K7 does not depend on t0 and t, then t0 = T . Remarks. The space E may be both complex and real. In the latter case, while checking conditions (4.4) and (4.7) it is necessary to consider the complexiOcation of the space E and the corresponding extensions of operators A and B. 2. The statement of Lemma 4.1 on the solvability of problem (4.10)–(4.11) on an interval [0; t0 ] is a particular case of Theorem 7 from [13]. If there is a priori estimate (4.13) then the solution may be continued on the whole segment [0; T ] by standard procedure. 4.3. Operator estimates We will need the following estimates for the operators introduced in Section 3. Lemma 4.2. The following estimates take place: For l = 1; 2; u ∈ HV2 ; v ∈ HVl+1 ; ∈ HMl+1 :
F1 (u; v) l 6 K8 u 2 v l+1 ;
(4.14)
F(u; ) l 6 K8 u 2 l+1 : For l = 2; 3;
u ∈ HV3 ;
v ∈ HVl+1 ;
(4.15) ∈ HMl+1 :
|(F1 (u; v); v)l | 6 K9 ∇u 2 ∇v l−1 ; |(F(u; ); )l | 6 K9 ∇u 2 ∇ l−1 : For l = 0; 1 and > =
7 4
(4.16)
or l = > = 2:
G(u1 ; 1 ) − G(u2 ; 2 ) l 6 K10 ( ∇u1 − ∇u2 l + 1 − 2 l ) ×( ∇u1 > + ∇u2 > + 1 > + 2 > ); where u1 ;
u2 ∈ HV>+1 ;
1 ; 2 ∈ HM> .
(4.17)
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For u ∈ HV4 ; ∈ HM3 ( u 3 + 2 )( ∇u 3 + 3 );
G(u; ) 3 6 K10
(4.18)
where K10 and K10 depend continuously on ( ∇u1 > + ∇u2 > + 1 > + 2 > ) and on ( u 3 + 2 ), respectively.
Lemma 4.3. Let l = 0; 1; 2 and 0 (0) = (1) If
11 8 ; 0 (1)
= 78 ; 0 (2) = 12 .
B0 0 (l) v l ; B0 0 (l) w l ¡ 1 then the following estimates take place:
(A(v) − A(w))h l 6 K11 ∇B0 0 (l) (v − w) l−1 ∇h l+1 ;
(A(v) − A(w))h l 6 K11 B0 0 (l) (v − w) l B0 h l :
(4.19)
Furthermore, for l = 0 there is another variant of the second estimate
(A(v) − A(w))h 0 6 K11 v − w 3 h 3 : 2
(4.20)
(2) There exist K12 ; K13 ; K14 ; K15 ¿ 0 such that provided B0 0 (l) a l ¡ K12 (A0 v; B0 v)l ¿ K13 B0 v 2l
(4.21)
and provided u 3 ; v 3 ¡ K14 (A(u)u − A(v)v; B0 (u − v))1 ¿ 12 K13 ; B0 (u − v) 21 ;
(4.22)
((A(u) − I )u; B0 u)2 ¿ K15 ∇u 23 :
(4.23)
The proofs of these two lemmas will be carried out in the appendix. 4.4. Uniqueness lemma Lemma 4.4. There exists a constant K16 ¿ 0, independent on T and ;, such that if a solution (u1 ; 1 ) of problem (3.16), (4.2), (3.18) exists in the class u1 ∈ C 1 ([0; T ]; HV1 ) ∩ C([0; T ]; HV3 ); 1 ∈ C 1 ([0; T ]; HM1 ) ∩ C([0; T ]; HM3 )
(4.24)
and
u1 (t) 3 ¡ K16 ;
t ∈ [0; T ]
(4.25)
then it is unique in class (4.24). Proof. Assume that in addition to (u1 ; 1 ) there is another solution (u2 ; 2 ) of problem (3.16), (4.2), (3.18).
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Suppose Orst that
u2 (t) 3 6 K16 ;
t ∈ [0; T ]
(4.26)
(K16 will be deOned below). The following identity takes place: 1 (N1 (); u)m + (N2 (u); )m = 0; 2 u ∈ HVm+1 ;
∈ HMm+1 ;
m = 0; 1; : : : :
(4.27)
Indeed, using Green formula we have 1 (N1 (); u)m + (N2 (u); )m 2
n
@ij @uj 1 @ui ui ; = + ij ; + @xj m 2 @xj @xi m i; j=1
n
@ij @ui ui ; + ij ; = @xj m @xj m i; j=1
n
=
i; j=1;| |6m
Rn
@ij @ui D ui D + D ij D @xj @xj
d x = 0:
Let w = u1 − u2 ; ∗ = 1 − 2 . Substitute (u1 ; 1 ) and (u2 ; 2 ) into (3.16) and take the di6erence of the obtained equalities. Treat (4.2) in the same way. We have dw(t) = A(u2 )u2 − A(u1 )u1 + u1 − u2 + F1 (w; u1 ) + F1 (u2 ; w) + N1 (∗ ); dt d∗ (t) = −A; (∗ ) + F(w; 1 ) + F(u2 ; ∗ ) + N2 (w) + G(u1 ; 1 ) − G(u2 ; 2 ): dt Taking the scalar product of the Orst equation with B0 w(t) in HV1 and of the second 1 B0 ∗ (t) in HM1 at every t ∈ [0; T ], adding the obtained equalities and taking one with 2 into account (4.27) we get
dw 1 d∗ ; B0 w + ; B0 ∗ dt 2 dt 1 1 = − (A(u1 )u1 − A(u2 )u2 ; B0 w)1 + (w; B0 w)1 + (F1 (w; u1 ); w)2 + (F1 (u2 ; w); w)2 +
1 [ − (A; (∗ ); B0 ∗ )1 + (F(w; 1 ); ∗ )2 2
+(f(u2 ; ∗ ); ∗ )2 + (G(u1 ; 1 ) − G(u2 ; 2 ); ∗ )2 ]: Obviously, (A; (∗ ); B0 ∗ )1 =
∗
− ;S∗ ; ∗
2
1 ¿ ∗ 2 :
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Now, using estimates (4.14)–(4.17), (4.22) (we can assume K16 to be small enough so that estimate (4.22) is also valid) we obtain d 1 1 d
w 22 +
∗ 22
∗ 22 + K14 w 23 + 2 dt 2 dt 6 K17 [ w 22 (1 + u1 3 + u2 3 ) + w 2 1 3 ∗ 2 + u2 3 ∗ 22 + ∗ 2 ( ∗ 2 + w 3 )( 1 2 + 2 2 + u1 3 + u2 3 )]: It is obvious that there exists K18 such that K14 w 23 + K18 ∗ 22 − K17 ∗ 2 w 3 ( 1 2 + 2 2 + u1 3 + u2 3 ) ¿ 0: Therefore, d 1 d
∗ 22
w 22 + dt 2 dt
1
∗ 22 + Kn 17[ w 22 (1 + u1 3 + u2 3 ) 6 K18 − 2 + w 2 1 3 ∗ 2 + ∗ 22 u2 3 + ∗ 22 ×( 1 2 + 2 2 + u1 3 + u2 3 )]
1 6 K w 22 + ∗ 22 : 2 Since w(0) = 0; ∗ (0) = 0, by Gronwall lemma w ≡ 0; ≡ 0. If (4.26) is not valid for some t, let t1 be the inOmum of such t. It is evident that
u2 (t1 ) 3 = K16 . But on the other hand, we have just proved that on [0; t1 ] the solution is unique. Hence u1 (t1 ) = u2 (t1 ); 1 (t1 ) = 2 (t1 ) what contradicts (4.25). 4.5. A priori estimate Lemma 4.5. There exists a constant K19 ¿ 0 (generally speaking, it is small enough), independent on T and ; such that provided 1
a 3 + √ 0 3 + f L1 (0;T ;HV3 ) ¡ K19 2 there are the following estimates for every solution of (3.16), (4.2), (3.18) in class (4.3): 1 2
u(t) 23 + (t) 23 ¡ K19 ; (4.28) 2 T 3K 2
∇u 23 ds ¡ 19 : (4.29) K15 0 Proof. Denote by t1 = t1 (K19 ) the inOmum of those t at which (4.28) is not valid (K19 will be deOned later).
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1 B0 (t) Taking the scalar product of (3.16) with B0 u(t) in HV2 and of (4.2) with 2 2 in HM at every t ∈ [0; t1 ], adding the obtained equalities and taking into account (4.27) we get
du 1 d ; B0 u + ; B0 dt 2 dt 2 2
= − (A(u)u; B0 u)2 + (u; B0 u)2 + (F1 (u; u); B0 u)2 +
1 [ − (A; (); B0 )2 2
+(F(u; ); B0 )2 + (G(u; ); B0 )2 ] + (f; B0 u)2 : Then we have 1 d 1 1 d
u 23 +
23 6 −((A(u) − I )u; B0 u)2 + (F1 (u; u); u)3 −
23 2 dt 4 dt 2 +
1 1 (F(u; ); )3 + (G(u; ); )3 + (f; u)3 : 2 2
Using (4.16), (4.18), (4.23) (we can assume K19 to be small enough so that estimate (4.23) is also valid) we obtain 1 d 1 d 1 K9
u 23 +
23 6 −K15 ∇u 23 −
23 + K9 ∇u 32 +
∇u 2 23 2 dt 4 dt 2 2 +
1 K ( ∇u 3 + 3 )( u 3 + 2 ) 3 + f 3 u 3 : 2 10
For K19 small enough it yields 1 d 1 1 d
u 23 +
23 6 f 3 u 3 − K15 ∇u 23 : 2 dt 4 dt 2 Let
(t) = 1 d 2 dt
2
||u||23 (t) +
1 2 2 ||||3 (t)
(t) 6 f 3 (t);
. Then from (4.30) it follows that
t ∈ [0; t1 ];
d (t) 6 f 3 : dt Integrate from 0 to t1 : (t1 ) 6
||a||23
1 + ||0 ||23 + 2
1 6 a 3 + √ 0 3 + 2
t1
0
0
T
f 3 (s) ds
f 3 (s) ds ¡ K19 ;
what contradicts the deOnition of t1 . Thus, (4.28) is valid for all t ∈ [0; T ].
(4.30)
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D.A. Vorotnikov, V.G. Zvyagin / Nonlinear Analysis 58 (2004) 631 – 656
Integrating (4.30) from 0 to T we obtain 1 1 1 1
u 23 (T ) + 23 (T ) − a 23 − 0 23 2 4 2 4 T 1 6 ( f 3 u 3 − K15 ∇u 23 ) ds: 2 0 This inequality and (4.28) imply T T 1 1 1 2 2 2 K15
∇u 3 ds 6 a 3 + 0 n 3 +
f 3 u 3 ds 2 2 4 0 0 T 1 2 3 2 6 K19 + K19
f 3 ds 6 K19 2 2 0 and we obtain estimate (4.29). 4.6. Properties of the operator A0 Lemma 4.6. Under the conditions and notations from Lemma 4.3 there exist constants K20 , K21 , K22 such that provided B0 0 (l) a l ¡ K12 ; ∈ C; Re ¿ 0 the following estimates are valid:
(A0 + I )v l 6 K20 () B0 v l ;
(4.31)
(A0 + I )v l ¿ K21 B0 v l :
(4.32)
Under the same conditions and l = 1; 2 the operator A0 : HVl+2 D(A0 ) ⊂ HVl → HVl is invertible and
(A0 + I )−1 6
K22 : 1 + ||
(4.33)
Remarks. 1. Here we have to use complexiOcations of Sobolev spaces and di6erential operators (see Remark 1 after Lemma 4.1). 2. The proof of this lemma will be carried out in the appendix. 4.7. Proof of Theorem 4.1 Let us put E = HV1 × HM1 ;
v0 = (a; 0 );
f1 (t; (u; )) = (f(t) + F(u) + N1 (); F(u; ) + N2 (u) + G(u; )); B(u; ) = (B0 u; B0 );
D(B) = HV3 × HM3 ;
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645
A(u; )(v1 ; v2 ) = (A(u)v1 ; A; v2 ); A0 (v1 ; v2 ) = (A0 v1 ; A; v2 ); 7 = ; =1 8 in Lemma 4.1 and let R be small enough. Then estimate (4.4) is a simple property of operator I − ", estimate (4.6) follows from (4.31) and (4.32), estimate (4.7) follows from (4.33), estimate (4.8) follows from (4.19) and estimate (4.9) follows from (4.14), (4.15) and (4.17). Thus, the conditions of Lemma 4.1 hold. Hence, for K2 small enough, system (3.16), (4.2), (3.18) has a solution (u; ) in class (4.24) on some interval [0; t0 ]. Moreover, from regularity properties of solutions of abstract parabolic equations [13, p. 324] it follows that for some A1 ¿ 0 the function A0A1 (u; ) : [0; t0 ] → E is continuously di6erentiable with respect to t and the function A0A1 A(u; )(u; ) : [0; t0 ] → E is continuous. But from (4.6) it follows (see [8, Section 14]) that for all A ¡ A1 the operator B1−A1 +A [A(u; )]−1+A1 A0−A1 is bounded. It yields the estimate
B1=2−A1 +A [A(u; )]A1 (u; ) 2 ¡ K;
t ∈ [0; t0 ];
(4.34)
where K does not depend on t. Let us put now E = HV2 × HM2 ; = 12 ;
=
1 2
D(B) = HV4 × HM4 ; + A2 (0 ¡ A2 ¡ A)
in Lemma 4.1 and let v0 ; f1 ; A; B have the given above form. Estimates from Lemmas 4.2, 4.3, 4.6 imply again that the conditions of Lemma 4.1 hold. Hence, problem (3.16), (4.2), (3.18) has a solution in class (4.3) on some interval [0; t∗ ]. Furthermore, estimate (4.28) with K19 small enough enables to apply Lemma 4.4. Therefore this solution is unique in class (4.24) on [0; t∗ ], i.e. it coincides with (u; ). Then estimate (4.34) and the boundedness of the operator B [A(u; )]−A1 B−(1=2)+A1 −A imply that estimate (4.13) is valid for all t ∈ [0; t∗ ]. Hence, t∗ = t0 , i.e. the solution (u; ) belongs to class (4.3) on the segment [0; t0 ]. From estimate (4.28) it follows that this solution is a priori bounded by a constant independent on t and t0 :
B(u; )(t) HV1 ×HM1 6 K;
t ∈ [0; t0 ]:
Therefore, we can consider t0 to be equal to T . The proof is fulOlled. 5. Proof of the main theorems 5.1. Proof of Theorem 3.1 Evidently, there exist sequences am → a m→∞
in
HV3 ; am ∈ HV4 ;
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D.A. Vorotnikov, V.G. Zvyagin / Nonlinear Analysis 58 (2004) 631 – 656
in
0m → 0 m→∞
fm → f m→∞
in
HM3 ;
0m ∈ HM4 ;
L1 (0; T ; HV3 );
fm ∈ L1 (0; T ; HV3 ) ∩ C 1 ([0; T ]; HV2 ):
Without loss of generality all triplets (am ; 0m ; fm ) satisfy estimate (3.19). Consider problems (3.16), (4.2), (3.18) with the data am ; 0m ; fm and ; = m1 for all natural m. By Theorem 4.1 for K1 small enough such problems have a unique solution (um ; m ) in class (4.3) and all these solutions are bounded uniformly with respect to m by estimates (4.28), (4.29). Therefore, without loss of generality we may assume that um → u∗
in
L∞ (0; T ; HV3 ) ∗ −weakly;
m → ∗
in
L∞ (0; T ; HM3 ) ∗ −weakly;
um → u∗
in
L2 (0; T ; HV4 ) weakly:
Let us show that the sequence (um ; m ) is fundamental in C([0; T ]; HV2 × HM2 ). Let wij = ui − uj ; ij = i − j . Apply the same procedure as in the proof of Lemma 4.4: substitute (ui ; i ) into (3.16), (4.2) with ; = 1i and (uj ; j ) into (3.16), (4.2) with ; = 1j , take the di6erences of the corresponding equations, take the scalar product of the Orst of the obtained equations with wij (t) in HV2 and of the second one with 2ij in HM2 at every t ∈ [0; T ], and taking into account (4.27) add the obtained equalities:
dwij 1 dn ij ; wij + ; ij dt 2 dt 2 2 = − (A(ui )ui − A(uj )uj ; B0 wij )1 + (wij ; wij )2 +(F1 (wij ; ui ); wij )2 + (F1 (uj ; wij ); wij )2 − 1 + 2
Sj Si − ; ij i j
2
1 (ij ; ij )2 2
+ (F(wij ; i ); ij )2 + (F(uj ; ij ); ij )2
+(G(ui ; i ) − G(uj ; j ); ij )2 + (fi − fj ; wij )2 : Let K1 be small enough. Using the estimates and arguing as in the proof of Lemma 4.4 we conclude: 1 d d
wij 22 +
ij 22 dt 2 dt
1 1 1 2 2 6 K wij 2 + ij 2 + fi − fj 2 + i 3 ij 3 + j 3 ij 3 2 i j
1 1 1 : 6 K wij 22 + ij 22 + fi − fj 2 + + 2 i j
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Integrate from 0 to t 1 1
wij 22 + ij 22 6 ai − aj 22 + 0i − 0j 22 2 2 T
1 1 1 2 2 ds
wij 2 (s) + ij 2 (s) + fi − fj 2 (s) + + +K i j 2 0 and by Gronwall lemma wij 22 (t) + t ∈ [0; T ]. Thus, um → u∗
in
C([0; T ]; HV2 );
m → ∗
in
C([0; T ]; HM2 ):
1
ij 22 (t) → 0 uniformly with respect to min(i; j)→∞ 2
This implies 1 Sm → 0 in C([0; T ]; HM0 ); m ∗ in C([0; T ]; HM0 ); A1=m m → N1 (m ) → N1 (∗ )
in
C([0; T ]; HV1 );
N2 (um ) → N2 (u∗ )
in
C([0; T ]; HM1 );
du∗ dum → in the sense of distributions; dt dt dm d∗ → in the sense of distributions: dt dt From estimates (4.14), (4.15), (4.17) we have also ∗ ) in C([0; T ]; HV1 ); m ) → F(u F(u F(um ; m ) → F(u∗ ; ∗ )
in
C([0; T ]; HM1 );
G(um ; m ) → G(u∗ ; ∗ )
in
C([0; T ]; HM1 ):
Furthermore, using (4.19), (4.20) we obtain vrai max A(um )um − A(u∗ )u∗ 0 (t) t∈[0;T ]
6 vrai max ( A(0)(um − u∗ )(t) 0 t∈[0;T ]
+ (A(u∗ ) − A(0))(um − u∗ ) 0 (t) + (A(um ) − A(u∗ ))um 0 (t)) 6 Kvrai max ( um − u∗ 2 + u∗ 11=4 um − u∗ 2 + um − u∗ 3=2 um 3 ): t∈[0;T ]
Therefore, A(um )um → A(u∗ )u∗
in
L∞ (0; T ; HV0 ):
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D.A. Vorotnikov, V.G. Zvyagin / Nonlinear Analysis 58 (2004) 631 – 656
Passing to the limit we conclude that the pair (u∗ ; ∗ ) is a solution of problem (3.16)– (3.18). Estimate (4.19) yields that T T T 2 2
A(u∗ )u∗ 2 6 2
(A(u∗ ) − A(0))u∗ 2 + 2
A(0)u∗ 22 0
0
6K
0
0
T
( u∗ 23 u∗ 24 + u∗ 24 ):
∗) ∈ Therefore, A(u∗ )u∗ belongs to L2 (0; T ; HV2 ). From estimate (4.14) we have F(u L∞ (0; T ; HV2 ). Note that N1 (∗ ) ∈ L∞ (0; T ; HV2 ) and f ∈ L2 (0; T ; HV2 ). Substituting (u∗ ; ∗ ) into (3.16), we conclude that dudt∗ ∈ L2 (0; T ; HV2 ). But u∗ ∈ L2 (0; T ; HV4 ) and by Lemma III.1.2 from [15] we have u∗ ∈ C([0; T ]; HV3 ). Substituting u∗ ; ∗ into (3.17) and taking into account estimates (4.14), (4.15) and (4.17) we conclude that all terms in (3.17) except ddt∗ belong to C([0; T ]; HM1 ). Hence, ∗ ∈ C 1 ([0; T ]; HM1 ). The uniqueness of the solution may be proved in exactly the same way as Lemma 4.4. 5.2. Proof of Theorem 2.1 Let us note that estimates (4.17), (4.18) in the particular case G(u; ) = 5(E(u)) have the form
5(E(u1 )) − 5(E(u2 )) l 6 K23 ( u1 >+1 + u2 >+1 ) u1 − u2 l+1 ;
(5.1)
5(E(u)) 3 6 K23 ( u 3 ) u 4 ;
(5.2)
where K23 , K23 continuously depend on the corresponding norms and l; > are as in estimate (4.17). Let 0 = 10 ; K0 = K1 . Then (2.3) implies estimate (3.19). Hence, by Theorem 3.1 there exists a unique solution (u; ) of problem (3.16)–(3.18) in class (3.20), (3.21). Taking into account notations (3.5)–(3.15) we conclude that (3.4) is valid and n @u @u P ui − 0 Su − Div (5(E) + ) − f0 = 0: + @t @xi i=1
From Prepositions I.1.1 and I.1.2 from [15] it follows that there exists a unique p from class (2.6) such that equalities (3.3) and (2.1) hold. Let p = 1 = ;
s = −pI + 20 E(u) + 5(E(u));
T = p + s :
Obviously, the triplet (u; T; p) is a solution of problem (1.1)–(1.9), (2.1)–(2.2). Since u belongs to class (2.4), estimates (5.1) and (5.2) yield that 5(E(u)) belongs to class (2.7). Hence, pI + s belongs to class (2.7), and T belongs to (2.5).
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5.3. The case r ¿ 1 The case r ¿ 1 is investigated exactly in the same way as the case r = 1. During the operator treatment, problem (3.16)–(3.18) is replaced by the problem du (5.3) + A(u)u = F(u) + N1 () + f; dt di i (5.4) + = F(u; i ) + N2 (u) + Gi (u; i ); dt u(0) = a; where
i (0) = i0 ;
i = 1; : : : ; r;
(5.5)
1 (; ∇u) : Gi (u; ) = − W − W − a(E + E) +
Then one considers the following system: di + A; i = F(u; i ) + N2 (u) + Gi (u; i ) (5.6) dt and the auxiliary problem (5.3), (5.6), (5.5). Then the analogues of Theorem 4.1 and Lemmas 4.4, 4.5 are proved and the passage to the limit is carried out just as in the case r = 1. Acknowledgements The work was partially supported by Grants 04-01-00081 of Russian Foundation of Basic Research and VZ-010-0 of the Ministry of Education of Russia and CRDF. Appendix A A.1. Some inequalities Below we need some well-known inequalities which hold both for scalar and for vector functions. The following statement takes place: for n = 2; 3 and 3p − 6 6m 2p one has H m ⊂ Lp ; If m ¿
3 2,
u Lp 6 K(m; p) u m :
the functions from H
u(x) 6 K(m) u m ;
m
(A.1)
are uniformly bounded:
x ∈ Rn :
It is a particular case of the Sobolev embedding theorem [15].
(A.2)
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D.A. Vorotnikov, V.G. Zvyagin / Nonlinear Analysis 58 (2004) 631 – 656
Let us mention also an inequality of HUolder type:
uv 0 6 K(p; q) u Lp v Lq
(A.3)
provided 1 1 1 + = : p q 2 These inequalities for n = 2; 3 imply the following well-known inequalities:
uv 0 6 K u 2 v 0 ;
(A.4)
uv 0 6 K u 1 v 1 ;
(A.5)
uv 1 6 K u 1 v 2 ;
(A.6)
uv 2 6 K u 2 v 2 :
(A.7)
A.2. Proof of Lemma 4.2 Estimates (4.14)–(4.16) are proved in the usual way (similar estimates are present, e. g. in [7, p. 302]). We shall prove them here for the reader’s convenience. Using (A.6) we have n n @v @v
F1 (u; v) 1 6
ui 2 ui @xi 6 K @xi 6 K u 2 v 2 : 1 1 i=1
i=1
Using (A.7) we have
n n @v @v
F1 (u; v) 2 6
ui 2 ui @xi 6 K @xi 6 K u 2 v 3 : 2
i=1
2
i=1
Estimate (4.14) is proved. Similarly one proves (4.15). Let us prove now estimate (4.16). We have, using the Leibnitz rule n
@v |(F1 (u; v); v)2 | 6 ui D @xi ; D v 0 | |62 i=1
n
@v 6 ui D @xi ; D v 0 | |62 i=1
+
n
ui D @v ; D v @xi 0
0¡| |62 i=1
+
n
| |=2;||=1;6 i=1
@v K( ; ) D ui D − ;D v : @xi 0
D.A. Vorotnikov, V.G. Zvyagin / Nonlinear Analysis 58 (2004) 631 – 656
651
Since div u = 0, integration by parts easily shows that the Orst sum is equal to zero. With the help of (A.5) and Cauchy–Buniakowski inequality we have that the second sum does not exceed the expression n @v K ∇ui 2 @xi ∇v 1 : 1
i=1
With the help of (A.4) we have that the third sum does not exceed the same expression. Estimate (4.16) for the case l = 2 is proved. The case l = 3 is examined analogously. Let us prove now estimate (4.17). It follows from conditions on coeQcients jk from Section 2.2 that the function 1 : RSn×n × RSn×n → RSn×n is C 3 -smooth and 1 (0; 0) = 0; ∇$ 1 (0; 0) = 0. Then the function g : RSn×n × Rn×n → RSn×n introduced by formula (3.2) is also 3 C -smooth and g(0; 0) = 0; ∇$ g(0; 0) = 0. We have
D (G(u1 ; 1 ) − G(u2 ; 2 )) 0 :
G(u1 ; 1 ) − G(u2 ; 2 ) l = | |6l
First we estimate the term with | | = 0. Using inequality (A.2) and the Lagrange theorem one has
G(u1 ; 1 ) − G(u2 ; 2 ) 0 = g(∇u1 ; 1 ) − g(∇u2 ; 2 ) L2 6
=
sup
$1 6 ∇u1 L∞ + ∇u2 L∞ $2 6 1 L∞ + 2 n L∞
sup
$1 6 ∇u1 L∞ + ∇u2 L∞ $2 6 1 L∞ + 2 L∞
∇$ g($1 ; $2 ) ( ∇(u1 − u2 ) L2 + 1 − 2 L2 )
∇$ g($1 ; $2 ) − ∇$ g(0; 0) ( ∇(u1 − u2 ) L2 + 1 − 2 L2 )
6 K( ∇u1 L∞ + ∇u2 L∞ + 1 L∞ + 2 L∞ )( ∇(u1 − u2 ) L2 + 1 − 2 L2 ) 6 K( ∇u1 7=4 + ∇u2 7=4 + 1 7=4 + 2 7=4 )( ∇(u1 − u2 ) 0 + 1 − 2 0 ): Now we estimate the Orst derivatives, using inequalities (A.1)–(A.3)
∇(G(u1 ; 1 ) − G(u2 ; 2 )) 0 = ∇$ g(∇u1 ; 1 )(∇2 u1 ; ∇1 ) − ∇$ g(∇u2 ; 2 )(∇2 u2 ; ∇2 ) 0 6 6 ∇$ g(∇u1 ; 1 )(∇2 (u1 − u2 ); ∇(1 − 2 )) 0 + (∇$ g(∇u1 ; 1 ) − ∇$ g(∇u2 ; 2 ))(∇2 u2 ; ∇2 ) 0 6 ∇$ g(∇u1 ; 1 ) L∞ ( ∇(u1 − u2 ) 1 + 1 − 2 1 ) + ∇$ g(∇u1 ; 1 ) − ∇$ g(∇u2 ; 2 ) L6 ( ∇2 u2 L3 + ∇2 L3 ) 6 ∇$ g(∇u1 ; 1 ) − ∇$ g(0; 0) L∞ ( ∇(u1 − u2 ) 1 + 1 − 2 1 )
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D.A. Vorotnikov, V.G. Zvyagin / Nonlinear Analysis 58 (2004) 631 – 656
+ ∇$ g(∇u1 ; 1 ) − ∇$ g(∇u2 ; 2 ) L6 ( ∇2 u2 L3 + ∇2 L3 ) 6 K( ∇u1 L∞ + 1 L∞ )( ∇(u1 − u2 ) 1 + 1 − 2 1 ) +K( ∇(u1 − u2 ) L6 + 1 − 2 L6 )( ∇2 u2 1=2 + ∇2 1=2 ) 6 K( ∇(u1 − u2 ) 1 + 1 − 2 1 )( ∇u1 7=4 + 1 7=4 + ∇u2 7=4 + 2 7=4 ): In the same way the terms D (G(u1 ; 1 ) − G(u2 ; 2 )) 0 with | | = 2 are estimated and we obtain estimate (4.17). Similarly one obtains estimate (4.18).
A.3. Proof of Lemma 4.3 Note that from the C 4 —smoothness of coeQcients ’i (see Section 2.2) it follows that the function 5 : RSn×n → RSn×n introduced by formula (3.1) is C 4 -smooth. Moreover, it is easy to check that 5(0) = 0; ∇$ 5(0) = 0. We have − A(w))h
(A(v) − A(w))h l = (A(v) l6K
− A(w))h
D (A(v) 0:
| |6l
First we estimate the term with | | = 0: − A(w))h
(A(v) 0
n @5ij @5ij @Ekl (h) 6 (E(v)) − (E(w)) @$kl @xi 0 @$kl i; j; k;l=1
n @5ij @5ij (E(v)) − (E(w)) @$kl @$kl L∞
6
i; j; k;l=1
@Ekl (h) 6 K E(v) − E(w) L ∇h 1 × ∞ @xi L2 6 K E(v) − E(w) 7=4 ∇h 1 6 K ∇B011=8 (v − w) −1 ∇h 1 : Furthermore, − A(w))h
(A(v) 0
n @5ij @5ij @Ekl (h) 6
(E(v)) − (E(w))
0 @$kl @$kl @xi i; j; k;l=1
(A.8)
D.A. Vorotnikov, V.G. Zvyagin / Nonlinear Analysis 58 (2004) 631 – 656
653
n @5ij @5ij 6 @$kl (E(v)) − @$kl (E(w)) L3 i; j; k;l=1
@Ekl (h) × @xi 6 K E(v) − E(w) L3 h 3 L6 6 K v − w 3=2 h 3
(A.9)
what implies (4.20). For | | = 1 we obtain − A(w))h)
D ((A(v) 0
n @5ij @5ij @Ekl (h) 6 (E(v)) − (E(w)) D @$kl @$kl @xi 0 i; j; k;l=1
@5ij @5n ij @Ekl (h) + (E(v)) − (E(w)) D @$kl @$kl @xi 0
n @5ij @5ij @5ij @Ekl (h) 6
D (E(v)) − (E(w)) L3
L 6 +
(E(v)) @$kl @$kl @xi @$kl i; j; k;l=1
@5ij @Ekl (h) − (E(w)) L∞ D
L 2 @$kl @xi n n @2 5ij 6 K
(E(v))D Ek1 l1 (v) @$kl @$n k1 l1 k1 ;l1 =1
i; j; k;l=1
−
6
2
@ 5ij (E(w))D Ek1 l1 (w) L3 ∇h 2 + E(v) − E(w) L∞ ∇h 2 @$kl @$n k1 l1
n
K
i; j; k;l=1
2 @ 5ij + @$kl @$k
n
k1 ;l1 =1
@2 5ij @$kl @$k
1 l1
(E(v))
L∞
D (Ek1 l1 (v) − Ek1 l1 (w)) L3
@2 5ij (E(v)) − (E(w)) D Ek1 l1 (w) L3 @$kl @$n k1 l1 1 l1 L∞
+ E(v) − E(w) L∞
6 K[
@2 5ij @2 5ij @2 5ij (E(v)) − (0) L∞ +
(0) L∞ @$kl @$n k1 l1 @$kl @$k1 l1 @$kl @$k1 l1
+ E(v) − E(w) L∞ ∇w 7=4 + E(v) − E(w) L∞ ] ∇h 2
∇(v − w) 7=4
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D.A. Vorotnikov, V.G. Zvyagin / Nonlinear Analysis 58 (2004) 631 – 656
6 K[( E(v) L∞ + K) ∇(v − w) 7=4 + E(v) − E(w) L∞ w 11=4 + E(v) − E(w) L∞ ] ∇h 2 6 K( v 11=4 ∇(v − w) 7=4 + ∇(v − w) 7=4 + ∇(v − w) 7=4 w 11=4 ) ∇h 2 6 K ∇B07=8 (v − w) 0 ∇h 2 :
(A.10)
For | | = 2 in a similar way one obtains 1=2 − A(w))h)
D ((A(v) 0 6 K ∇B0 (v − w) 1 ∇h 3 :
Estimates (A.8), (A.10), (A.11) yield estimate (4.19) for all l. Notations (3.5), (3.6) imply A(0) = I − 0 ". Therefore (A(0)v; B0 v)l ¿ 2K13 (B0 v; B0 v)l for some K13 ¿ 0. On the other hand, from estimate (4.19) it follows that
((A0 − A(0))v; B0 v)l 6 (A(a) − A(0))v l B0 v l 6 K11 B0 0 (l) a l B0 v 2l 6 K13 B0 v 2l for K12 small enough, what implies (4.21). Let u 3 ; v 3 ¡ K14 and K14 be small enough. Then estimate (4.21) yields (A(u)(u − v); B0 (u − v))1 ¿ K13 B0 (u − v) 21 But (4.19) implies |((A(u) − A(v))v; B0 (u − v))1 | 6 Kn 11 B07=8 (u − v) 1 B0 (u − v) 21 6
K13
B0 (u − v) 21 2
for K14 small enough and it yields (4.22). Now, note that ((A(0) − I )u; B0 u)2 = −(Su; B0 u)2 ¿ 2K15 ∇u 23 for some K15 ¿ 0. And estimate (4.19) and the triangle inequality yield |((A(u) − A(0))u; B0 u)|2 6 K11 ∇B01=2 u 1 ∇u 3 (I − ")u 2 6 K( ∇u 2 ∇u 3 u 2 + u 3 ∇u 3 Su 2 ) 6 K15 ∇u 23 for K14 small enough what implies (4.23).
(A.11)
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A.4. Proof of Lemma 4.6 Let B0 0 (l) a l ¡ K12 . Then estimate (4.19) implies
(A(a) − A(0))v l 6 K B0 v l : But for all ∈ C; Re ¿ 0
(A(0) + I )v l = (−0 " + ( + 1)I )v l 6 K21 () B0 v l and these two estimates yield (4.31). From (4.21) we have |((A0 + I )v; B0 v)l | ¿ |Re((A0 + I )v; B0 v)l | ¿ K13 (B0 v; B0 v)l :
(A.12)
This estimate and the Cauchy–Buniakowski inequality imply (4.32). From (4.32) it follows that Ker (A0 + I ) = {0}. Let us show that Im(A0 + I ) is dense in HVl . If not, then in HVl there is a nonzero vector h orthogonal to Im(A0 + I ). Let C = B0−1 h ∈ D(A0 ). Then (A.12) implies Re ((A0 + I )C; B0 C)l ¿ K13 (B0 C; B0 C)l ¿ 0:
(A.13)
But since (A0 + I )C ∈ Im (A0 + I ); B0 C = h, the left-hand side of (A.13) is equal to zero. Hence, Im (A0 + I ) is dense in HVl . Let us show now that Im (A0 + I ) is closed. Let hi → h0 ; hi ∈ Im (A0 + I ). i→∞
We have to show that h0 ∈ Im (A0 +I ). Note that there exist Ci such that (A0 +I )Ci = hi . Then the sequence (A0 +I )Ci converges in HVl . Then from (4.32) it follows that B0 Ci converges also in HVl : B0 Ci → $0 . Let C0 = B0−1 $0 ∈ D(A0 ). We have: B0 (Ci − C0 ) → 0. And estimate (4.31) yields (A0 + I )(Cn − C0 ) → 0, i.e. h0 = (A0 + I )C0 . Thus, A0 is a surjection. It remains to show (4.33). For || ¿ 1; u ∈ D(A0 ) we obtain 1 1 1 (u; u)l = Re 2 (u; u)l 6 Re 2 (u; u)l + Re 2 (A0 u; B0 u)l−1 || || || +Re = Re
1 1 (B0 u; A0 u)l−1 + Re 2 (A0 u; A0 u)l 2 || ||
1 ((A0 + I )u; (A0 + I )u)l ; ||2
whence
u l 6
1 2
(A0 + I )u l 6
(A0 + I )u l : || 1 + ||
(A.14)
Similarly,
u l 6 K (A0 + I )u l 6
2 K (A0 + I )u n l 1 + ||
Estimates (A.14), (A.15) imply (4.33).
(A.15)
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References [1] Ju.Ja. Agranovich, P.E. Sobolevskii, Motion of a nonlinear viscoelastic Huid, Dokl. Akad. Nauk. SSSR 314 (3) (1990) 521–525 (in Russian). [2] Yu.Ya. Agranovich, P.E. Sobolevskii, Motion of nonlinear viscoelastic Huid, Nonlinear Anal. TMA 32 (6) (1998) 755–760. [3] G. Astarita, G. Marucci, Principles of Non-Newtonian Fluid Hydromechanics, McGraw Hill, New York, 1974. [4] C. GuillopPe, J.C. Saut, Existence results for the fow of viscoelastic Huids with a di6erential constitutive law, Nonlinear Anal. TMA 15 (9) (1990) 849–869. [5] C. GuillopPe, J.C. Saut, Mathematical problems arising in di6erential models for viscoelastic Huids, in: J.F. Rodrigues, A. Sequeira (Eds.), Mathematical Topics in Fluid Mechanics, Longman, Harlow, 1993, pp. 64–92. [6] I. Gyarmati, Non-equilibrium thermodynamics, Field Theory and Variational Principles, Springer, Heidelberg, 1970. [7] T. Kato, Nonstationary Hows of viscous and ideal Huids in R3 , J. Funct. Anal. 9 (1972) 296–305. [8] M.A. Krasnoselskii, P.P. Zabreiko, E.I. Pustylnik, P.E. Sobolevskii, Integral Operators in the Spaces of Integrable Functions, Nauka, Moscow, 1966 (in Russian). [9] V.G. Litvinov, Motion of the Nonlinear-Viscous Fluid, Nauka, Moscow, 1982 (in Russian). [10] J.G. Oldroyd, On the formation of rheological equations of state, Proc. R. Soc. London A 200 (1950) 523–541. [11] J.G. Oldroyd, Non-Newtonian How of Huids and solids, in: F.R. Eirich (Ed.), Rheology: Theory and Applications, Academic Press, New York, 1956. [12] M. Reiner, Reology, in: S. Flugge (Ed.) Handbuch der Physik Bd. VI, Springer, Berlin, 1958. [13] P.E. Sobolevskii, On the equations of parabolic type in a Banach space, Trudy Moskov. Mat. Obshch. 10 (1961) 297–350. [14] R. Talhouk. Existence locale et unicitPe d’Pecoulements de Huides viscoPelastiques dans des domaines non bornPes, C.R. Acad. Sci. Paris, t.328, Ser. I, 1999, pp. 87–92. [15] R. Temam, Navier–Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1979. [16] C. Truesdell, A First Course in Rational Continuum Mechanics, The John Hopkins University, Baltimore, 1972. [17] W.L. Wilkinson, Non-Newtonian Fluids, Pergamon Press, Oxford, 1960.
www.elsevier.com/locate/na
On the solvability of the initial-value problem for the motion equations of nonlinear viscoelastic medium in the whole space D.A. Vorotnikov, V.G. Zvyagin∗ Research Institute of Mathematics, Voronezh State University, Universitetskaya Pl. 1, 394006 Voronezh, Russia Received 18 March 2004; accepted 1 May 2004
Abstract The paper deals with the initial-value problem for the system of motion equations of a wide class of incompressible nonlinear viscoelastic mediums, i.e. mediums possessing both nonlinear viscous and viscoelastic properties, in the whole space R2 or R3 . The existence and uniqueness of global in time solutions of this problem for small data are obtained. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Nonlinear viscoelastic medium; Frame-indi6erent derivative; Initial-value problem; Nonlinear evolution equation in a Banach space
1. Introduction It is known [6] that the motion of an incompressible medium with constant density = const is determined by the system of di6erential equations in the form of Cauchy n @u @u = Div T + f0 ; (t; x) ∈ [0; T] × Rn ; ui (1.1) + @t @xi i=1
div u = 0; (t; x) ∈ [0; T ] × Rn :
(1.2)
Here u is the velocity vector, f0 is the body force, T is the stress tensor (all of them depend on a point x of the space Rn ; n = 2; 3 and on a moment of time t). ∗
Corresponding author. Fax: +7-0732-208755. E-mail address: [email protected] (V.G. Zvyagin).
0362-546X/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.05.012
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The divergence div is taken with respect to the variable x. The divergence Div of a n @ tensor is the vector with the coordinates (Div )j = i=1 @xiji . Without loss of generality we can consider the density to be equal to 1. The type of a medium is determined by the choice of the constitutivelaw of the @u @ui relation between and the strain velocity tensor E(u)=(Eij (u)); Eij (u)= 12 @x + @xji . j For instance, one class of media is connected with the Stokes conjecture that the deviator of the stress tensor in every point is completely determined by the strain velocity tensor in the same point at the same moment of time. It is the conception of linear- and nonlinear-viscous Huid [9]. Prandtl and Eiring models [17] are examples of models of nonlinear-viscous Huids. However this conception is not satisfactory for all incompressible mediums. In particular, it is not suitable for media “with memory”: concrete, various polymers and solutions of polymers, the earth’s crust, etc. To take into account the e6ects of memory one may introduce time derivatives into the constitutive law. In the case of using this method the models of Maxwell, Je6reys [5,12], Oldroyd [10,11], Larson, Giesekus, Phan-Thien and Tanner [4,5], Spriggs [3] and a lot of other ones have appeared. The mathematical investigation of a part of these models was carried out in [4,14]. The models of nonlinear viscoelastic mediums, i.e. the media possessing both nonlinear viscosity and viscoelasticity, were investigated in [1,2]. In this work we study the motion equations for a much wider class of nonlinear viscoelastic mediums. This class contains a large number of existing models of linear and nonlinear-viscous, viscoelastic and nonlinear viscoelastic Huids and mediums, including all the models listed above. Namely, we research nonlinear viscoelastic media with the following combined constitutive law: T = s + p :
(1.3)
The tensor s is a nonlinear-viscous constituent of the stress tensor: s = −pI + (E(u));
(1.4)
where p is the pressure function, I is the identity matrix. The tensor p may consist of several viscoelastic (Maxwell) constituents with various relaxation times r k; (1.5) p = k=1
k + k
Da k + k ( k ; E) = 2k E: Dt
(1.6)
Here and k are arbitrary functions with values in the space of symmetric matrixes. The expression DDta A is the objective (frame-indi6erent) Oldroyd derivative of a tensor [3,5]. For a function A(x; t) with values in the set of matrices n × n it is deOned by the formula: n Da A @A @A ui + AW − WA − a(EA + AE) (1.7) = + Dt @t @xi i=1
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633
@uj @ui is the vorticity tensor, a is some and in this expression W = (Wij ); Wij = 12 @x − @xi j number, k ¿ 0 are the relaxation times, k ¿ 0 are the viscosities, k = 1; : : : ; r. It follows from the principle of material frame-indi6erence (see [3,16, Section 1 of chapter VI]), that without loss of generality of the model the functions and k ; k = 1; : : : ; r can be represented as (E) = ’1 E + ’2 E2 ;
(1.8)
k (; E) = 0k I + 1k E + 2k E2 + 3k + 4k 2 + 5k (E + E) + 6k (E2 + E2 ) + 7k (E2 + 2 E) + 8k (E2 2 + 2 E2 ); where ’1 ; ’2 and
jk 2
(1.9)
are scalar functions
’i = ’i (Tr (E ); det E);
i = 1; 2;
jk = jk (Tr E2 ; Tr E3 ; Tr(); Tr(2 ); Tr(3 ); Tr(E); Tr(2 E); Tr(E2 ); Tr(2 E2 )); k = 1; : : : ; r;
j = 0; : : : ; 8:
In this paper, we prove the global existence and uniqueness for small data of the solutions of the initial-value problem for the motion equations for the described above class of nonlinear viscoelastic mediums in the whole space Rn ; n = 2; 3. Let us mention some results concerning certain special cases of the considered problem. GuillopPe and Saut [4] proved the local in time existence and global existence for small data of the Dirichlet initial-boundary value problem for the system of motion equations of a viscoelastic Huid in a bounded domain in the case ’1 ≡ const; ’2 ≡ 0, i ≡ 0; i = 1; : : : ; r, and also for several speciOc functions i . Talhouk [14] generalized their local existence result (for r = 1; 1 ≡ 0) onto the case of unbounded domains. The motion equations of a nonlinear viscoelastic Huid were investigated by Agranovich and Sobolevskii [1,2] provided that the time derivative (1.7) was replaced by the partial derivative @t@ , what essentially narrows the class of mediums satisfying the model (see [11]). The plan of the paper is the following. In the second section the basic notations are introduced, the statement of the problem is carried out and the basic result is formulated. In the third section the operator treatment of the considered problem is realized. In the fourth section an auxiliary problem depending on a parameter is introduced and investigated. The existence of solutions of this problem and a uniform a priori estimate are proved. In the Ofth section the passage to the limit as the parameter tends to zero is carried out and the solution of the original problem is obtained. In the appendix some technical lemmas are proved. 2. Notations and the main result 2.1. Basic notations We will use the following notations:
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Denote by Rn×n the space of matrices of the order n × n with the following scalar product: for A = (Aij ); B = (Bij ) (A; B)Rn×n =
n
Aij Bij :
i; j=1
Denote by RSn×n its subspace of symmetric matrices. Denote by Rn×n×n the space of ordered collections of n matrices of the order n × n with the following scalar product: for A = (A1 ; : : : ; An ); B = (B1 ; : : : ; Bn ) (A; B)Rn×n×n =
n
(Ai ; Bi )Rn×n :
i=1
@p @p The symbol grad p will stand for the gradient @x of a function ; : : : ; @x n 1 n p : R → R. The symbol ∇u will stand for the Jacobi matrix of a vector function u : Rn → Rn . The symbol ∇ will denote the ordered collection of the Jacobi matrices of the columns of a matrix function : Rn → Rn×n . The symbol the " will denote n @2 @ n @1 Laplacian i=1 @x2 . The symbol D will denote the derivative @x 1 ; : : : ; @x nn where i 1 = ( 1 ; : : : ; n ) is a multi-index. The symbol ∇$ will stand for the Frechet derivative of functions or matrix functions of one or two matrix arguments ’ : RSn×n → R, : RSn×n × RSn×n → RSn×n , etc. The partial derivative of a function with the matrix argument $ = ($ij ) with respect to an element $ij will be denoted as @$@ij . Let us remind that if an operator B : D(B) → E with a dense in a Banach space E range of deOnition D(B) satisOes condition (4.4), it generates the analytical semigroup e−Bt ; t ¿ 0 then one can deOne its fractional powers [8,13]: ∞ 1 B = s− −1 e−Bt ds( ¡ 0); &(− ) 0 where & is the Euler function B = (B− )−1 ( ¿ 0);
B0 = I:
We shall use the function spaces of Sobolev type HVm = {u ∈ H m (Rn ; Rn ); div u = 0} and HMm = H m (Rn ; RSn×n ). The scalar product and the Euclidean norm in both spaces will be denoted as (·; ·)m ; · m . Let us remind that the scalar product in these spaces can be deOned by the equality (u; v)m = ((I − ")m=2 u; (I − ")m=2 v)L2 : In the case of natural m there is also another formula (u; v)m = (D u; D v)L2 : | |6m
The symbol P will stand for the Leray projection, Pu = u − "−1 ∇(div u). It is an orthogonal projection from H m (Rn ; Rn ) onto HVm .
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The symbols C([0; T ]; X ); L2 (0; T ; X ), etc. will denote the Banach spaces of continuous, quadratically integrable or other functions on the interval [0; T ] with values in some Banach space X . The symbol K will stand for various positive constants. 2.2. Statement of the initial-value problem and the main result Let us provide the statement of the initial-value problem for system (1.1)–(1.9) describing the motion of a nonlinear viscoelastic medium. For this purpose we have to add two conditions to (1.1)–(1.9). Generally, the pressure p may be determined up to an arbitrary scalar function of time. For deOniteness the following condition is imposed: p(t; x) d x ≡ 0; (2.1) 3
where 3 is some Oxed bounded domain in Rn . The initial conditions have the following form: u(0; x) = a(x);
k (0; x) = k0 (x);
x ∈ Rn ;
k = 1; : : : ; r:
(2.2)
Let us turn now to the formulation of the existence and uniqueness theorem for a solution of problem (1.1)–(1.9), (2.1)–(2.2). Let 0 = ’1 (0;0) 2 . Assume that 0 ¿ 0. This is a natural condition since the physical meaning of 0 is a viscosity parameter. Let us also assume that ’i and jk are C 4 - and C 3 -smooth functions, respectively, and 0k (4) = 1k (4) = 3k (4) = 0;
@ 0k (4) =0 @ Tr()
(4 stands for the point (0; 0; 0; 0; 0; 0; 0; 0; 0)). This assumption is also natural in the considered model since the functions k in it correspond to “nonlinear” e6ects, i.e. e6ects of the second order and higher. Therefore the coeQcients 1k and 3k at the Orst order terms E and should be “of the Orst order”, i.e. they should vanish in the point 4, and the coeQcient 0k at the zero-order term I should be of the second order, i.e. it should vanish in 4 and its partial derivative with respect to the “linear” argument Tr() should vanish in 4. Remarks. 1. Instead of the C 4 —smoothness of ’1 and ’2 it is suQcient for these functions to be di6erentiable in zero provided the function : RSn×n → RSn×n is C 4 -smooth or has locally Lipschitz third derivatives. 2. All these assumptions take place for the mentioned above models of Oldroyd, Larson, Giesekus, Phan-Thien and Tanner, Spriggs, Prandtl, Eiring and their combinations. Now we can formulate the main result.
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Theorem 2.1. Given, a ∈ HV3 ; k0 ∈ HM3 ; k = 1; : : : ; r; f0 ∈ L1 (0; T ; H 3 (Rn ; Rn )) ∩ L2 (0; T ; H 2 (Rn ; Rn )), there exists a constant K0 ¿ 0, independent on T, such that provided r
a 3 +
k0 3 + f0 L1 (0;T ;H 3 ) ¡ K0 (2.3) k=1
problem (1.1)–(1.9), (2.1)–(2.2) has a solution in the class u ∈ L2 (0; T ; HV4 ) ∩ C([0; T ]; HV3 ) ∩ W21 (0; T ; HV2 )
(2.4)
3 T ∈ L2 (0; T; HM; loc );
(2.5)
3 (Rn ; R)): p ∈ L2 (0; T ; Hloc
(2.6)
Furthermore, we have the following information about the constituents of the stress tensor: s + pI ∈ L2 (0; T ; HM3 ) ∩ C([0; T ]; HM2 ) ∩ W21 (0; T ; HM1 ) p ; k ∈ L∞ (0; T ; HM3 ) ∩ C([0; T ]; HM2 ) ∩ C 1 ([0; T ]; HM1 ); This solution is unique in class (2.4)–(2.8). If f0 ∈ L1 (0; +∞; H 3 (Rn ; Rn )) ∩ L2 (0; +∞; H 2 (Rn ; Rn )) and r
a 3 +
k0 3 + f0 L1 (0;+∞;H 3 ) ¡ K0
(2.7) k = 1; : : : ; r:
(2.8)
(2.9)
k=1
then problem (1.1)–(1.9), (2.1)–(2.2) has a unique solution in class (2.4)–(2.8) for every T ¿ 0. 3. Operator treatment of the problem In this section problem (1.1)–(1.9), (2.1)–(2.2), describing the motion of a nonlinear viscoelastic medium, will be rewritten as a Cauchy problem in a Banach space. Below for simplicity we shall consider r = 1 (concerning r ¿ 1, see Section 5.3). Let for brevity = 1 , = 1 , = 1 =1 . Let also 5(E) = (E) − 20 E:
(3.1)
Denote by g the function 1 (; E(u)) : (3.2) Note that Div pI = grad p and under condition (1.2) 2 Div E(u) = Su. We have from (1.1)–(1.7), (3.1)–(3.2) n @u @u ui + grad p − 0 Su − Div (5(E) + ) = f0 ; (3.3) + @t @xi i=1 n @ @ + + ui + g(; ∇u) = 2E: (3.4) @t @xi g(; ∇u) = W − W − a(E + E) +
i=1
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Let 5 = (5ij ). Let us introduce the following notations: (A(u)v) j =
n @5ij @Ekl (v) (E(u)) ; @$kl @xi
j = 1; : : : ; n;
(3.5)
i; k;l=1
(u)v + v − 0 Sv; A(u)v = −P A F1 (u; v) = −P
n i=1
ui
@v ; @xi
(3.6) (3.7)
n @ F(u; ) = − ui ; @xi
(3.8)
F(u) = F1 (u; u) + u;
(3.9)
G(u; ) = −g(; ∇u);
(3.10)
N1 () = P(Div );
(3.11)
N2 (u) = 2E(u);
(3.12)
B0 = I − ";
(3.13)
A0 = A(a);
(3.14)
f = Pf0 :
(3.15)
i=1
Consider the problem du + A(u)u = F(u) + N1 () + f; dt d + = F(u; ) + N2 (u) + G(u; ); dt u(0) = a;
(0) = 0 :
(3.16) (3.17) (3.18)
The statement of Theorem 2.1 will be deduced from the following result. Theorem 3.1. Given a ∈ HV3 ; 0 ∈ HM3 ; f ∈ L1 (0; T ; HV3 ) ∩ L2 (0; T ; HV2 ), there exists a constant K1 ¿ 0, independent on T, such that provided
a 3 + 0 3 + f L1 (0;T ;HV3 ) ¡ K1
(3.19)
problem (3.16)–(3.18) has a unique solution in the class u ∈ L2 (0; T ; HV4 ) ∩ C([0; T ]; HV3 ) ∩ W21 (0; T ; HV2 );
(3.20)
∈ L∞ (0; T ; HM3 ) ∩ C([0; T ]; HM2 ) ∩ C 1 ([0; T ]; HM1 ):
(3.21)
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4. Auxiliary problem 4.1. Solvability of an auxiliary problem Before proving Theorems 2.1 and 3.1, we investigate the solvability of an auxiliary problem. Introduce a family of operators A; () = − ;S; ; ¿ 0 (4.1) and consider the following equation: d (4.2) + A; = F(u; ) + N2 (u) + G(u; ): dt Theorem 4.1. Given a ∈ HV4 ; 0 ∈ HM4 ; f ∈ L1 (0; T ; HV3 ) ∩ C 1 ([0; T ]; HV2 ), there exists a constant K2 ¿ 0, independent on T and ;, such that provided
a 3 + 0 3 + f L1 (0;T ;HV3 ) ¡ K2 there is a unique solution of problem (3.16), (4.2), (3.18) in the class u ∈ C 1 ([0; T ]; HV2 ) ∩ C([0; T ]; HV4 ); ∈ C 1 ([0; T ]; HM2 ) ∩ C([0; T ]; HM4 ):
(4.3)
In order to prove this theorem we need some lemmas. 4.2. On an equation in a Banach space Lemma 4.1. Let ; ; R; M be some numbers, 0 6 ¡ 6 1. Let B : D(B) → E be a linear operator in a Banach space E; D(B) = E, M ; ∈ C; Re ¿ 0: (4.4)
(B + I )−1 6 1 + || For every v0 such that v0 ∈ D(B );
B v0 ¡ R;
(4.5)
let a linear operator A(v0 ) = A0 be de>ned on D(A0 ) = D(B) and for all v ∈ D(B): K3 A0 v 6 Bv 6 K4 A0 v ;
(4.6)
where K3 ; K4 do not depend on v; v0 . Let also M
(A0 + I )−1 6 ; Re ¿ 0: 1 + ||
(4.7)
For every v ∈ D(B ) such that B v ¡ R let a linear operator A(v)(·) be de>ned on D(B) and
(A(v) − A(w))A0−1 6 K5 B (v − w) ;
(4.8)
where K5 does not depend on v; w ∈ D(B ) such that B v ; B w ¡ R.
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Let f1 : [0; T ] × D(B ) → E and
f1 (t; v) − f1 (s; w) 6 K6 ( B (v − w) + |t − s|);
(4.9)
where K6 does not depend on t; s and v; w ∈ D(B ) such that B v ; B w ¡ R. Then for all v0 ∈ D(B) satisfying (4.5), there exists a solution v of the problem dv (4.10) + A(v)v = f1 (t; v); dt (4.11) v(0) = v0 in the class v ∈ C 1 ([0; t0 ]; E);
Bv ∈ C([0; t0 ]; E);
(4.12)
where t0 = t0 (v0 ; f1 ) ¿ 0 is some number. If for every such solution it is a priori known that it is bounded in the following sense:
B v(t) ¡ K7 ;
t ∈ [0; t0 ];
(4.13)
where K7 does not depend on t0 and t, then t0 = T . Remarks. The space E may be both complex and real. In the latter case, while checking conditions (4.4) and (4.7) it is necessary to consider the complexiOcation of the space E and the corresponding extensions of operators A and B. 2. The statement of Lemma 4.1 on the solvability of problem (4.10)–(4.11) on an interval [0; t0 ] is a particular case of Theorem 7 from [13]. If there is a priori estimate (4.13) then the solution may be continued on the whole segment [0; T ] by standard procedure. 4.3. Operator estimates We will need the following estimates for the operators introduced in Section 3. Lemma 4.2. The following estimates take place: For l = 1; 2; u ∈ HV2 ; v ∈ HVl+1 ; ∈ HMl+1 :
F1 (u; v) l 6 K8 u 2 v l+1 ;
(4.14)
F(u; ) l 6 K8 u 2 l+1 : For l = 2; 3;
u ∈ HV3 ;
v ∈ HVl+1 ;
(4.15) ∈ HMl+1 :
|(F1 (u; v); v)l | 6 K9 ∇u 2 ∇v l−1 ; |(F(u; ); )l | 6 K9 ∇u 2 ∇ l−1 : For l = 0; 1 and > =
7 4
(4.16)
or l = > = 2:
G(u1 ; 1 ) − G(u2 ; 2 ) l 6 K10 ( ∇u1 − ∇u2 l + 1 − 2 l ) ×( ∇u1 > + ∇u2 > + 1 > + 2 > ); where u1 ;
u2 ∈ HV>+1 ;
1 ; 2 ∈ HM> .
(4.17)
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For u ∈ HV4 ; ∈ HM3 ( u 3 + 2 )( ∇u 3 + 3 );
G(u; ) 3 6 K10
(4.18)
where K10 and K10 depend continuously on ( ∇u1 > + ∇u2 > + 1 > + 2 > ) and on ( u 3 + 2 ), respectively.
Lemma 4.3. Let l = 0; 1; 2 and 0 (0) = (1) If
11 8 ; 0 (1)
= 78 ; 0 (2) = 12 .
B0 0 (l) v l ; B0 0 (l) w l ¡ 1 then the following estimates take place:
(A(v) − A(w))h l 6 K11 ∇B0 0 (l) (v − w) l−1 ∇h l+1 ;
(A(v) − A(w))h l 6 K11 B0 0 (l) (v − w) l B0 h l :
(4.19)
Furthermore, for l = 0 there is another variant of the second estimate
(A(v) − A(w))h 0 6 K11 v − w 3 h 3 : 2
(4.20)
(2) There exist K12 ; K13 ; K14 ; K15 ¿ 0 such that provided B0 0 (l) a l ¡ K12 (A0 v; B0 v)l ¿ K13 B0 v 2l
(4.21)
and provided u 3 ; v 3 ¡ K14 (A(u)u − A(v)v; B0 (u − v))1 ¿ 12 K13 ; B0 (u − v) 21 ;
(4.22)
((A(u) − I )u; B0 u)2 ¿ K15 ∇u 23 :
(4.23)
The proofs of these two lemmas will be carried out in the appendix. 4.4. Uniqueness lemma Lemma 4.4. There exists a constant K16 ¿ 0, independent on T and ;, such that if a solution (u1 ; 1 ) of problem (3.16), (4.2), (3.18) exists in the class u1 ∈ C 1 ([0; T ]; HV1 ) ∩ C([0; T ]; HV3 ); 1 ∈ C 1 ([0; T ]; HM1 ) ∩ C([0; T ]; HM3 )
(4.24)
and
u1 (t) 3 ¡ K16 ;
t ∈ [0; T ]
(4.25)
then it is unique in class (4.24). Proof. Assume that in addition to (u1 ; 1 ) there is another solution (u2 ; 2 ) of problem (3.16), (4.2), (3.18).
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Suppose Orst that
u2 (t) 3 6 K16 ;
t ∈ [0; T ]
(4.26)
(K16 will be deOned below). The following identity takes place: 1 (N1 (); u)m + (N2 (u); )m = 0; 2 u ∈ HVm+1 ;
∈ HMm+1 ;
m = 0; 1; : : : :
(4.27)
Indeed, using Green formula we have 1 (N1 (); u)m + (N2 (u); )m 2
n
@ij @uj 1 @ui ui ; = + ij ; + @xj m 2 @xj @xi m i; j=1
n
@ij @ui ui ; + ij ; = @xj m @xj m i; j=1
n
=
i; j=1;| |6m
Rn
@ij @ui D ui D + D ij D @xj @xj
d x = 0:
Let w = u1 − u2 ; ∗ = 1 − 2 . Substitute (u1 ; 1 ) and (u2 ; 2 ) into (3.16) and take the di6erence of the obtained equalities. Treat (4.2) in the same way. We have dw(t) = A(u2 )u2 − A(u1 )u1 + u1 − u2 + F1 (w; u1 ) + F1 (u2 ; w) + N1 (∗ ); dt d∗ (t) = −A; (∗ ) + F(w; 1 ) + F(u2 ; ∗ ) + N2 (w) + G(u1 ; 1 ) − G(u2 ; 2 ): dt Taking the scalar product of the Orst equation with B0 w(t) in HV1 and of the second 1 B0 ∗ (t) in HM1 at every t ∈ [0; T ], adding the obtained equalities and taking one with 2 into account (4.27) we get
dw 1 d∗ ; B0 w + ; B0 ∗ dt 2 dt 1 1 = − (A(u1 )u1 − A(u2 )u2 ; B0 w)1 + (w; B0 w)1 + (F1 (w; u1 ); w)2 + (F1 (u2 ; w); w)2 +
1 [ − (A; (∗ ); B0 ∗ )1 + (F(w; 1 ); ∗ )2 2
+(f(u2 ; ∗ ); ∗ )2 + (G(u1 ; 1 ) − G(u2 ; 2 ); ∗ )2 ]: Obviously, (A; (∗ ); B0 ∗ )1 =
∗
− ;S∗ ; ∗
2
1 ¿ ∗ 2 :
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Now, using estimates (4.14)–(4.17), (4.22) (we can assume K16 to be small enough so that estimate (4.22) is also valid) we obtain d 1 1 d
w 22 +
∗ 22
∗ 22 + K14 w 23 + 2 dt 2 dt 6 K17 [ w 22 (1 + u1 3 + u2 3 ) + w 2 1 3 ∗ 2 + u2 3 ∗ 22 + ∗ 2 ( ∗ 2 + w 3 )( 1 2 + 2 2 + u1 3 + u2 3 )]: It is obvious that there exists K18 such that K14 w 23 + K18 ∗ 22 − K17 ∗ 2 w 3 ( 1 2 + 2 2 + u1 3 + u2 3 ) ¿ 0: Therefore, d 1 d
∗ 22
w 22 + dt 2 dt
1
∗ 22 + Kn 17[ w 22 (1 + u1 3 + u2 3 ) 6 K18 − 2 + w 2 1 3 ∗ 2 + ∗ 22 u2 3 + ∗ 22 ×( 1 2 + 2 2 + u1 3 + u2 3 )]
1 6 K w 22 + ∗ 22 : 2 Since w(0) = 0; ∗ (0) = 0, by Gronwall lemma w ≡ 0; ≡ 0. If (4.26) is not valid for some t, let t1 be the inOmum of such t. It is evident that
u2 (t1 ) 3 = K16 . But on the other hand, we have just proved that on [0; t1 ] the solution is unique. Hence u1 (t1 ) = u2 (t1 ); 1 (t1 ) = 2 (t1 ) what contradicts (4.25). 4.5. A priori estimate Lemma 4.5. There exists a constant K19 ¿ 0 (generally speaking, it is small enough), independent on T and ; such that provided 1
a 3 + √ 0 3 + f L1 (0;T ;HV3 ) ¡ K19 2 there are the following estimates for every solution of (3.16), (4.2), (3.18) in class (4.3): 1 2
u(t) 23 + (t) 23 ¡ K19 ; (4.28) 2 T 3K 2
∇u 23 ds ¡ 19 : (4.29) K15 0 Proof. Denote by t1 = t1 (K19 ) the inOmum of those t at which (4.28) is not valid (K19 will be deOned later).
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1 B0 (t) Taking the scalar product of (3.16) with B0 u(t) in HV2 and of (4.2) with 2 2 in HM at every t ∈ [0; t1 ], adding the obtained equalities and taking into account (4.27) we get
du 1 d ; B0 u + ; B0 dt 2 dt 2 2
= − (A(u)u; B0 u)2 + (u; B0 u)2 + (F1 (u; u); B0 u)2 +
1 [ − (A; (); B0 )2 2
+(F(u; ); B0 )2 + (G(u; ); B0 )2 ] + (f; B0 u)2 : Then we have 1 d 1 1 d
u 23 +
23 6 −((A(u) − I )u; B0 u)2 + (F1 (u; u); u)3 −
23 2 dt 4 dt 2 +
1 1 (F(u; ); )3 + (G(u; ); )3 + (f; u)3 : 2 2
Using (4.16), (4.18), (4.23) (we can assume K19 to be small enough so that estimate (4.23) is also valid) we obtain 1 d 1 d 1 K9
u 23 +
23 6 −K15 ∇u 23 −
23 + K9 ∇u 32 +
∇u 2 23 2 dt 4 dt 2 2 +
1 K ( ∇u 3 + 3 )( u 3 + 2 ) 3 + f 3 u 3 : 2 10
For K19 small enough it yields 1 d 1 1 d
u 23 +
23 6 f 3 u 3 − K15 ∇u 23 : 2 dt 4 dt 2 Let
(t) = 1 d 2 dt
2
||u||23 (t) +
1 2 2 ||||3 (t)
(t) 6 f 3 (t);
. Then from (4.30) it follows that
t ∈ [0; t1 ];
d (t) 6 f 3 : dt Integrate from 0 to t1 : (t1 ) 6
||a||23
1 + ||0 ||23 + 2
1 6 a 3 + √ 0 3 + 2
t1
0
0
T
f 3 (s) ds
f 3 (s) ds ¡ K19 ;
what contradicts the deOnition of t1 . Thus, (4.28) is valid for all t ∈ [0; T ].
(4.30)
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D.A. Vorotnikov, V.G. Zvyagin / Nonlinear Analysis 58 (2004) 631 – 656
Integrating (4.30) from 0 to T we obtain 1 1 1 1
u 23 (T ) + 23 (T ) − a 23 − 0 23 2 4 2 4 T 1 6 ( f 3 u 3 − K15 ∇u 23 ) ds: 2 0 This inequality and (4.28) imply T T 1 1 1 2 2 2 K15
∇u 3 ds 6 a 3 + 0 n 3 +
f 3 u 3 ds 2 2 4 0 0 T 1 2 3 2 6 K19 + K19
f 3 ds 6 K19 2 2 0 and we obtain estimate (4.29). 4.6. Properties of the operator A0 Lemma 4.6. Under the conditions and notations from Lemma 4.3 there exist constants K20 , K21 , K22 such that provided B0 0 (l) a l ¡ K12 ; ∈ C; Re ¿ 0 the following estimates are valid:
(A0 + I )v l 6 K20 () B0 v l ;
(4.31)
(A0 + I )v l ¿ K21 B0 v l :
(4.32)
Under the same conditions and l = 1; 2 the operator A0 : HVl+2 D(A0 ) ⊂ HVl → HVl is invertible and
(A0 + I )−1 6
K22 : 1 + ||
(4.33)
Remarks. 1. Here we have to use complexiOcations of Sobolev spaces and di6erential operators (see Remark 1 after Lemma 4.1). 2. The proof of this lemma will be carried out in the appendix. 4.7. Proof of Theorem 4.1 Let us put E = HV1 × HM1 ;
v0 = (a; 0 );
f1 (t; (u; )) = (f(t) + F(u) + N1 (); F(u; ) + N2 (u) + G(u; )); B(u; ) = (B0 u; B0 );
D(B) = HV3 × HM3 ;
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645
A(u; )(v1 ; v2 ) = (A(u)v1 ; A; v2 ); A0 (v1 ; v2 ) = (A0 v1 ; A; v2 ); 7 = ; =1 8 in Lemma 4.1 and let R be small enough. Then estimate (4.4) is a simple property of operator I − ", estimate (4.6) follows from (4.31) and (4.32), estimate (4.7) follows from (4.33), estimate (4.8) follows from (4.19) and estimate (4.9) follows from (4.14), (4.15) and (4.17). Thus, the conditions of Lemma 4.1 hold. Hence, for K2 small enough, system (3.16), (4.2), (3.18) has a solution (u; ) in class (4.24) on some interval [0; t0 ]. Moreover, from regularity properties of solutions of abstract parabolic equations [13, p. 324] it follows that for some A1 ¿ 0 the function A0A1 (u; ) : [0; t0 ] → E is continuously di6erentiable with respect to t and the function A0A1 A(u; )(u; ) : [0; t0 ] → E is continuous. But from (4.6) it follows (see [8, Section 14]) that for all A ¡ A1 the operator B1−A1 +A [A(u; )]−1+A1 A0−A1 is bounded. It yields the estimate
B1=2−A1 +A [A(u; )]A1 (u; ) 2 ¡ K;
t ∈ [0; t0 ];
(4.34)
where K does not depend on t. Let us put now E = HV2 × HM2 ; = 12 ;
=
1 2
D(B) = HV4 × HM4 ; + A2 (0 ¡ A2 ¡ A)
in Lemma 4.1 and let v0 ; f1 ; A; B have the given above form. Estimates from Lemmas 4.2, 4.3, 4.6 imply again that the conditions of Lemma 4.1 hold. Hence, problem (3.16), (4.2), (3.18) has a solution in class (4.3) on some interval [0; t∗ ]. Furthermore, estimate (4.28) with K19 small enough enables to apply Lemma 4.4. Therefore this solution is unique in class (4.24) on [0; t∗ ], i.e. it coincides with (u; ). Then estimate (4.34) and the boundedness of the operator B [A(u; )]−A1 B−(1=2)+A1 −A imply that estimate (4.13) is valid for all t ∈ [0; t∗ ]. Hence, t∗ = t0 , i.e. the solution (u; ) belongs to class (4.3) on the segment [0; t0 ]. From estimate (4.28) it follows that this solution is a priori bounded by a constant independent on t and t0 :
B(u; )(t) HV1 ×HM1 6 K;
t ∈ [0; t0 ]:
Therefore, we can consider t0 to be equal to T . The proof is fulOlled. 5. Proof of the main theorems 5.1. Proof of Theorem 3.1 Evidently, there exist sequences am → a m→∞
in
HV3 ; am ∈ HV4 ;
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D.A. Vorotnikov, V.G. Zvyagin / Nonlinear Analysis 58 (2004) 631 – 656
in
0m → 0 m→∞
fm → f m→∞
in
HM3 ;
0m ∈ HM4 ;
L1 (0; T ; HV3 );
fm ∈ L1 (0; T ; HV3 ) ∩ C 1 ([0; T ]; HV2 ):
Without loss of generality all triplets (am ; 0m ; fm ) satisfy estimate (3.19). Consider problems (3.16), (4.2), (3.18) with the data am ; 0m ; fm and ; = m1 for all natural m. By Theorem 4.1 for K1 small enough such problems have a unique solution (um ; m ) in class (4.3) and all these solutions are bounded uniformly with respect to m by estimates (4.28), (4.29). Therefore, without loss of generality we may assume that um → u∗
in
L∞ (0; T ; HV3 ) ∗ −weakly;
m → ∗
in
L∞ (0; T ; HM3 ) ∗ −weakly;
um → u∗
in
L2 (0; T ; HV4 ) weakly:
Let us show that the sequence (um ; m ) is fundamental in C([0; T ]; HV2 × HM2 ). Let wij = ui − uj ; ij = i − j . Apply the same procedure as in the proof of Lemma 4.4: substitute (ui ; i ) into (3.16), (4.2) with ; = 1i and (uj ; j ) into (3.16), (4.2) with ; = 1j , take the di6erences of the corresponding equations, take the scalar product of the Orst of the obtained equations with wij (t) in HV2 and of the second one with 2ij in HM2 at every t ∈ [0; T ], and taking into account (4.27) add the obtained equalities:
dwij 1 dn ij ; wij + ; ij dt 2 dt 2 2 = − (A(ui )ui − A(uj )uj ; B0 wij )1 + (wij ; wij )2 +(F1 (wij ; ui ); wij )2 + (F1 (uj ; wij ); wij )2 − 1 + 2
Sj Si − ; ij i j
2
1 (ij ; ij )2 2
+ (F(wij ; i ); ij )2 + (F(uj ; ij ); ij )2
+(G(ui ; i ) − G(uj ; j ); ij )2 + (fi − fj ; wij )2 : Let K1 be small enough. Using the estimates and arguing as in the proof of Lemma 4.4 we conclude: 1 d d
wij 22 +
ij 22 dt 2 dt
1 1 1 2 2 6 K wij 2 + ij 2 + fi − fj 2 + i 3 ij 3 + j 3 ij 3 2 i j
1 1 1 : 6 K wij 22 + ij 22 + fi − fj 2 + + 2 i j
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Integrate from 0 to t 1 1
wij 22 + ij 22 6 ai − aj 22 + 0i − 0j 22 2 2 T
1 1 1 2 2 ds
wij 2 (s) + ij 2 (s) + fi − fj 2 (s) + + +K i j 2 0 and by Gronwall lemma wij 22 (t) + t ∈ [0; T ]. Thus, um → u∗
in
C([0; T ]; HV2 );
m → ∗
in
C([0; T ]; HM2 ):
1
ij 22 (t) → 0 uniformly with respect to min(i; j)→∞ 2
This implies 1 Sm → 0 in C([0; T ]; HM0 ); m ∗ in C([0; T ]; HM0 ); A1=m m → N1 (m ) → N1 (∗ )
in
C([0; T ]; HV1 );
N2 (um ) → N2 (u∗ )
in
C([0; T ]; HM1 );
du∗ dum → in the sense of distributions; dt dt dm d∗ → in the sense of distributions: dt dt From estimates (4.14), (4.15), (4.17) we have also ∗ ) in C([0; T ]; HV1 ); m ) → F(u F(u F(um ; m ) → F(u∗ ; ∗ )
in
C([0; T ]; HM1 );
G(um ; m ) → G(u∗ ; ∗ )
in
C([0; T ]; HM1 ):
Furthermore, using (4.19), (4.20) we obtain vrai max A(um )um − A(u∗ )u∗ 0 (t) t∈[0;T ]
6 vrai max ( A(0)(um − u∗ )(t) 0 t∈[0;T ]
+ (A(u∗ ) − A(0))(um − u∗ ) 0 (t) + (A(um ) − A(u∗ ))um 0 (t)) 6 Kvrai max ( um − u∗ 2 + u∗ 11=4 um − u∗ 2 + um − u∗ 3=2 um 3 ): t∈[0;T ]
Therefore, A(um )um → A(u∗ )u∗
in
L∞ (0; T ; HV0 ):
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D.A. Vorotnikov, V.G. Zvyagin / Nonlinear Analysis 58 (2004) 631 – 656
Passing to the limit we conclude that the pair (u∗ ; ∗ ) is a solution of problem (3.16)– (3.18). Estimate (4.19) yields that T T T 2 2
A(u∗ )u∗ 2 6 2
(A(u∗ ) − A(0))u∗ 2 + 2
A(0)u∗ 22 0
0
6K
0
0
T
( u∗ 23 u∗ 24 + u∗ 24 ):
∗) ∈ Therefore, A(u∗ )u∗ belongs to L2 (0; T ; HV2 ). From estimate (4.14) we have F(u L∞ (0; T ; HV2 ). Note that N1 (∗ ) ∈ L∞ (0; T ; HV2 ) and f ∈ L2 (0; T ; HV2 ). Substituting (u∗ ; ∗ ) into (3.16), we conclude that dudt∗ ∈ L2 (0; T ; HV2 ). But u∗ ∈ L2 (0; T ; HV4 ) and by Lemma III.1.2 from [15] we have u∗ ∈ C([0; T ]; HV3 ). Substituting u∗ ; ∗ into (3.17) and taking into account estimates (4.14), (4.15) and (4.17) we conclude that all terms in (3.17) except ddt∗ belong to C([0; T ]; HM1 ). Hence, ∗ ∈ C 1 ([0; T ]; HM1 ). The uniqueness of the solution may be proved in exactly the same way as Lemma 4.4. 5.2. Proof of Theorem 2.1 Let us note that estimates (4.17), (4.18) in the particular case G(u; ) = 5(E(u)) have the form
5(E(u1 )) − 5(E(u2 )) l 6 K23 ( u1 >+1 + u2 >+1 ) u1 − u2 l+1 ;
(5.1)
5(E(u)) 3 6 K23 ( u 3 ) u 4 ;
(5.2)
where K23 , K23 continuously depend on the corresponding norms and l; > are as in estimate (4.17). Let 0 = 10 ; K0 = K1 . Then (2.3) implies estimate (3.19). Hence, by Theorem 3.1 there exists a unique solution (u; ) of problem (3.16)–(3.18) in class (3.20), (3.21). Taking into account notations (3.5)–(3.15) we conclude that (3.4) is valid and n @u @u P ui − 0 Su − Div (5(E) + ) − f0 = 0: + @t @xi i=1
From Prepositions I.1.1 and I.1.2 from [15] it follows that there exists a unique p from class (2.6) such that equalities (3.3) and (2.1) hold. Let p = 1 = ;
s = −pI + 20 E(u) + 5(E(u));
T = p + s :
Obviously, the triplet (u; T; p) is a solution of problem (1.1)–(1.9), (2.1)–(2.2). Since u belongs to class (2.4), estimates (5.1) and (5.2) yield that 5(E(u)) belongs to class (2.7). Hence, pI + s belongs to class (2.7), and T belongs to (2.5).
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5.3. The case r ¿ 1 The case r ¿ 1 is investigated exactly in the same way as the case r = 1. During the operator treatment, problem (3.16)–(3.18) is replaced by the problem du (5.3) + A(u)u = F(u) + N1 () + f; dt di i (5.4) + = F(u; i ) + N2 (u) + Gi (u; i ); dt u(0) = a; where
i (0) = i0 ;
i = 1; : : : ; r;
(5.5)
1 (; ∇u) : Gi (u; ) = − W − W − a(E + E) +
Then one considers the following system: di + A; i = F(u; i ) + N2 (u) + Gi (u; i ) (5.6) dt and the auxiliary problem (5.3), (5.6), (5.5). Then the analogues of Theorem 4.1 and Lemmas 4.4, 4.5 are proved and the passage to the limit is carried out just as in the case r = 1. Acknowledgements The work was partially supported by Grants 04-01-00081 of Russian Foundation of Basic Research and VZ-010-0 of the Ministry of Education of Russia and CRDF. Appendix A A.1. Some inequalities Below we need some well-known inequalities which hold both for scalar and for vector functions. The following statement takes place: for n = 2; 3 and 3p − 6 6m 2p one has H m ⊂ Lp ; If m ¿
3 2,
u Lp 6 K(m; p) u m :
the functions from H
u(x) 6 K(m) u m ;
m
(A.1)
are uniformly bounded:
x ∈ Rn :
It is a particular case of the Sobolev embedding theorem [15].
(A.2)
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D.A. Vorotnikov, V.G. Zvyagin / Nonlinear Analysis 58 (2004) 631 – 656
Let us mention also an inequality of HUolder type:
uv 0 6 K(p; q) u Lp v Lq
(A.3)
provided 1 1 1 + = : p q 2 These inequalities for n = 2; 3 imply the following well-known inequalities:
uv 0 6 K u 2 v 0 ;
(A.4)
uv 0 6 K u 1 v 1 ;
(A.5)
uv 1 6 K u 1 v 2 ;
(A.6)
uv 2 6 K u 2 v 2 :
(A.7)
A.2. Proof of Lemma 4.2 Estimates (4.14)–(4.16) are proved in the usual way (similar estimates are present, e. g. in [7, p. 302]). We shall prove them here for the reader’s convenience. Using (A.6) we have n n @v @v
F1 (u; v) 1 6
ui 2 ui @xi 6 K @xi 6 K u 2 v 2 : 1 1 i=1
i=1
Using (A.7) we have
n n @v @v
F1 (u; v) 2 6
ui 2 ui @xi 6 K @xi 6 K u 2 v 3 : 2
i=1
2
i=1
Estimate (4.14) is proved. Similarly one proves (4.15). Let us prove now estimate (4.16). We have, using the Leibnitz rule n
@v |(F1 (u; v); v)2 | 6 ui D @xi ; D v 0 | |62 i=1
n
@v 6 ui D @xi ; D v 0 | |62 i=1
+
n
ui D @v ; D v @xi 0
0¡| |62 i=1
+
n
| |=2;||=1;6 i=1
@v K( ; ) D ui D − ;D v : @xi 0
D.A. Vorotnikov, V.G. Zvyagin / Nonlinear Analysis 58 (2004) 631 – 656
651
Since div u = 0, integration by parts easily shows that the Orst sum is equal to zero. With the help of (A.5) and Cauchy–Buniakowski inequality we have that the second sum does not exceed the expression n @v K ∇ui 2 @xi ∇v 1 : 1
i=1
With the help of (A.4) we have that the third sum does not exceed the same expression. Estimate (4.16) for the case l = 2 is proved. The case l = 3 is examined analogously. Let us prove now estimate (4.17). It follows from conditions on coeQcients jk from Section 2.2 that the function 1 : RSn×n × RSn×n → RSn×n is C 3 -smooth and 1 (0; 0) = 0; ∇$ 1 (0; 0) = 0. Then the function g : RSn×n × Rn×n → RSn×n introduced by formula (3.2) is also 3 C -smooth and g(0; 0) = 0; ∇$ g(0; 0) = 0. We have
D (G(u1 ; 1 ) − G(u2 ; 2 )) 0 :
G(u1 ; 1 ) − G(u2 ; 2 ) l = | |6l
First we estimate the term with | | = 0. Using inequality (A.2) and the Lagrange theorem one has
G(u1 ; 1 ) − G(u2 ; 2 ) 0 = g(∇u1 ; 1 ) − g(∇u2 ; 2 ) L2 6
=
sup
$1 6 ∇u1 L∞ + ∇u2 L∞ $2 6 1 L∞ + 2 n L∞
sup
$1 6 ∇u1 L∞ + ∇u2 L∞ $2 6 1 L∞ + 2 L∞
∇$ g($1 ; $2 ) ( ∇(u1 − u2 ) L2 + 1 − 2 L2 )
∇$ g($1 ; $2 ) − ∇$ g(0; 0) ( ∇(u1 − u2 ) L2 + 1 − 2 L2 )
6 K( ∇u1 L∞ + ∇u2 L∞ + 1 L∞ + 2 L∞ )( ∇(u1 − u2 ) L2 + 1 − 2 L2 ) 6 K( ∇u1 7=4 + ∇u2 7=4 + 1 7=4 + 2 7=4 )( ∇(u1 − u2 ) 0 + 1 − 2 0 ): Now we estimate the Orst derivatives, using inequalities (A.1)–(A.3)
∇(G(u1 ; 1 ) − G(u2 ; 2 )) 0 = ∇$ g(∇u1 ; 1 )(∇2 u1 ; ∇1 ) − ∇$ g(∇u2 ; 2 )(∇2 u2 ; ∇2 ) 0 6 6 ∇$ g(∇u1 ; 1 )(∇2 (u1 − u2 ); ∇(1 − 2 )) 0 + (∇$ g(∇u1 ; 1 ) − ∇$ g(∇u2 ; 2 ))(∇2 u2 ; ∇2 ) 0 6 ∇$ g(∇u1 ; 1 ) L∞ ( ∇(u1 − u2 ) 1 + 1 − 2 1 ) + ∇$ g(∇u1 ; 1 ) − ∇$ g(∇u2 ; 2 ) L6 ( ∇2 u2 L3 + ∇2 L3 ) 6 ∇$ g(∇u1 ; 1 ) − ∇$ g(0; 0) L∞ ( ∇(u1 − u2 ) 1 + 1 − 2 1 )
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D.A. Vorotnikov, V.G. Zvyagin / Nonlinear Analysis 58 (2004) 631 – 656
+ ∇$ g(∇u1 ; 1 ) − ∇$ g(∇u2 ; 2 ) L6 ( ∇2 u2 L3 + ∇2 L3 ) 6 K( ∇u1 L∞ + 1 L∞ )( ∇(u1 − u2 ) 1 + 1 − 2 1 ) +K( ∇(u1 − u2 ) L6 + 1 − 2 L6 )( ∇2 u2 1=2 + ∇2 1=2 ) 6 K( ∇(u1 − u2 ) 1 + 1 − 2 1 )( ∇u1 7=4 + 1 7=4 + ∇u2 7=4 + 2 7=4 ): In the same way the terms D (G(u1 ; 1 ) − G(u2 ; 2 )) 0 with | | = 2 are estimated and we obtain estimate (4.17). Similarly one obtains estimate (4.18).
A.3. Proof of Lemma 4.3 Note that from the C 4 —smoothness of coeQcients ’i (see Section 2.2) it follows that the function 5 : RSn×n → RSn×n introduced by formula (3.1) is C 4 -smooth. Moreover, it is easy to check that 5(0) = 0; ∇$ 5(0) = 0. We have − A(w))h
(A(v) − A(w))h l = (A(v) l6K
− A(w))h
D (A(v) 0:
| |6l
First we estimate the term with | | = 0: − A(w))h
(A(v) 0
n @5ij @5ij @Ekl (h) 6 (E(v)) − (E(w)) @$kl @xi 0 @$kl i; j; k;l=1
n @5ij @5ij (E(v)) − (E(w)) @$kl @$kl L∞
6
i; j; k;l=1
@Ekl (h) 6 K E(v) − E(w) L ∇h 1 × ∞ @xi L2 6 K E(v) − E(w) 7=4 ∇h 1 6 K ∇B011=8 (v − w) −1 ∇h 1 : Furthermore, − A(w))h
(A(v) 0
n @5ij @5ij @Ekl (h) 6
(E(v)) − (E(w))
0 @$kl @$kl @xi i; j; k;l=1
(A.8)
D.A. Vorotnikov, V.G. Zvyagin / Nonlinear Analysis 58 (2004) 631 – 656
653
n @5ij @5ij 6 @$kl (E(v)) − @$kl (E(w)) L3 i; j; k;l=1
@Ekl (h) × @xi 6 K E(v) − E(w) L3 h 3 L6 6 K v − w 3=2 h 3
(A.9)
what implies (4.20). For | | = 1 we obtain − A(w))h)
D ((A(v) 0
n @5ij @5ij @Ekl (h) 6 (E(v)) − (E(w)) D @$kl @$kl @xi 0 i; j; k;l=1
@5ij @5n ij @Ekl (h) + (E(v)) − (E(w)) D @$kl @$kl @xi 0
n @5ij @5ij @5ij @Ekl (h) 6
D (E(v)) − (E(w)) L3
L 6 +
(E(v)) @$kl @$kl @xi @$kl i; j; k;l=1
@5ij @Ekl (h) − (E(w)) L∞ D
L 2 @$kl @xi n n @2 5ij 6 K
(E(v))D Ek1 l1 (v) @$kl @$n k1 l1 k1 ;l1 =1
i; j; k;l=1
−
6
2
@ 5ij (E(w))D Ek1 l1 (w) L3 ∇h 2 + E(v) − E(w) L∞ ∇h 2 @$kl @$n k1 l1
n
K
i; j; k;l=1
2 @ 5ij + @$kl @$k
n
k1 ;l1 =1
@2 5ij @$kl @$k
1 l1
(E(v))
L∞
D (Ek1 l1 (v) − Ek1 l1 (w)) L3
@2 5ij (E(v)) − (E(w)) D Ek1 l1 (w) L3 @$kl @$n k1 l1 1 l1 L∞
+ E(v) − E(w) L∞
6 K[
@2 5ij @2 5ij @2 5ij (E(v)) − (0) L∞ +
(0) L∞ @$kl @$n k1 l1 @$kl @$k1 l1 @$kl @$k1 l1
+ E(v) − E(w) L∞ ∇w 7=4 + E(v) − E(w) L∞ ] ∇h 2
∇(v − w) 7=4
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6 K[( E(v) L∞ + K) ∇(v − w) 7=4 + E(v) − E(w) L∞ w 11=4 + E(v) − E(w) L∞ ] ∇h 2 6 K( v 11=4 ∇(v − w) 7=4 + ∇(v − w) 7=4 + ∇(v − w) 7=4 w 11=4 ) ∇h 2 6 K ∇B07=8 (v − w) 0 ∇h 2 :
(A.10)
For | | = 2 in a similar way one obtains 1=2 − A(w))h)
D ((A(v) 0 6 K ∇B0 (v − w) 1 ∇h 3 :
Estimates (A.8), (A.10), (A.11) yield estimate (4.19) for all l. Notations (3.5), (3.6) imply A(0) = I − 0 ". Therefore (A(0)v; B0 v)l ¿ 2K13 (B0 v; B0 v)l for some K13 ¿ 0. On the other hand, from estimate (4.19) it follows that
((A0 − A(0))v; B0 v)l 6 (A(a) − A(0))v l B0 v l 6 K11 B0 0 (l) a l B0 v 2l 6 K13 B0 v 2l for K12 small enough, what implies (4.21). Let u 3 ; v 3 ¡ K14 and K14 be small enough. Then estimate (4.21) yields (A(u)(u − v); B0 (u − v))1 ¿ K13 B0 (u − v) 21 But (4.19) implies |((A(u) − A(v))v; B0 (u − v))1 | 6 Kn 11 B07=8 (u − v) 1 B0 (u − v) 21 6
K13
B0 (u − v) 21 2
for K14 small enough and it yields (4.22). Now, note that ((A(0) − I )u; B0 u)2 = −(Su; B0 u)2 ¿ 2K15 ∇u 23 for some K15 ¿ 0. And estimate (4.19) and the triangle inequality yield |((A(u) − A(0))u; B0 u)|2 6 K11 ∇B01=2 u 1 ∇u 3 (I − ")u 2 6 K( ∇u 2 ∇u 3 u 2 + u 3 ∇u 3 Su 2 ) 6 K15 ∇u 23 for K14 small enough what implies (4.23).
(A.11)
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655
A.4. Proof of Lemma 4.6 Let B0 0 (l) a l ¡ K12 . Then estimate (4.19) implies
(A(a) − A(0))v l 6 K B0 v l : But for all ∈ C; Re ¿ 0
(A(0) + I )v l = (−0 " + ( + 1)I )v l 6 K21 () B0 v l and these two estimates yield (4.31). From (4.21) we have |((A0 + I )v; B0 v)l | ¿ |Re((A0 + I )v; B0 v)l | ¿ K13 (B0 v; B0 v)l :
(A.12)
This estimate and the Cauchy–Buniakowski inequality imply (4.32). From (4.32) it follows that Ker (A0 + I ) = {0}. Let us show that Im(A0 + I ) is dense in HVl . If not, then in HVl there is a nonzero vector h orthogonal to Im(A0 + I ). Let C = B0−1 h ∈ D(A0 ). Then (A.12) implies Re ((A0 + I )C; B0 C)l ¿ K13 (B0 C; B0 C)l ¿ 0:
(A.13)
But since (A0 + I )C ∈ Im (A0 + I ); B0 C = h, the left-hand side of (A.13) is equal to zero. Hence, Im (A0 + I ) is dense in HVl . Let us show now that Im (A0 + I ) is closed. Let hi → h0 ; hi ∈ Im (A0 + I ). i→∞
We have to show that h0 ∈ Im (A0 +I ). Note that there exist Ci such that (A0 +I )Ci = hi . Then the sequence (A0 +I )Ci converges in HVl . Then from (4.32) it follows that B0 Ci converges also in HVl : B0 Ci → $0 . Let C0 = B0−1 $0 ∈ D(A0 ). We have: B0 (Ci − C0 ) → 0. And estimate (4.31) yields (A0 + I )(Cn − C0 ) → 0, i.e. h0 = (A0 + I )C0 . Thus, A0 is a surjection. It remains to show (4.33). For || ¿ 1; u ∈ D(A0 ) we obtain 1 1 1 (u; u)l = Re 2 (u; u)l 6 Re 2 (u; u)l + Re 2 (A0 u; B0 u)l−1 || || || +Re = Re
1 1 (B0 u; A0 u)l−1 + Re 2 (A0 u; A0 u)l 2 || ||
1 ((A0 + I )u; (A0 + I )u)l ; ||2
whence
u l 6
1 2
(A0 + I )u l 6
(A0 + I )u l : || 1 + ||
(A.14)
Similarly,
u l 6 K (A0 + I )u l 6
2 K (A0 + I )u n l 1 + ||
Estimates (A.14), (A.15) imply (4.33).
(A.15)
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