On the existence and uniqueness of solution to problems of fluid dynamics on a sphere

J. Math. Anal. Appl. 388 (2012) 627–644

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

On the existence and uniqueness of solution to problems of fluid dynamics on a sphere Yuri N. Skiba ∗ Centro de Ciencias de la Atmósfera, Universidad Nacional Autónoma de México, Av. Universidad # 3000, Ciudad Universitaria, C.P. 04510, México, D.F., Mexico

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 5 February 2011 Available online 25 October 2011 Submitted by P.G. Lemarie-Rieusset

Orthogonal projectors and fractional derivatives on a two-dimensional unit sphere are introduced. Hilbert and Banach spaces of smooth functions on the sphere and some embedding assertions are given. The unique solvability of a nonstationary problem of vortex dynamics of viscous incompressible fluid on a rotating sphere is shown. The existence of a weak solution to stationary problem is proved too, and a condition guaranteeing the uniqueness of solution is also given. © 2011 Elsevier Inc. All rights reserved.

Keywords: Incompressible fluid dynamics Sphere Unique solvability

1. Introduction The three-dimensional Euler and Navier–Stokes equations are the fundamental equations for the numerical simulation of the general circulation of the atmosphere and global climatic processes. However, since the characteristic length scale for the horizontal atmospheric motion is much larger than the characteristic length scale of the vertical atmospheric motions, the shallow-water equations is widely used as a good approximation for the large-scale atmospheric motions [47]. The shallowwater equations support both fast (gravity) waves and slow (Rossby–Haurwitz) waves [12]. The barotropic vorticity equation is obtained from the shallow water model as a result of filtering the surface gravity waves. The unique weak solvability of the barotropic vorticity equation for ideal fluid on a sphere was shown by Szeptycki [66] under the conditions that the initial stream function ψ 0 (x) ∈ W22 ( S ) ∩ L∞ ( S ), the external vorticity source F (t , x) ∈ L2 (0, T ; W12 ( S )) ∩ L∞ ( Q ). Then there exists the unique solution ψ(t , x) such that

  ψ ∈ L∞ 0, T ; W22 ( S ) ,

ψ ∈ L∞ ( Q ),

  ψt ∈ L∞ 0, T ; W12 ( S ) .

Using the theory of integral equations, the global existence, uniqueness and regularity properties of classical solutions of the vorticity equation on a rotating sphere was proved by Chern, Colin and Kaper [12, Theorems 1–3]. The Schauder fixed-point theorem is used on a set defined by Hölder norms. Ben-Artzi [7] constructed for the Navier–Stokes flow a global-in-time solution when the initial vorticity is integrable. Unlike Giga, Miyakawa and Osada [23] his proof does not appeal to the Nash estimate, but relies on a simple property of the heat equation which makes it easy to prove the continuous dependence of the solution on the initial vorticity. Brezis [10] proves the uniqueness of a continuous solution in L1 (R2 ) for the vorticity equation for planar Navier–Stokes flow provided that the solution is locally bounded. The space L2 (R2 ) is naturally associated with the Navier–Stokes equation, since it is the energy space, because the square of the L2 norm of u is the total (kinetic) energy of the fluid, which is nonincreasing with time. As it mentioned by Gallagher and Gallay [22], the first mathematical result on the Cauchy problem is due to Leray [43] who proved that, for any initial

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Y.N. Skiba / J. Math. Anal. Appl. 388 (2012) 627–644

data u 0 ∈ L2 (R2 ), the Navier–Stokes system has a unique global solution u ∈ C0 ([0, +∞), L2 (R2 )) such that u (·, 0) = u 0 and ∇ u ∈ L2 ((0, +∞), L2 (R2 )). The unique solvability of two-dimensional Navier–Stokes equation with a measure as initial vorticity was considered in the interesting paper by Gallagher and Gallay [22], in which the authors study the uniqueness of solutions to the Cauchy problem for the vorticity equation in the whole plane R2 when the initial vorticity is taken from M (R2 ), the space of finite Radon measures. The existence of a global solution to the problem was established earlier (Giga, Miyakawa and Osada [23], Kato [36]), but the uniqueness for an arbitrary finite Radon measure as initial vorticity has been proved by Gallagher and Gallay [22]. To be precise, the authors prove that, any solution of the 2D Navier–Stokes equation, whose vorticity distribution is uniformly bounded in L1 (R2 ) for positive times is entirely determined by the trace of the vorticity at t = 0, which is a finite measure. The analysis of classes of functions, in which there exists unique solution of the vorticity equation on a rotating sphere, is of special theoretical interest. Such analysis becomes particularly important in the study of stability of solutions. All known results on the unique solvability of 2D Navier–Stokes equations in a bounded domain of Euclidean space R2 differ mainly in (i) technique, (ii) s of common form paces of generalized functions under consideration and (iii) procedure to construct approximate solutions [6,41,45,62,67]. This work is devoted to the study of unique solvability of a generalized nonstationary problem for the vorticity equation of incompressible viscous fluid on a rotating sphere. The vorticity equation considered here takes into account the Rayleigh friction of the form σ ψ, the term 2ψλ describing the rotation of sphere, the external vorticity source (forcing) F (x), and the turbulent viscosity term of common form ν (−)s+1 ψ , where s  1 is an arbitrary real number. The case s = 1 corresponds to the classical form used in Navier–Stokes equations (see Ladyzhenskaya [41], Temam [67,68], Szeptycki [65,66], Dymnikov and Filatov [17], Ilyin and Filatov [26,27]), while the case s = 2 was considered by Simmons, Wallace and Branstator [52], Dymnikov and Skiba [18–20], Skiba [60], etc. The turbulent term of such form for natural numbers s is applied by Lions [45] for studying the solvability of Navier–Stokes equations in a limited area by the artificial viscosity method. Here we consider only real solutions. Note that for s = 1, σ = 0, ψ0 ∈ W22 ( S ), F ∈ L2 (0, T ; W12 ( S )) ∩ L∞ (0, T ; L2 ( S )), the theorem of uniqueness and existence of solution ψ ∈ L∞ (0, T ; W22 ( S )) such that ψt ∈ L∞ (0, T ; L2 ( S )) and ψ ∈ L2 (0, T ; W12 ( S )) was proved by Szeptycki [66, Theorem 3.1]. The rotation of a sphere is not considered by Szeptycki. It is also shown in [66] that if additionally ψ0 ∈ L∞ ( S ) and F ∈ L∞ ( Q ) then ψ ∈ L∞ ( Q ). The unique solvability of general1 ized problem for stream function ψ from L2 (0, T ; W22 ( S )) ∩ C (0, T ; W12 ( S )), s = 1, σ = 0, ψ0 ∈ W32 ( S ), F ∈ L2 (0, T ; W− 2 ( S ))

2 n and F t ∈ L2 (0, T ; W− 2 ( S )) was proved by Ilyin and Filatov [26] (W2 ( S ) is the Sobolev space of the functions orthogonal to a constant on a sphere; see also Dymnikov and Filatov [17] and Ilyin and Filatov [27]). The rotation of a sphere was taken into account in [26]. The existence and uniqueness of solution of problem (40), (41) for s = 1 and s = 2 was proved in [61]. In this work, we consider a common case for arbitrary real number s  1. In the works [17,26,27], a function on a sphere is treated as the trace of the corresponding function of R3 . Unlike this, in [60,61] and in the present work, the functional spaces are introduced directly on a sphere.

2. Orthogonal projectors and fractional derivatives Let S = {x ∈ R3 : |x| = 1} be a unit sphere in the three-dimensional Euclidean space; we denote by C∞ ( S ) the set of infinitely differentiable functions on S and by



 f , g =

f (x) g (x) dS

(1)

S

and

f =  f , f 1/2

(2)

the inner product and norm in C∞ ( S ), respectively. Here x = (λ, μ) is a point on the sphere, dS = dλ dμ is an element of sphere surface, μ = sin φ ; μ ∈ [−1, 1], φ is the latitude, λ ∈ [0, 2π ) is the longitude and g is the complex conjugate of function g. It is known [48], that spherical harmonics

 Y nm (λ, μ) =

2n + 1 (n − m)! 4π

1/2

(n + m)!

P nm (μ)e imλ ,

n  0, |m|  n

(3)

form orthogonal basis in C∞ ( S ) [25,48]:





Y nm , Y lk = δmk δnl

where

δmk =

1, 0,

if m = k, if m = k

(4)

(5)

Y.N. Skiba / J. Math. Anal. Appl. 388 (2012) 627–644

629

is the Kronecker delta, and

P nm (μ) =

n (1 − μ2 )m/2 dn+m  2 μ −1 2n n! dμn+m

(6)

is the associated Legendre function of degree n and zonal wavenumber m. Let n and m be integer, n  0, and |m|  n. Each spherical harmonic Y nm (λ, μ) is the eigenfunction of spectral problem

−Y nm = χn Y nm ,

|m|  n

(7)

corresponding to the eigenvalue

χn = n(n + 1)

(8)

of multiplicity 2n + 1. Here

   ∂  1 ∂2 2 ∂ − = − 1−μ − ∂μ ∂μ 1 − μ2 ∂λ2

(9)

is symmetric and positive definite Laplace operator on S. For each integer n  0, the span of 2n + 1 spherical harmonics Y nm (λ, μ) (|m|  n) forms a generalized (2n + 1)dimensional eigenspace

Hn = {ψ : −ψ = χn ψ}

(10)

corresponding to the eigenvalue χn [48]. The subspace Hn is invariant not only with respect to the Laplace operator but also to any transformations of the SO(3) group of rotations of sphere about its arbitrary axis [25,48]. In order to simplify notation we will widely use a multi-index α ≡ (m, n) ≡ (mα , nα ) for the wavenumber (m, n):

α ≡ (−m, n) ≡ (−mα , nα ), Y α ≡ Y nm ,

α (k)

≡

P α ≡ P nm ,

n ∞



and

n=k m=−n

N

α (k)

χα ≡ χn = n(n + 1), ≡

n N



(11)

.

n=k m=−n

We now introduce operators of projection and fractional differentiation (fractional derivatives) of functions on the sphere [1,21,60].

 = {ψnm } = {ψα } of Fourier coefficients Definition 1. We denote by Φ the space of sequences ψ

  ψnm = ψ, Y nm

(12)

∈ C∞ ( S ) .

of functions ψ Due to the Parseval identity [48], the mapping C∞ ( S ) → Φ is isometric isomorphism, conserving the norm and inner product of elements. Therefore we will identify the spaces C∞ ( S ) and Φ . Since (−)k ψ ∈ C∞ ( S ) for every ψ ∈ C∞ ( S ) and any natural k, then due to the formula

(−)k ψ =

n ∞



k

n(n + 1) ψnm Y nm

(13)

n=1 m=−n

 ∈ Φ tend to zero as n → ∞ faster than the sequence 1/nk for any degree k. the Fourier coefficients ψnm of every sequence ψ Definition 2. We define Φ ∗ as a space dual to Φ and consisting of formal Fourier series h ∼ h = {hα } of Fourier coefficients hα satisfies the condition sequence 





α hα Y α such that the

ψα h α < ∞

α ( 0)

 ∈ Φ . Note that symbol ∼ asserts nothing about the convergence of series. Thus, Φ ∗ represents a space of continufor all ψ ous linear functionals

, h(ψ) ≡ ψ h =



ψα h α

α ( 0) m on Φ , and, due to isomorphism, can be identified with the space of such sequences  h = {hm n } whose elements hn increase not faster than some degree n [37].

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Y.N. Skiba / J. Math. Anal. Appl. 388 (2012) 627–644

Definition 3. The closure of C∞ ( S ) in norm (2) is the Hilbert space

  L2 ( S ) = ψ ∈ Φ ∗ : ψ < ∞ of generalized functions on S with inner product (1). The space L2 ( S ) is the direct orthogonal sum of subspaces Hn [48]:

L2 ( S ) =

∞ 

Hn .

(14)

n =0

Definition 4. Let ω be an angle between two unit radius-vectors x1 , x2 corresponding to points x1 , x2 ∈ S. Then x1 ·x2 = cos ω is the scalar product of vectors x1 and x2 [48]. A function z(x · y ) depending only on the distance ρ (x, y ) = arccos (x · y ) = ω between two points x, y of sphere is called the zonal function. The convolution of a function ψ ∈ L2 ( S ) and a zonal function Z (x · y ) ∈ L2 ( S ) is defined by



1

(ψ ∗ z)(x) =

ψ( y ) Z (x · y ) dS ( y )

4π

(15)

S

(see [25]). Let n  0. Orthogonal projector Y n : L2 ( S ) → Hn of L2 ( S ) on the subspace Hn of homogeneous spherical polynomials of degree n is introduced by

Y n (ψ; x) = (2n + 1)(ψ ∗ P n )(x).

(16)

In order to show that (16) is really the projector we first prove that n

Y n (ψ; x) =

ψnm Y nm (x)

(17)

m=−n

and hence, Y n (ψ) ∈ Hn . Indeed,

P n (x1 · x2 ) =

n

4π 2n + 1

Y nm (x1 )Y nm (x2 )

m=−n

for two radius-vectors x1 and x2 [48]. Therefore, due to Definition 4, and formulas (16) and (12) we have

2n + 1

Y n (ψ; x) =



ψ( y ) P n (x · y ) dS ( y )

4π S

 =

ψ( y )

Y nm (x)Y nm ( y ) dS ( y )

m=−n

S n

=

n



n



Y nm (x) ψ( y ), Y nm ( y ) =

m=−n

ψnm Y nm (x).

m=−n

Using this result we get

 n   2n + 1 m (2n + 1) Y n (ψ) ∗ P n (x) = ψn Y nm ( y ) P n (x · y ) dS ( y ) 4π

=

n

m=−n

ψnm

m=−n

n

k=−n

S



Y nk (x)

Y nm ( y )Y nk ( y ) dS ( y ) = Y n (ψ; x) S

and hence, Y n (Y n (ψ)) = Y n (ψ) for all functions ψ ∈ L ( S ). For the sake of brevity we will sometimes write simply Y n (ψ) instead of Y n (ψ; x). Obviously, any function from subspace H0 is constant: 2

Y 0 (ψ) =

1



ψ( y ) dS ( y ) = Const.

4π S

(18)

Y.N. Skiba / J. Math. Anal. Appl. 388 (2012) 627–644

631

Let N  0. We introduce finite dimensional subspaces P N and P0N of spherical polynomials of degree n  N as direct orthogonal sums of subspaces Hn :

PN =

N 

P0N =

Hn ,

N 

n =0





Hn = ψ ∈ P N : Y 0 (ψ) = 0 .

(19)

n =1

Thus, Y 0 (ψ) = 0 for any ψ ∈ P0N . Definition 5. We define an orthogonal projector T N : L2 ( S ) → P N as ∞

T N ψ(x) =

Y n (ψ; x) = (ψ ∗ S N )(x)

(20)

n =0

where

S N (x · y ) =

N

(2n + 1) P n (x · y ) n =0

is the convolution kernel [40]. Thus, if ψ(x) ∈ L2 ( S ) then T N ψ is the triangular truncation of Fourier series of ψ using the spherical harmonics as basic functions. Note that the Parseval–Steklov identities



ψ 2 =

|ψα |2 =



(21)

n =0

α ( 0)

ψ, h =

∞

  Y n (ψ)2 ,

ψα h α =

∞





Y n (ψ), Y n (h)

(22)

n =0

α ( 0)

hold for any functions ψ, h ∈ L2 ( S ) [48]. Due to (21), each function ψ(x) ∈ L2 ( S ) is represented by its own Fourier–Laplace series

ψ(x) =

∞

Y n (ψ; x) ≡

n ∞



ψnm Y nm (x).

(23)

n=0 m=−n

n =0

Definition 6. Let s > 0 and ψ(x) ∈ C∞ ( S ). A spherical operator Λs = (−)s/2 of real order s is defined by means of equations





s/2



s/2

Y n Λs ψ = χn Y n (ψ) = n(n + 1)

Y n (ψ)

(24)

valid for any natural number n. Thus, Λs is a multiplier operator which is completely defined by infinite set of multiplicators {χn }n∞=0 [11,29,32]. We will consider Λs as a derivative of real order s of functions on a sphere, besides, s /2

Λs ψ(x) =

∞

n =1

χns/2 Y n (ψ; x) ≡



χαs/2 ψα Y α (x).

(25)

α (1 )

In particular, Λ2n = (−)n for any natural n, and operator Λ can be interpreted as the square root of nonnegative and symmetric Laplace operator (9). It is well known that the main disadvantage of local derivatives ∂ n /∂λn and ∂ n /∂ μn is that they depend on the choice of coordinate system (i.e., on sphere rotation). The new derivatives Λs and projectors Y n are invariant with respect to any element of the group SO(3) of sphere rotations [64], and hence are free from this disadvantage. 3. Hilbert spaces In this section we introduce a family of Hilbert spaces Hs of generalized functions (distributions) on a sphere that depends on a real parameter s, besides, a function ψ ∈ Hs for some s if its sth fractional derivative belongs to the space L2 ( S ) [21,29,33,34,46,60].

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Y.N. Skiba / J. Math. Anal. Appl. 388 (2012) 627–644

Definition 7. We denote by C∞ 0 ( S ) the space of infinitely differentiable functions which are orthogonal to any constant on a sphere:

  ∞ C∞ 0 ( S ) = ψ ∈ C ( S ): Y 0 (ψ) = 0 . Note that operator Λs may be defined on functions C∞ 0 ( S ) by means of (25) for every real degree s. Definition 8. For any real s, we introduce in C∞ 0 ( S ) the inner product · , ·s and norm · s in the following way: ∞ 

   s ψ, hs = Λs ψ, Λs h = χns Y n (ψ), Y n (h) ≡ χα ψ α h α , n =1

(26)

α (1 )

∞ 1/2 1/2

  s  2 1/2 s s 2    ψ s = Λ ψ = ψ, ψs = χn Y n (ψ) ≡ χα |ψα | . n =1

(27)

α (1 )

s Definition 9. Let s be a real. The Hilbert spaces obtained by closing the space C∞ 0 ( S ) in the norms (27) we denote as H . 0 For the sake of brevity we will keep the symbols · , · and · for the inner product and norm in H (see (1) and (2)). It is shown in [1] that

r s 0 −s Φ = C∞ ⊂ H−r ⊂ Φ ∗ 0 (S) ⊂ H ⊂ H ⊂ H ⊂ H

(28)

s ∗

are continuous if 0 < s < r, and the dual space (H ) coincides with H−s for all s  0. Let s, r be real numbers. Operator Λr : C∞ ( S ) → C∞ ( S ) is symmetric: 0

 r    Λ ψ, h s = ψ, Λr h s ,

0

and therefore can be extended for elements from Hs . Definition 10. An element z ∈ Hs is called the rth derivative Λr ψ of function ψ ∈ Hs if

   z, hs = z, Λr h s r holds for all h ∈ C∞ 0 ( S ) where Λ h is defined by (25).

The following assertion establish embedding estimates for functions of spaces Hs (see (28)). Lemma 1. (See [60].) Let s be real, r > 0, and ψ ∈ Hs+r . Then ψ ∈ Hs and

ψ s  2−r /2 ψ s+r ,   ψ s+r = Λr ψ  .

(29) (30)

s

Corollary 1. Let s and r be real numbers. The mapping Λr : Hs+r → Hs is isometry and isomorphism. In particular, at r = −2s, the operator Λ−2s : H−s → Hs is isometric isomorphism. Lemma 2 (Poincare inequality). (See [61].) For any ψ ∈ H1 ,

√ √ ψ  1/ 2 ∇ψ = 1/ 2 Λψ .

(31)

In fact, a more general assertion than Lemma 2 is valid: Lemma 3. (See [60].) Let r , s and t be real numbers, r < t, and a =

 r    Λ ψ   ar −t Λt ψ  . s s

√

2. Then for any ψ ∈ Hs+t ,

(32)

It is easy to prove some inequalities for functions of spaces Hs . Indeed,

ψ, hs =





χα(s+r )/2 ψα χα(s−r )/2 hα



α (1 )

due to (26), and hence, the generalized Schwartz inequality

Y.N. Skiba / J. Math. Anal. Appl. 388 (2012) 627–644

  ψ, hs   ψ s+r h s−r

633

(33)

and estimate

ψ 2s  ψ s+r ψ s−r

(34)

hold for any ψ ∈ Hs+r and h ∈ Hs−r , where s is a real and r > 0. In particular, we have

  ψ, h  ψ s h −s

(35)

for all ψ ∈ Hs and h ∈ H−s (s  0). We now obtain two interpolating inequalities well known in the theory of periodic functions [8,38,49]. Let r, s and t be real numbers; t  r < s; a, ρ > 0 and let Ψ ∈ Hs . If we take a = ρ 1/(s−r ) χα in the inequality

1  as−r + at −r (χα is the multiplicator (8)) then we get s drα  ρ dα + ρ p /( p −1) χαt .

Here

p=

r −t s−t

< 1.

(36)

Taking into account Definition 8, we obtain the first interpolating inequality (ρ is an arbitrary positive number; see also [1], formula (4.18)):

ψ r  ρ ψ s + ρ p /( p −1) ψ t .

(37)

If we now take

ρ = ψ sp−1 ψ t1− p in (37) then we derive the second interpolating inequality 1− p

ψ r  2 ψ t

p

ψ s .

(38)

We now give the common form of continuous linear functional on Hs . Lemma 4. (See [61].) Let s  0. Then for all h ∈ H−s a sesquilinear form G (ψ) = ψ, h is a continuous linear functional on Hs with a norm

G = h −s = sup ψ, h. ψ s =1

Conversely, every continuous linear functional on Hs has the form ψ, h where element h ∈ H−s is uniquely determined. Note that for each s ∈ R, the functions −s/2

W s,α = Λ−s Y α = χα

Yα

(39)

form orthonormal basis in Hs , and for any ψ ∈ Hs ,

ψ=

N

α (1 )

where

q s,α W s,α =

N

ψα Y α ,

s/2

q s,α = χα ψα

α (1 )

χα and Y α are determined by (11).

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Y.N. Skiba / J. Math. Anal. Appl. 388 (2012) 627–644

4. Dynamics of viscous non-divergent fluid on a rotating sphere Let us consider the nonlinear nonstationary problem for the relative vorticity ψ(t , x) in a viscous two-dimensional incompressible rotating fluid on unit sphere S:

∂ ψ + J (ψ, ψ + 2μ) = −σ ψ + ν (−)s+1 ψ + F , ∂t ψ(0, x) = ψ0 (x).

(40) (41)

The problem takes into account a forcing (an external vorticity source) F (x) and Rayleigh friction σ ψ in the planetary boundary layer. Eq. (1) is written in a non-dimensional form using geographical coordinate system (λ, μ) with the pole N being on the axis of rotation of sphere S. Here ψ is the stream function, ψ is the relative vorticity, ψ + 2μ, is the absolute vorticity,

J (ψ, h) =

∂ψ ∂ h ∂ψ ∂ h = ( − n × ∇ψ) · ∇ h ∂λ ∂ μ ∂ μ ∂λ

(42)

 is the outward unit normal vector at each point of sphere S. We denote scalar and vector products of is the Jacobian, n vectors by symbols “·” and “×”, respectively. Besides,





∇h =

  ∂h ∂h , 1 − μ2 ∂μ 1 − μ2 ∂λ 1

(43)

is the gradient of function h,  is the spherical Laplace operator (9) and symbols ψt , ψλ and ψμ denote partial derivatives of ψ with respect to t, λ and μ, respectively. The velocity vector field

 × ∇ψ v = n

(44)

having the components



u = − 1 − μ2 ψμ ,

v=

1 1 − μ2

ψλ

(45)

is solenoidal:

∇ · v = 0.

(46)

The term J (ψ, 2μ) = 2ψλ takes into account the rotation of sphere. We will consider the turbulent viscosity term of common form ν (−)s+1 ψ , where s  1 is an arbitrary real number. The case s = 1 corresponds to classical form used in Navier–Stokes equations [17,27,41,65–67], while the case s = 2 was considered in [18–20,52,60]. The turbulent term of such form for natural numbers s is applied in [45] for studying the solvability of Navier–Stokes equations in a limited area by the artificial viscosity method. Eq. (40) is obtained by applying the operator curl to the equations of motion of 2D fluid. Since the sphere is a smooth manifold without any edge, such transformation implies that if ψ is a solution of problem (1), (2) then ψ + const is also the solution for any const. In order to eliminate this constant the problem (1), (2) will be considered in classes of functions which are orthogonal to a constant on the sphere:

Y 0 (ψ) = 0,

Y 0 ( F ) = 0.

(47)

5. Properties of the Jacobian We now study some properties of the Jacobian (2.1.3) which will be needed us in the following sections. Let all functions under consideration be complex-valued. It is clear that

J (ψ, h) = − J (h, ψ),

J (ψ, ψ) = 0.

(48)

Let n be a natural number, and let r be a real number. Since s/2

Λs Y n (ψ) = χn Y n (ψ)

(49)

then





J ψ, Λs ψ = 0 for any homogeneous spherical polynomial ψ of degree n (ψ ∈ Hn ). Obviously,

(50)

Y.N. Skiba / J. Math. Anal. Appl. 388 (2012) 627–644

J (ψ, h) = 0

635

(51)

for any zonal functions ψ(x) = ψ(z · x) and h(x) = h(z · x), where z is a pole of sphere S. Note that a smooth vector field  × ∇ψ is solenoidal, and therefore, n



J (ψ, h) = ∇ · h( n × ∇ψ)

due to (42).  defined on S has a compact support K ⊂ S, that is, X  (x) = 0 if x ∈ Suppose that a smooth vector field X / K . Then



∇ · X dS = 0

(52)

S

(see [25]). Using the theorem on the partition of unity [16], we obtain



J (ψ, h) dS = 0.

(53)

S

It is easy to show that the relation

J (hψ, f ) = h J (ψ, f ) − ψ J ( f , h)

(54)

is valid for all continuously differentiable functions ψ , f and h on S. Integrating (54) over S and using (53) and (48) we obtain that











J (ψ, f ), h = J ( f , h), ψ = − J (ψ, h), f



(55)

holds for sufficiently smooth complex-valued functions ψ , f and h on S. Let C be the set of complex numbers, and let ψ ∈ C∞ ( S ), ψ : S → C. Then defining the gradient operator of superposition G (ψ) = G ◦ ψ of two functions as

∇ G (ψ) =

∂ G (ψ)∇ψ ∂ψ

we get







J ψ, G (ψ) = ( n × ∇ψ) ·

∂G ∇ψ ∂ψ



at each point x ∈ S. Therefore, due to (55),





J (ψ, h), G (ψ) = 0.

(56)

Lemma 5. Let r be a real number, and ψ, h ∈ C∞ ( S ). Then







J (ψ, h), ψ r = 0,



J (ψ, μ), Λr ψ = 0.

(57)

Indeed, the first relation follows from (56). Further,



), Λr ψ

J (ψ, μ



1  2π = −1

  1  2π ∂ψ r 1 ∂  r /2 2 Λ ψ dλ dμ = Λ ψ d λ d μ = 0. ∂λ 2 ∂λ −1

0

0

∂ We used here the commutativity of operators Λr and ∂λ . Indeed,

Λr

  ∂ r m ∂ m r /2 Λ Yn Y = Λr imY nm = imχn Y nm = ∂λ n ∂λ

for each basic function (spherical harmonic) Y nm . Definition 11. We denote by L p ( S ) the completion of continuous function on sphere S in the norm

1/ p

 |ψ| p dS

ψ L p ( S ) = S

(58)

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Y.N. Skiba / J. Math. Anal. Appl. 388 (2012) 627–644

if p = ∞, and in the norm

  ψ L∞ ( S ) = sup vraiψ(x)

(59)

x∈ S

if p = ∞. Applying Schwartz inequality one can obtain from (42) that

   J (ψ, h) =



1/2 |∇ψ|2 |∇ h|2 dS

 ∇ψ L4 ( S ) ∇ h L4 ( S ) .

(60)

S

Lemma 6. (See [21].) Let 1 < p < ∞. For each function ψ such that Λψ ∈ L p ( S ), the following norms are equivalent:

∇ψ L p ( S )  Λψ L p ( S ) . We now give important results on continuous embedding for spaces L p ( S ) and Hs . Lemma 7. (See [5].) Let p > 1 and 0 < s < 2/ p. Let r be such that s = 2(1/ p − 1/r ), and ψ ∈ L p ( S ). Then Λ−s ψ ∈ Lr ( S ) and

 −s  Λ ψ 

Lr ( S )

 C ψ L p ( S ) .

(61)

In the particular case, that p = 2 we have Lemma 8. Let s ∈ (0, 1). If number r is such that s = 1 − 2/r then each function ψ of Hs belongs to Lr ( S ), and

ψ Lr ( S )  C ψ s .

(62)

Corollary 2. The embedding operator H1/2 → L4 ( S ) is compact. Indeed, since sphere S is a compact set, the continuous embedding H1/2 into L40 ( S ) is compact. Lemma on continuous embedding of H1 into L4 ( S ) was established in [65] (see also [26]). A compact embedding of H1 into L p ( S ) was proved in [60] by using Theorem 7.1 from work [42] and theorem about the partition of unity [16]:

ψ L p ( S )  K ψ 1−α Λψ α ,

K = max{2, p /2},

α = 1 − 2/ p , p ∈ [2, ∞).

(63)

Thus, Lemma 8 and Corollary 2 contain more precise estimates. Using Lemmas 7, 8 and (27), we obtain

  ∇ψ L4 ( S )  C 1 Λψ L4 ( S )  C 2 Λψ 1/2  C Λ3/2 ψ . The last inequality together with (60) imply Lemma 9. Let ψ, h ∈ H3/2 . Then J (ψ, h) belongs to L2 ( S ), and

      J (ψ, h)  C Λ3/2 ψ Λ3/2 h.

(64)

It follows from Lemma 9 that  J (ψ, h), g  is the continuous form in H3/2 × H3/2 × H0 . In particular,

   J (ψ, h)  M ψ h .

(65)

6. Unique solvability of nonstationary problem for the vorticity equation This section is devoted to the study of unique solvability of a generalized nonstationary problem for the vorticity equation of incompressible viscous fluid on a rotating sphere. Except self-interest, the analysis of classes of functions, in which there exists a solution of the problem, is particularly important in the study of stability of solutions. The fluid dynamics stability has been studied by many researchers [4,14,15,24,30,31,50,51,53–61,63,67,68]. In these investigations common results and methods of the mathematical theory of stability were broadly applied [2,3,13,28,39,44].

Y.N. Skiba / J. Math. Anal. Appl. 388 (2012) 627–644

637

Definition 12. Let p ∈ [1, ∞). We denote by L p (0, T ; X) the Banach space of measurable functions ψ : (0, T ) → X, image of which is in X and such that

 T ψ L p (0,T ;X) =

  ψ(t ) p dt

1/ p < ∞.

X

(66)

0

If p = ∞ then the norm in L∞ (0, T ; X) is defined by

  ψ L∞ (0,T ;X) = sup vraiψ(t )X . t ∈(0, T )

Hereafter, Q = (0, T ) × S and L p ( Q ) = L p (0, T ; L p ( S )). Lemma 10. Let 1  p  ∞, 0 < s < 1 and s = 1 − 2/r. If ψ ∈ L p (0, T ; Hs ) then ψ ∈ L p (0, T ; Lr ( S )) and

ψ L p (0,T ;Lr ( S ))  C ψ L p (0,T ;Hs ) . To prove Lemma 10, it is sufficient to build the inequality (62) in the pth degree, and then integrate the result over interval (0, T ). Lemma 11. (See [60].) Let p and α be constants from the conditions (63), and let ψ ∈ L∞ (0, T ; H0 )∩L p α (0, T ; H1 ). Then ψ ∈ L p ( Q ) and

ψ L p ( Q )  C ψ L pα (0,T ;H1 ) . In particular, if p = 4 then

α = 1/2 and p α = 2. Indeed, due to (63), we have

ψ L p ( S )  K p ψ p (1−α ) ∇ψ p α  C ψ 1 . pα

p

Integrating the last inequality over (0, T ) and using Lemma 10 we obtain the required estimate. Theorem 1. Let s  1, ν > 0 and σ  0. Suppose that initial field ψ0 ∈ L2 ( S ), and forcing F (t , x) ∈ L2 (0, T ; H−s ). Then nonstationary problem (40), (41) has unique weak solution ψ(t , x) ∈ L∞ (0, T ; H2 ) such that

    ψ(t , x) ∈ L∞ 0, T ; H0 ∩ L2 0, T ; Hs ,   ψ(0, x) = ψ0 (x), ψt ∈ L2 0, T ; H−s ,

(67)

and

t

 

ψt , h − J (ψ, h), ψ + 2μ + σ ψ, h dt − ν

0

t 0



Λ

s +2

s



t

ψ, Λ h dt =

 F , h dt

(68)

0

holds for all t ∈ (0, T ) and h ∈ L (0, T ; H ). 2

s

Remark 1. Here we considered only real solutions, and therefore, used (55) in (68). The existence and uniqueness of solution of problem (40), (41) for s = 1 and s = 2 was proved in [61]. Here we consider a common case for arbitrary real number s  1. Proof of Theorem 1. I. The existence of weak solution. We now prove the existence of weak solution by using Galerkin method. We take the functions W α (x) = W s,α (x) (see (39)) as a basis in Hs . 1. We shall look for an approximate solution ψ N of problem (40), (41) from P0N (see (19)) in the following form:

ψ N (t , x) =

N

qαN (t ) W α (x).

(69)

α (1 )

The functions qαN (t ) will be found from the conditions

d dt

         ψ N , W α − J ψ N , W α , ψ N + 2μ + σ ψ N , W α − ν Λs+2 ψ N , Λs W α =  F , W α 

(70)

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Y.N. Skiba / J. Math. Anal. Appl. 388 (2012) 627–644

α (α = (m, n), 0  |m|  n  N). We take ψ N (0, x) = ψ0N (x) as initial condition. Moreover,   ψ N − ψ0  → 0. ψ0N ∈ P0N , 0

which hold for all

(71)

The integration with respect to t is eliminated here (see justification for this step in [41]). It follows from the theory of ordinary differential equations that system (70) has unique solution for t  0 under arbitrary initial conditions [3]. 2. We now obtain a priori estimates of solutions ψ N . Let us multiply (70) by qαN (t ) and sum the resulting expressions over α in the same manner as in (69). Due to (57) we obtain

1 d 2 dt

       Λψ N 2 + σ Λψ N 2 + ν Λs+1 ψ N 2 = − F , ψ N .

(72)

In order to estimate the right-hand side of (72) we will use inequality (35), Lemma 1 and ε -inequality. This gives

√ √        ν 2   F , ψ N   F −s ψ N   1/ 2 F −s ψ N  = 1/ 2 F −s Λs+1 ψ N   Λs+1 ψ N  + C F 2−s . s s +1 2

Using the last inequality in (72) and (30) we get

d dt

     Λψ N 2 + 2σ Λψ N 2 + ν Λψ N 2  C F 2

(73)

−s

s

where C is a constant independent of N. Integrating (73) over time from 0 to T , and taking into account that F (t , x) ∈ L2 (0, T ; H−s ), and ψ N (0) are uniformly bounded due to (71), we obtain the uniform estimate

   √  Λψ N (t ) + σ Λψ N 

L2 ( Q )

+

 √  ν Λψ N 

L2 (0, T ;Hs )

 C.

χα−1  W α in (70), and use (57) then, after similar transformations, we obtain     √  √  ψ N (t ) + σ ψ N  2 + ν ψ N L2 (0,T ;Hs )  C . L (Q )

(74)

If we replace W α by

(75)

3. We now show the uniform boundness of ∂∂t ψ N . Applying (42), Hölder inequality and Lemma 6, we have

T N

 





J ψ N , h , ψ N + 2μ dt

G (h) ≡ 0

     C ∇ψ N L4 ( Q ) ψ N + 2μL4 ( Q ) ∇ h L2 ( Q )      C Λψ N  4 ψ N + 2μ 4 h 2 s L (Q )

L (Q )

L (0, T ;H )

for all h ∈ L2 (0, T ; Hs ). Due to (74) and (75), functions Λψ N (t , x) and ψ N (t , x) satisfy Lemma 11 for p = 4 and and therefore,

  Λψ N 

L4 ( Q )

 C,

  ψ N  4  C. L (Q )

(76)

α = 1/2, (77)

Thus,

 N  G (h)  C h

(78)

L2 (0, T ;Hs )

for all h ∈ L2 (0, T ; Hs ). It follows from (78) that for each fixed function ψ N there exists an element J N ∈ L2 (0, T ; H−s ) such that the continuous linear functional can be represented as

T N

G (h) =





J N , h dt

(79)

0

(see Lemma 4). Due to (76),

 N J 

L2 (0, T ;H−s )

     C Λψ N L4 ( Q ) ψ N + 2μL4 ( Q ) .

(80)

Then estimates (77) imply that





J N are uniformly bounded in L2 0, T ; H−s .

(81)

Y.N. Skiba / J. Math. Anal. Appl. 388 (2012) 627–644

639

Using (70) and Definition 5 we have

∂ ψ N = − T N J N − σ ψ N + ν (−)s+1 ψ N + T N F . ∂t The norm of projector T N : L2 (0, T ; H−s ) → L2 (0, T ; H−s ) is less than unit (see (20)), and therefore, taking into account (74), (75), (81) and the condition F (t , x) ∈ L2 (0, T ; H−s ) we obtain

  ∂ ψ N are uniformly bounded in L2 0, T ; H−s . ∂t

(82)

4. We now prove the convergence of approximate solutions ψ N as N → ∞. According to estimates (74), (75) and (82), which are uniform in N, we can select a subsequence from {ψ N } (we will use the same notation for this subsequence) such that

  ψ N converges to ψ ∗-weakly in L∞ 0, T ; H2 ,   ψ N converges to ψ weakly in L2 0, T ; Hs ,

(83) (84)

  ∂ ∂ ψ N converges to ψ weakly in L2 0, T ; H−s ∂t ∂t

(85)

(see [8], Theorems 3 and 3*). Note that a Banach space is closed with respect to weak convergence [35]. It follows from Theorem 5.1 (see [45]) where we take B 0 = Hs , B = H0 and B 1 = H−s that

  ψ N − ψ  2 → 0 as N → ∞. L (Q )

(86)

Due to (84), (85) and Lemma 1.2 from [38] (where X = H−s ),

  ψ0N converges to ψ(0) weakly in L2 0, T ; Hs .

(87)

Hence, taking into account (71) we obtain

ψ(0) = ψ0 .

(88)

We now prove that

t

 

  J ψ , h , ψ N + 2μ dt →

t

N

0





J (ψ, h), ψ + 2μ dt

(89)

0

for any h ∈ L2 (0, T ; Hs ). For this purpose, first prove that



J (ψ, h) ∈ L2 0, T ; H−s



(90)

for any h ∈ L2 (0, T ; Hs ). Indeed, due to (77) and (86), the estimate Λψ L4 ( Q )  C is valid for the limit function ψ , and application of Hölder inequality implies the estimate

   J (ψ, h)

L4/3 ( Q )

 C Λψ L4 ( Q ) Λh L2 ( Q )  C Λψ L4 ( Q ) Λh L2 (0,T ;Hs )

(91)

which holds for all h ∈ L2 (0, T ; Hs ). Further, for each s  1, any function f (t , x) of set M = L2 (0, T ; Hs ) ∩ L∞ (0, T ; H0 ) satisfies Lemma 11 for p = 4. Thus, for all f (t , x) of M we have











J (ψ, h) f d Q   J (ψ, h)L4/3 ( Q ) f L4 ( Q )  C  J (ψ, h)L4/3 ( Q ) f L2 (0, T ;Hs ) .

(92)

Q

Since M is dense in L2 (0, T ; Hs ), and the spaces L2 (0, T ; H−s ) and L2 (0, T ; Hs ) are dual, we obtain (90). In particular, J (ψ, W α ) ∈ L2 (0, T ; H−s ) for all W α ∈ Hs and

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Y.N. Skiba / J. Math. Anal. Appl. 388 (2012) 627–644

     N     N  γ ≡ J ψ , W α ψ + 2μ − J (ψ, W α )(ψ + 2μ) d Q  Q

 

      N      J ψ − ψ, W α ψ N + 2μ  d Q +  J (ψ, W α ) ψ N − ψ d Q   

Q

Q

            C Λ ψ N − ψ L4 ( Q ) W α L2 (0,T ;Hs ) ψ N + 2μL4 ( Q ) +  J (ψ, W α ) ψ N − ψ d Q  Q

≡ γ1 + γ2 .

(93)

γ1 in (93) tends to zero due to (77), estimate    N      Λ ψ − ψ  4  C Λ ψ N − ψ L2 (0,T ;H1 )  C ψ N − ψ L2 ( Q ) L (Q )

The first term

and assertion (86). Since J (ψ , W α ) ∈ L2 (0, T ; H−s ) (see (90)), the second term γ2 in (93) tends to zero due to (84). Thus, (89) is valid. Therefore, if we consider the limit of Eq. (70) as N → ∞ and use (83)–(86), we obtain that function ψ satisfies the equation

    ψt , W α  − J (ψ, W α ), ψ + 2μ + σ ψ, W α  − ν Λs+2 ψ, Λs W α =  F , W α .

(94)

Then (94) will also be valid if we replace W α by arbitrary function g ∈ H , and hence, equality (68) holds for all elements h ∈ L2 (0, T ; Hs ). s

 and ψ  are two generalized solutions of problem (67), (68), and let II. The uniqueness of weak solution. Suppose that ψ  −ψ . Then for all t  T the function g satisfies the following equation: g=ψ

t

   gt , h + σ  g , h − ν Λs+2 g , Λs h dτ =

0









 + 2μ + J (ψ , h),  g dτ . J ( g , h), ψ

(95)

0

Since  gt ∈ L

2

(0, T ; H−s ), we can put h

1 2

t

 Λ g (t )2 + σ

t

= g in (95). Then J ( g , g ) = 0, Λ g (0) = 0, and

t Λ g 2 dτ + ν

0

 s+1 2 Λ g  d τ =

0

t





, g ),  g dτ . J (ψ

0

We used here the fact that if g ∈ L (0, T ; H ) and  gt ∈ L2 (0, T ; H−s ) then everywhere [45]. We now estimate the right-hand side of Eq. (96): 2

(96)

s

1 d Λ g (t ) 2 2 dt

= Λ gt , Λ g  =  gt , g  almost

 t  t        L4 ( S ) ∇ g L4 ( S )  g dτ  J (ψ , g ),  g dτ   ∇ ψ   0

0

 L∞ (0,T ;H0 )  C ψ

t Λ g 1/2  g 3/2 dτ 0

t C

 3/2 Λ g 1/2 Λs+1 g  dτ

0

t ν 0

 s+1 2 Λ g  dτ + C

t Λ g 2 dτ . 0

It was successively used here the Hölder inequality, Lemma 6, estimate (63), boundness of solution ψ in the norm of space L∞ (0, T ; H0 ), Lemma 3, and Young inequality [41]. From (96) we obtain the inequality

  Λ g (t )2  C

t Λ g 2 dτ

(97)

0

which implies g (t , x) ≡ 0 [41]. The theorem is proved.

2

Y.N. Skiba / J. Math. Anal. Appl. 388 (2012) 627–644

641

7. Solvability of stationary vorticity equation In this section, it is proved the existence of a generalized solution ψ(x) to the stationary vorticity equation of incompressible viscous fluid on a rotating sphere (see (40)):

J (ψ, ψ + 2μ) = −σ ψ + ν (−)s+1 ψ + F (x).

(98)

It is also proved that for a sufficiently large turbulent diffusion coefficient ν , Eq. (98) has unique solution. The dependence of the solution smoothness on the exponent turbulent viscosity term ν (−)s+1 ψ is shown as well. We will need the following: Lemma 12. Let s be a real and r > 0. Then the bounded operator Λ−r : Hs → Hs is the compact operator if r > 0, the Hilbert–Schmidt operator if r > 1, and the nuclear operator if r > 2. −r /2

−r /2

≡ χn in the canonical expansion (25) of operator Λ−r tend to zero as Proof. The multiplicators χα Then (see [48]), the operator Λ−r is compact. Furthermore, the series ∞





 Λ−r Y α 2 = χα−r =

α (1 )

n =1

α (1 )

2n + 1

[n(n + 1)]r

diverges if r  1 and converges if r > 1. It follows immediately from its comparison with the series

2n + 1

[n(n

+ 1)]r

Thus the series

=

1 nr −1 (n



−r /2

α (1) X α

+ 1)r

+

α → ∞ (n → ∞).

1 nr (n

+ 1)r −1

converges if r > 2.



nn

−r and the relation

.

2

Corollary 3. Let s be a real and r > 0. Then each set bounded in H s+r is compact in H s . This result immediately follows from the equality

  ψn − ψm s = Λ−r (ψn − ψm )s+r . Note that Corollary 3 is the particular case of Rellich’s lemma [9]. Theorem 2. Let s  1, ν > 0 and ψ(x) ∈ Hs+2 of Eq. (98) such that



σ  0. Suppose that the vorticity source F (x) ∈ H−s . Then there exists at least one weak solution







ν Λs+2 ψ, Λs h − σ ψ, h + J (ψ, h), ψ + 2μ =  F , h

(99)

holds for all h ∈ H−s . If in addition,





ν 2 > 21−s M  F (x)−s

(100)

then this problem has unique solution (here M is the constant from (65)). Remark 2. The case s = 1 and σ = 0 was considered in [26] (see also [17,27]). For s = 1 and s = 2 the theorem was proved in [60,61]. Here we consider a common case for arbitrary real number s  1. Proof of Theorem 2. I. The existence of weak solution. We will use the Galerkin method again. 1. Let us define an approximate solution (see (39)) from P0N (see (19)) in the following form:

ψ N (x) =

N

qαN W α (x) =

α (1 )

N

ψαN Y α (x)

α (1 )

that satisfies the equation









 





ν Λs+2 ψ N , Λs W α − σ ψ N , W α + J ψ N , W α , ψ N + 2μ =  F , W α  for all W α (x) (α = (m, n), 0  |m|  n  N). We now prove the existence of solution ψ let us define a mapping V : P0N → P0N so that

(101) N

of problem (101). With this aim,

642

Y.N. Skiba / J. Math. Anal. Appl. 388 (2012) 627–644

N

V ψ N (x) =

v αN Y α (x)

α (1 )

where









 





v αN = ν Λs+2 ψ N , Λs W α − σ ψ N , W α + J ψ N , W α , ψ N + 2μ −  F , W α .

(102)

One can get





V ψN,ψN =

N

v αN ψαN

α (1 )

 2     = ν Λs+2 ψ N , Λs ψ N + σ Λψ N  + F , ψ N  2    ν Λs+1 ψ N  − F −s ψ N s       as+2 ν ψ N  − F −s Λs ψ N . The estimate (35) and Lemma 3 with a =

(103)

√

2 are used here. Thus functional  V ψ N , ψ N  is nonnegative for all ψ N such that

 N ψ  = a0  C F −s

(104)

where C = 1/(as+2 ν ). Let us fix some a0 satisfying (104). Then (due to Lemma 4.3 from [45]) there exists such a function ψ N that ψ N  a0 and V ψ N = 0, i.e., v αN = 0 for all α . In this case, (102) implies (101). 2. We now obtain a priori estimates of approximate solution ψ N . Let us multiplying (101) by qαN and sum the resulting expressions over α . Taking into account (48) and (25) we obtain



2



2





ν Λs+1 ψ N  + σ Λψ N   F −s ψ N s .

(105)

The second term in the left-hand side of inequality (105) can be neglected. Then using again Lemma 3 and dividing the result by Λψ N s we obtain the uniform (in N) boundness of functions Λψ N in Hs :

  Λψ N   C F −s , s

where C = 1/ν

√

2.

(106)

Let us replace W α by χα−1  W α in (101), multiply the equation obtained by qαN and sum over α taking account of (57) and (35). Then, applying the transformations analogous to those used in deriving the inequality (106), we obtain the uniform (in N) boundness of functions ψ N in Hs :

  ψ N   1 F −s . s

(107)

ν

3. We now prove the convergence of approximate solutions ψ N as N → ∞. According to (106) and (107), it can be selected a subsequence from {ψ N } (we will use the same notation for this subsequence) such that

ψ N converges to ψ weakly in Hs+2 , N

(108)

s

ψ converges to ψ weakly in H .

(109)

It follows from (109) and Corollary 3 that for any r < s,

  ψ N − ψ  → 0 as N → ∞. r

(110)

We now prove that

 









J αN ≡ J ψ N , W α , ψ N + 2μ → J α ≡ J (ψ, W α ), ψ + 2μ as N → ∞. Indeed,

 N         J − J α    J ψ N − ψ, W α , ψ N + 2μ  +  J (ψ, W α ), ψ N − ψ  ≡ γ1 + γ2 . α Due to Lemmas 6 and 8, and estimates (106) and (107),





γ1  C ψ N − ψ .

(111)

Y.N. Skiba / J. Math. Anal. Appl. 388 (2012) 627–644

643

γ2 → 0. Thus (110) implies (111). Therefore, passing to the limit and using (108)–(111) we obtain that    ν Λs+2 ψ, Λs W α − σ ψ, W α  + J (ψ, W α ), ψ + 2μ =  F , W α  (112)

And due to (109),



holds for every W α , and hence, equality (99) holds for all elements h ∈ Hs .

 −ψ . Then for all h ∈ Hs ,  and ψ  be two solutions of problem (99), and let g = ψ II. The uniqueness of weak solution. Let ψ function g satisfies the following equation:













 + 2μ + J (ψ , h),  g = 0. ν Λs+2 g , Λs h − σ  g , h + J ( g , h), ψ

(113)

Putting h = g in (113) and taking into account (48), (65), Lemmas 1 and 3, and estimate ψ s  F −s /ν (see (107)), we get



2





, g ),  g  0 = ν Λs+1 g  + σ Λ g 2 + J (ψ









  g 2  ν −1 ν 2 2s−1 − M F −s  g 2 . ν a2(s−1) − M ψ

Thus, if condition (100) is fulfilled then  g = 0, i.e., g (x) ≡ 0, and the solution is unique. The theorem is proved. Remark 3. Restriction on the coefficient ν can be relaxed if one take into account the term inequality to the term containing the Jacobian.

2

σ Λ g 2 and apply the Young’s

Acknowledgments This work was supported by the projects FOSEMARNAT 2004-01-160 (CONACyT, Mexico) and PAPIIT IN105608-3 (UNAM, Mexico), and by the grant 14539 of National System of Investigators (CONACyT, Mexico). Comments from anonymous referee, who made valuable suggestions, were also much appreciated.

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