On the spectral properties for the linearized problem around space-time periodic states of the compressible Navier-Stokes equations
aa r X i v : . [ m a t h . A P ] F e b On the spectral properties for the linearized problemaround space-time periodic states of the compressibleNavier-Stokes equations
Mohamad Nor Azlan , Shota Enomoto , Yoshiyuki Kagei Abstract
This paper studies the linearized problem for the compressible Navier-Stokes equation around space-time periodic state in an infinite layer of R n ( n = 2 , t → ∞ . Keywords : Compressible Navier-Stokes equation, infinite layer, periodicstates, linearized stability.
This paper is concerned with the stability of space-time periodic state to thecompressible Navier-stokes equation ∂ ˜ t ˜ ρ + div ˜ x (˜ ρ ˜ v ) = 0 , (1.1)˜ ρ ( ∂ ˜ t ˜ v + ˜ v · ∇ ˜ x ˜ v ) − µ ∆ ˜ x ˜ v − ( µ + µ ′ ) ∇ ˜ x div ˜ x ˜ v + ∇ ˜ x ˜ p (˜ ρ ) = ˜ ρ ˜ G (1.2)in an n dimensional infinite layer ˜Ω = R n − × (0 , d ) , for n = 2 ,
3. Here, ˜ ρ =˜ ρ (˜ x, ˜ t ) and ˜ v = ⊤ (˜ v (˜ x, ˜ t ) , · · · , ˜ v n (˜ x, ˜ t )) denote the unknown density and thevelocity at time ˜ t ≥ x ∈ ˜Ω, respectively. ˜ p (˜ ρ ) is the pressure, asmooth function of ρ and satisfies ˜ p ′ ( ρ ∗ ) > ρ ∗ > µ and µ ′ are the viscosity coefficients that areassumed to be constants satisfying µ > , n µ + µ ′ ≥ . Kuala Lumpur, MALAYSIA ([email protected]) General Education Department, National Institute of Technology, Toba College, Mie 517-8501, JAPAN ([email protected]) Department of Mathematics, Tokyo Institute of Technology, Tokyo 152-8551, JAPAN([email protected]) µ ′ µ satisfies µ ′ µ ≤ µ (1.3)for a certain constant µ >
0. ˜ G = ˜ G (˜ x, ˜ t ) is a given external force satisfying˜ G (˜ x ′ + π ˜ α i e ′ i , ˜ x n , ˜ t ) = ˜ G (˜ x ′ , ˜ x n , ˜ t ), ˜ G (˜ x ′ , ˜ x n , ˜ t + T ) = ˜ G (˜ x ′ , ˜ x n , ˜ t ) , where ˜ α i ( i =1 , · · · , n −
1) are positive constants and e ′ i = ⊤ (0 , · · · , , i , , · · · , ∈ R n − .The system (1.1)-(1.2) is considered under the boundary condition and initialcondition ˜ v | ˜ x n =0 ,d = 0 , (1.4)(˜ ρ, ˜ v ) | ˜ t =0 = (˜ ρ , ˜ v ) . (1.5)One can see that if ˜ G is sufficiently small, the system (1.1)-(1.2) with (1.4)and (1.5) has a space-time periodic state ˜ u p = ⊤ (˜ ρ p , ˜ v p ).Our main interest is the spectral properties of the linearized evolution oper-ator around the space-time periodic state ˜ u p .The large time behavior of solutions around the motionless state in un-bounded domains has been investigated in detail. See, e.g., [5, 10, 11, 16, 17,19, 20, 21, 22, 23, 24, 26] for the cases of multi-dimensional whole space, halfspace, exterior domains and infinite layer. On the other hand, the stability ofnontrivial flow (e.g., parallel flows and periodic flows) has been paid much atten-tion. A difficulty in the mathematical analysis appears due to the non-uniformvelocity field of parallel flows and periodic patterns, which makes the hyperbolicaspect of the equations (1.1)-(1.2) stronger, and thus, the stability analysis isgetting more difficult compared with that of the motionless state.The stability of parallel flows in an infinite layer has been studied in [2, 3,4, 12, 13, 14]. As for the stability of time periodic parallel flows, it was provedin [3, 4] that the asymptotic leading part of the evolution operator is given bya product of some time periodic function and n − x ′ is useful since the parallel flow is uniform in x ′ variable.In the case of the periodic steady states, since the periodic steady states isnon-uniform in x ′ variable, the analysis by Fourier wave decomposition doesnot work well. In [7], the Bloch wave decomposition was employed by focusingon the spatial periodicity. It was proved in [7] that the linearized semigroupbehaves like a semigroup generated by an n − u ( t ) = ⊤ ( φ, w ) = ⊤ ( γ ( ρ − ρ p ) , v − v p ) takes the following form: ∂ t φ + div( φv p ) + γ div( ρ p w ) = f ( u ) , (1.6) ∂ t w − νρ p ∆ w − ˜ νρ p ∇ div v + ∇ (cid:0) p ′ ( ρ p ) γ ρ p φ (cid:1) + 1 γ ρ p ( ν ∆ v p + ˜ ν ∇ div v p ) φ + v p · ∇ w + w · ∇ v p = g ( u ) , (1.7)on Ω = R n − × (0 , w | ∂ Ω = 0 , (1.8)( φ, w ) | t =0 = ( φ , w ) . (1.9)Here u p = ⊤ ( φ p , w p ) denotes the non-dimensionalization of ˜ u p = ⊤ ( ˜ φ p , ˜ w p ),and ν , ˜ ν and γ are non-dimensional parameters. The terms f ( u ) and g ( u ) arenon-linear terms given by f ( u ) =div( φw ) ,g ( u ) = − w · ∇ w − φρ p ( γ ρ p + φ ) (cid:0) ν ∆ w + ˜ ν ∇ div w − νφγ ρ p ∆ v p − ˜ νφγ ρ p ∇ div v p (cid:1) + φγ ρ p ∇ (cid:0) p (1) (cid:0) ρ p , φγ (cid:1) φγ (cid:1) + φ γ ρ p ( γ ρ p + φ ) ∇ (cid:0) p (cid:0) ρ p + φγ (cid:1)(cid:1) + 1 ρ p ∇ (cid:0) p (2) (cid:0) ρ p , φγ (cid:1) φ γ (cid:1) , where p (1) ( ρ p , φ ) = Z p ′ ( ρ p + θφ ) dθ,p (2) ( ρ p , φ ) = Z (1 − θ ) p ′′ ( ρ p + θφ ) dθ. We consider the linearized problem for (1.6)-(1.9) which can be written as ∂ t u + L ( t ) u = 0 , u | t = s = u (1.10)on Ω = R n − × (0 , u ( t ) = ⊤ ( φ ( t ) , w ( t )) ∈ D ( L ( t )); and L ( t ) is theoperator on L (Ω) × L (Ω) of the form L ( t ) = div( v p ( t ) · ) γ div( ρ p ( t ) · ) ∇ (cid:0) p ′ ( ρ p ( t )) γ ρ p ( t ) · (cid:1) − νρ p ( t ) ∆ − ˜ νρ p ( t ) ∇ div ! + γ ρ p ( ν ∆ v p + ˜ ν ∇ div v p ) v p · ∇ + ⊤ ( ∇ v p ) ! D ( L ( t )) = { u = ⊤ ( φ, w ) ∈ L (Ω); w ∈ H (Ω) , L ( t ) u ∈ L (Ω) } . We denote by U ( t, s ) the solution operator for (1.10). Since L ( t ) has spaceperiodic coefficients, the spectral properties of U ( t, s ) can be analyzed by usingthe Bloch transform as in [7]. If we apply the Bloch transform to U ( t, s ), we havea family { U η ′ ( t, s ) } η ′ ∈ Q ∗ of solution operators, where Q ∗ = Π n − i =1 [ − α i , α i ); andeach U η ′ ( t, s ) is the solution operator for the problem ∂ t u + L η ′ ( t ) u = 0 , u | t = s = u (1.11)on Ω per = Π n − j =1 T παj × (0 , . Here T a = R /a Z ; and L η ′ ( t ) is an operator acting on functions on Ω per whichtakes the form L η ′ ( t ) = ∇ η ′ · ( v p ( t ) · ) γ ∇ η ′ · ( ρ p ( t ) · ) ∇ η ′ (cid:0) p ′ ( ρ p ( t )) γ ρ p ( t ) · (cid:1) − νρ p ( t ) ∆ η ′ − ˜ νρ p ( t ) ∇ η ′ ⊤ ∇ η ′ ! + γ ρ p ( ν ∆ v p + ˜ ν ∇ div v p ) v p · ∇ η ′ + ⊤ ( ∇ v p ) ! , where ∇ η ′ and ∆ η ′ are defined by ∇ η ′ = ∇ + i ˜ η ′ , ∆ η ′ = ∇ η ′ · ∇ η ′ , with ˜ η ′ = ⊤ ( η ′ , ∈ R n .As in [2, 3, 4], one can investigate the spectral properties of U η ′ ( t, s ) by theFloquet theory. Since L η ′ ( t + 1) = L η ′ ( t ) for all t , the large time behavior of U η ′ ( t, s ) is controlled by the spectrum of the monodromy operator U η ′ (1 , B η ′ , where B η ′ is an operator onthe time periodic function space X = L ( T ; L (Ω per ) × L (Ω per )) defined by D ( B η ′ ) = { u = ⊤ ( φ, w ) ∈ X ; w ∈ H ( T ; H − (Ω per ) ∩ L ( T ; H (Ω per )) ,φ ∈ C ( T ; L (Ω per )) , B η ′ u ∈ X } ,B η ′ u = ∂ t u + L η ′ u, u = ⊤ ( φ, w ) ∈ D ( B η ′ ) . The spectrum of − B η ′ gives Floquet exponents of the problem (1.11).Our main result is summarized as follows. If the external force G and Blochparameter η ′ are sufficiently small, then σ ( − B η ′ ) ∩ (cid:26) λ ∈ C ; Re λ > − β , | λ − πik | ≤ π , k ∈ Z (cid:27) = { λ η ′ ,k : k ∈ Z } , (1.12)4here λ η ′ ,k is a simple eigenvalue of − B η ′ satisfying λ η ′ ,k = 2 πik − i n − X j =1 a j η j − n − X j,k =1 a jk η j η k + O ( | η ′ | ) ( η ′ →
0) (1.13)with some constants a j , a jk ∈ R , where ( a jk ) ≤ j,k ≤ n − is positive definite.Our result, in fact, will yield the following asymptotic behavior of a partof the solution operator U ( t, s ). In a similar manner to [2, 3, 4], based on ourmain result and the Floquet theory, one can show that there exist a boundedprojection P ( t ) on L (Ω) × L (Ω) such that P ( t + 1) = P ( t ) and the followingestimates hold: k P ( t ) U ( t, s ) u k L (Ω) × L (Ω) ≤ C (1 + t − s ) − n − k u k L (Ω) × L (Ω) , k P ( t ) U ( t, s ) u − u (0) ( t ) H ( t − s ) σ k L (Ω) × L (Ω) ≤ C ( t − s ) − n − ( p − ) − k u k L p (Ω) × L p (Ω) (1 ≤ p ≤ . Here u (0) = u (0) ( x ′ , x n , t ) is some function πα i -periodic in x i ( i = 1 , · · · , n − t and H ( t ) σ is a solution of the linear heat equation ( ∂ t σ + P n − j =1 a j ∂ x j σ − P n − j,k =1 a jk ∂ x j ∂ x k σ = 0 ,σ | t =0 = σ . This paper is organized as follows. In Section 2, we transform the equations(1.1)-(1.2) into a non-dimensional form and introduce basic notation that is usedthroughout the paper. In Section 3, we first state the existence of a space-timeperiodic state and then state the main results of this paper. Section 4 is devotedto the proof of the main results. In Section 5, we give a proof of the existenceof a space-time periodic state.
In this section, we transform (1.1)-(1.2) into a non-dimensional form and intro-duce some function spaces and notations which are used throughout the paper.We rewrite the problem into the non-dimensional form. We introduce thefollowing non-dimensional variables:˜ x = dx, ˜ t = T t, ˜ ρ = ρ ∗ ρ, ˜ v = dT v, ˜ p = ρ ∗ ˜ p ′ ( ρ ∗ ) p, ˜ G = [ ˜ G ] ,T, ˜Ω per G, where [ ˜ G ] ,T, ˜Ω per = X j =0 T j − d − j ) − n Z T k ∂ j ˜ t ˜ G k H − j (˜Ω per ) d ˜ t . per = Π n − j =1 T π ˜ αj × (0 , d ); and k · k H m (˜Ω per ) denotes the usual H m -normover ˜Ω per (whose definition is given below).Under this change of variables, the domain ˜Ω is transformed intoΩ = R n × (0 , . The equations (1.1)-(1.2) are rewritten as ∂ t ρ + div( ρv ) = 0 , (2.1) ρ ( ∂ t v + v · ∇ v ) − ν ∆ v − ˜ ν ∇ div v + γ ∇ p ( ρ ) = SρG. (2.2)Here ν , ˜ ν , γ and S are non-dimensional parameters defined by ν = µTρ ∗ d , ˜ ν = ( µ + µ ′ ) Tρ ∗ d , γ = Td p ˜ p ′ ( ρ ∗ ) , S = T d [ ˜ G ] ,T, ˜Ω per . We note that p ′ (1) = 1 and [ G ] , , Ω per = 1 . Furthermore, the assumption (1.3) is written as ν ≤ ν + ˜ ν ≤ ν ∗ ν for some positive number ν ∗ .The boundary and initial conditions (1.4)-(1.5) are transformed into v | x n =0 , = 0 , (2.3)( ρ, v ) | t =0 = ( ρ , v ) . (2.4)We next introduce notation used throughout this paper. Let D be a domain.We denote by L p ( D ) (1 ≤ p ≤ ∞ ) the usual Lebesgue space on D and its normis denoted by k·k L p ( D ) . Let m be a nonnegative integer. H m ( D ) denotes the m -th order L -Sobolev space on D and its norm denoted by k · k H m ( D ) . C m ( D ) isdefined as the set of C m -functions having compact supports in D . Furthermore,we denote by H m ( D ) the completion of C ∞ ( D ) in H m ( D ) and the dual spaceof H m ( D ) is denoted by H − m ( D ).We simply write the set of all vector fields w = ⊤ ( w , · · · , w n ) on D as w j ∈ L p ( D ) (resp., H m ( D )) and its norm is denoted by k · k L p ( D ) (resp., k · k H m ( D ) ).For u = ⊤ ( φ, w ) with φ ∈ H k ( D ) and w ∈ H m ( D ), we define k u k H k ( D ) × H m ( D ) =( k φ k H k ( D ) + k w k H m ( D ) ) . When k = m , we simply write k u k H k ( D ) × H k ( D ) = k u k H k ( D ) .We set [[ f ( t )]] k = (cid:0) [ k ] X j =0 k ∂ jt f ( t ) k H k − j (Ω per ) (cid:1) , where [ k ] is the largest integer smaller than or equal to k .6he inner product of L is defined as( f, g ) = Z Ω per f ( x ) g ( x ) dx for f, g ∈ L (Ω per ). Here, g denotes the complex conjugate of g . Moreover, themean value of f = f ( x ) and g = g ( x, t ) over Ω per and Ω per × T is written as h f i = Z Ω per f ( x ) dx and hh g ii = Z h g ( t ) i dt, respectively. We next introduce a weighted inner product: hh u , u ii = Z h u ( t ) , u ( t ) i t dt for u j = ⊤ ( φ j , w j ) j = 1 ,
2, where h u ( t ) , u ( t ) i t = Z Ω per φ ( t ) φ ( t ) p ′ ( ρ p ( t )) γ ρ p ( t ) + w ( t ) · w ( t ) ρ p ( t ) dx. Here ρ p denotes the density of the space-time periodic state given in Proposition3.1 below. By Proposition 3.1, we see that ρ p ≥ ρ on Ω per × T for a positiveconstant ρ and that | ρ p ( x, t ) − | ≤ and | p ′ ( ρ p ( x, t )) − | ≤ for all ( x, t ) ∈ Ω per × T . Therefore, hh u , u ii defines an inner product.We finally define L ∗ (Ω per ) and H m ∗ (Ω per ) by L ∗ (Ω per ) := { φ ∈ L (Ω per ); h φ i = 0 } and H m ∗ (Ω per ) = H m (Ω per ) ∩ L ∗ (Ω per ) , respectively.We next introduce the Bogovskii lemma [1, 8]. Lemma 2.1 ([1, 8]) . There exist a bounded operator B : L ∗ (Ω per ) → H (Ω per ) such that for any f ∈ L ∗ (Ω per ) , div B f = f, k∇B f k L (Ω per ) ≤ C k f k L (Ω per ) , where C is a positive constant depending only on Ω per . Furthermore, if f = div g with g = ⊤ ( g , · · · , g n ) satisfying g ∈ H (Ω per ) , then kB (div g ) k L (Ω per ) ≤ C k g k L (Ω per ) .
7n terms of the Bogovskii operator B , we introduce the following inner prod-uct on L ∗ (Ω per ) × L (Ω per ). For each t ∈ T , we define (( u , u )) t by(( u , u )) t = h u , u i t − δ [( w , B φ ) + ( B φ , w )] , where δ is a positive constant. One can see that there exists a positive constant C such that if 0 < δ < Cγ , then (( · , · )) t defines an inner product satisfying12 k u k L (Ω per ) ,γ ≤ (( u, u )) t ≤ k u k L (Ω per ) ,γ , where k u k L (Ω per ) ,γ = 1 γ k φ k L (Ω per ) + k w k L (Ω per ) . We next introduce the Bloch transform. Let S ( R n − ) be the Schwartzspace on R n − . We define the Bloch transform T by( T ϕ )( x ′ , η ′ ) = 1(2 π ) n − | Q | X k ,...,k n − ∈ Z n − ˆ ϕ ( η ′ + n − X j =1 k j α j e ′ j ) e i P n − j =1 k j α j x ′ = 1 | Q ∗ | X l ,...l n − ∈ Z n − ϕ ( x ′ + n − X j =1 l j πα j e ′ j ) e − iη ′ · ( x ′ + P n − j =1 l j παj e ′ j ) for ϕ ∈ S ( R n − ), where ˆ ϕ denotes the Fourier transform of ϕ :ˆ ϕ ( ξ ′ ) = Z R n − ϕ ( x ′ ) e − iξ ′ · x ′ dx ′ ;and Q = n − Y i =1 (cid:2) − πα i , πα i (cid:1) , Q ∗ = n − Y i =1 (cid:2) − α i , α i (cid:1) . Let ϕ ( x ′ , η ′ ) be in C ∞ ( R n − × R n − ) such that ϕ ( x ′ , η ′ ) is Q -periodic in x ′ and ϕ ( x ′ , η ′ ) e iη ′ · x ′ is Q ∗ -periodic in η ′ . We define ( S ϕ )( x ′ ) by( S ϕ )( x ′ ) = 1 | Q ∗ | Z Q ∗ ϕ ( x ′ , η ′ ) e iη ′ · x ′ dη ′ , where x ′ ∈ R n − . Note that ϕ ( x ′ , η ′ + α j e ′ j ) = ϕ ( x ′ , η ′ ) e − iα j e ′ j · x ′ .The operators T and S have the following properties. See, e.g., [25] for thedetails. Proposition 2.2. (i) ( T ϕ )( x ′ , η ′ ) is Q -periodic in x ′ and ( T ϕ )( x ′ , η ′ ) e iη ′ .x ′ is Q ∗ -periodic in η ′ . (ii) T is uniquely extended to an isometric operator from L ( R n − ) to L ( Q ∗ ; L ( Q )) . (iii) S is the inverse operator of T . (iv) Let ϕ be Q -periodic in x ′ . Then it holds that T ( ψϕ ) = ψ T ϕ . (v) T ( ∂ x j ϕ ) = ( ∂ x j + iη j ) T ϕ and T defines an isomorphism from H m ( R n − ) to L ( Q ∗ ; H m ( Q )) . Main Results
In this section, we state the main results of this paper. We first state theexistence of the space-time periodic state of (2.1)-(2.3). We consider the timeperiodic problem for ∂ t ρ + div( ρv ) = 0 , (3.1) ρ ( ∂ t v + v · ∇ v ) − ν ∆ v − ˜ ν ∇ div v + γ ∇ p ( ρ ) = SρG (3.2)in Ω per under the boundary condition v | x n =0 , = 0 . (3.3) Proposition 3.1.
Let G ∈ ∩ j =0 C j ( T ; H − j (Ω per )) with [ G ] , , Ω per = 1 .There exist positive constants ν , γ , ε and a such that if ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o , then the following assertions hold.There exist a space-time periodic solution u p = ⊤ ( ρ p , v p ) = ⊤ (1 + φ p , v p ) ∈∩ j =0 C j ( T ; H − j (Ω per ) × H − j (Ω per )) ∩ H j ( T ; H − j (Ω per ) × H − j (Ω per )) of problem (3 . - (3 . satisfying h ρ p ( t ) i = 1 for each t ∈ T and ρ p = ρ p ( x, t ) ≥ ρ for a positive constant ρ . Furthermore, u p satisfies the following estimates γ [[ φ p ( t )]] + [[ v p ( t )]] ≤ C ν γ , (3.4) Z γ ν + ˜ ν [[ ∇ φ p ( s )]] + ( ν + ˜ ν ) k ∂ t φ k L (Ω per ) + ν ν + ˜ ν [[ v p ( s )]] dt ≤ C ν γ , (3.5) where C is a positive constant independent of ν , ˜ ν , γ and S . Remark 3.2. (i)
Since k φ p k L ∞ (Ω per ) ≪ if γ ν +˜ ν ≫ and ν ν +˜ ν ≫ , we have ρ p ∼ , and therefore p ′ ( ρ p ) ∼ . (ii) If γ ν +˜ ν ≫ , then the assumption on S in Proposition . implies S ≤ ε √ a min (cid:26) , ν γ (cid:27) . The proof of Proposition 3.1 is essentially the same as that given in [27].Since we solve the time periodic problem in H (Ω per ) × H (Ω per ) and we needto know the dependence of the estimates on the parameters ν , ˜ ν and γ , we willgive an outline of the proof of Proposition 3.1 in Section 5 below.Our main result is concerned with the spectrum of the linearized solutionoperator U ( t, s ) around the space-time periodic solution u p = ⊤ ( φ p , v p ).As was mentioned in the introduction, we apply Bloch transform to (1.10).By Proposition 2.2, we then obtain (1.11) and consider the spectrum of − B η ′ to obtain the Floquet exponents of (1.11) for | η ′ | ≪ heorem 3.3. There exist positive constants ν , γ , ε and a such that if νν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o , then the followingassertions hold. (i) There exists a positive constant r = r ( ν, ˜ ν, γ, ν , γ ) such that if | η ′ | ≤ r , then Σ := (cid:26) λ ∈ C ; Re λ > − β , | λ − πik | ≥ π , k ∈ Z (cid:27) ⊂ ρ ( − B η ′ ) , and for λ ∈ Σ k ( λ + B η ′ ) − F k L ( T ; L (Ω per ) × H (Ω per )) ≤ C k F k X . (ii) If | η ′ | ≤ r , then σ ( − B η ′ ) ∩ (cid:26) λ ∈ C ; Re λ > − β , | λ − πik | ≤ π , k ∈ Z (cid:27) = { λ η ′ ,k : k ∈ Z } , where λ η ′ ,k is a simple eigenvalue that satisfies λ η ′ ,k = 2 πik − i n − X j =1 a j η j − n − X j,k =1 a jk η j η k + O ( | η ′ | ) ( η ′ → with some constants a j , a jk ∈ R satisfying n − X j,k =1 a jk ξ j ξ k ≥ κ γ ν | ξ ′ | , for all ξ ′ = ⊤ ( ξ , · · · , ξ n − ) ∈ R n − and some positive constant κ . As a conse-quence Re λ η ′ ,k ≤ − κ γ ν | η ′ | . (3.6)We next consider eigenfunctions for eigenvalues λ η ′ , . We introduce theadjoint operator B ∗ η ′ defined by D ( B ∗ η ′ ) = { u = ⊤ ( φ, w ) ∈ X ; w ∈ H ( T ; H − (Ω per )) ∩ L ( T ; H (Ω per )) ,φ ∈ C ( T ; L (Ω per )) , B ∗ η ′ u ∈ X } ,B ∗ η ′ u = − ∂ ∗ t u + L ∗ η ′ ( · ) u, u = ⊤ ( φ, w ) ∈ D ( B ∗ η ′ ) , where ∂ ∗ t = γ ρ p p ′ ( ρ p ) ∂ t ( p ′ ( ρ p ) γ ρ p · ) ρ p ∂ t ( ρ p · ) ! , ∗ η ′ = − v p · ∇ η ′ (cid:0) p ′ ( ρ p ) ρ p · (cid:1) ρ p p ′ ( ρ p ) − γ ⊤ ∇ η ′ · ( ρ p · ) −∇ η ′ (cid:0) p ′ ( ρ p ) γ ρ p · (cid:1) − νρ p ∆ η ′ − ˜ νρ p ∇ η ′ ⊤ ( ∇ η ′ ) ! + γ p ′ ( ρ p ) ( ν ∆ v p + ˜ ν ∇ div v p )0 − div v p − ρ p v p · ∇ η ′ ( ρ p · ) + ∇ v p ! . Let u (0) and u (0) ∗ denote the eigenfunctions for the eigenvalue 0 of − B and − B ∗ satisfying hh u (0) , u (0) ∗ ii = 1 . It then follows that u η ′ = 12 πi Z | λ | = π ( λ + B η ′ ) − u (0) dλ and u ∗ η ′ = 12 πi Z | λ | = π ( λ + B ∗ η ′ ) − u (0) ∗ dλ are eigenfunctions for B η ′ and B ∗ η ′ associated with eigenvalue λ η ′ , and λ η ′ , ,respectively. Note that eigenfunctions for eigenvalues λ η ′ ,k are given by e πikt u η ′ and the same holds for the adjoint eigenfunctions.We have the following estimates for the eigenfunctions for u η ′ and u ∗ η ′ . Theorem 3.4.
Under the same assumptions of Theorem . the following es-timates hold uniformly for | η ′ | ≤ r and t ∈ T : k u η ′ ( t ) k H (Ω per ) ≤ C, k u η ′ ( t ) − u (0) ( t ) k H (Ω per ) ≤ C | η ′ | , k u ∗ η ′ ( t ) k H (Ω per ) ≤ C. Theorems 3.3 and 3.4 will be proved in Section 4.
In this section, we prove Theorems 3.3 and 3.4. To do so, we consider theresolvent problem ( λ + B η ′ ) u = F (4.1)for u ∈ D ( B η ′ ) with | η ′ | ≪
1, where F = ⊤ ( f, g ) is a given function.We expand B η ′ as B η ′ := B + n − X j =1 η j B (1) j + n − X j,k =1 η j η k B (2) j,k . B = ∂ t + div( v p · ) γ div( ρ p · ) ∇ (cid:0) p ′ ( ρ p ) γ ρ p · (cid:1) − νρ p ∆ − ˜ νρ p ∇ div ! + γ ρ p ( ν ∆ v p + ˜ ν ∇ div v p ) v p · ∇ + ⊤ ( ∇ v p ) ! ,B (1) j = i v jp γ ρ p ⊤ e j (cid:0) p ′ ( ρ p ) γ ρ p (cid:1) e j − ρ p (2 ν e j ⊗ e j ∂ x j + ˜ ν e j div − ˜ ν ∇ ( ⊤ e j )) + v jp ! ,B (2) j,k = (cid:18) ρ p ( νδ jk I n + ˜ ν e j ⊤ e k ) (cid:19) . We set M η ′ = n − X j =1 η j B (1) j + n − X j,k =1 η j η k B (2) j,k . We begin with investigating the spectral properties of B . For this purpose,we first consider the unique solvability for the time periodic problem ( ∂ t u + L ( t ) u = F, h φ ( t ) i = 0 , (4.2)when F = ⊤ ( f, g ) ∈ L ( T ; L ∗ (Ω per ) × H − (Ω per )). Proposition 4.1.
There exists positive constants ν , γ , ε and a such thatif ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o , then thefollowing assertions hold true. For any F ∈ L ( T ; L ∗ (Ω per ) × H − (Ω per )) ,there exists a unique time periodic solution u = ⊤ ( φ, w ) ∈ C ( T ; L ∗ (Ω per ) × L (Ω per )) ∩ L ( T ; L (Ω per ) × H (Ω per )) to (4 . . Furthermore, the solution u satisfies k u ( t ) k L (Ω per ) ,γ + Z t e − β ( t − s ) δ k φ ( s ) k L (Ω per ) ds + Z t e − β ( t − s ) (cid:16) ν k∇ w ( s ) k L (Ω per ) + ˜ ν k div w ( s ) k L (Ω per ) (cid:17) ds ≤ C − e − β Z (cid:18) δγ + δ ν (cid:19) k f ( s ) k L (Ω per ) + 1 ν k g ( s ) k H − (Ω per ) ds. (4.3) Here δ = C min n ν +˜ ν , νγ , γ o and β = C min (cid:8) ν + ˜ ν, δγ (cid:9) , where C is apositive constant independent of ν , ˜ ν , γ and S . To prove Proposition 4.1, we prepare the following lemma about the estimateof solution of the initial value problem for (4.2) under the initial condition u | t =0 = u = ⊤ ( φ , w ) , (4.4)when u ∈ L ∗ (Ω per ) × H (Ω per ) and F ∈ L ( T ; L ∗ (Ω per ) × H − (Ω per )).12 emma 4.2. There exists positive constants ν , γ , ε and a such that if ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o , then there exists aunique solution u = ⊤ ( φ, w ) ∈ C ([0 , ∞ ); L ∗ (Ω per ) × L (Ω per )) ∩ L ([0 , ∞ ); L ∗ (Ω per ) × H (Ω per )) to (4 . and (4 . . Furthermore, u satisfies k u ( t ) k L (Ω per ) ,γ + Z t e − β ( t − s ) δ k φ ( s ) k L (Ω per ) ds + Z t e − β ( t − s ) (cid:16) ν k∇ w ( s ) k L (Ω per ) + ˜ ν k div w ( s ) k L (Ω per ) (cid:17) ds ≤ e − β t k u k L (Ω per ) ,γ + C − e − β Z (cid:18) δγ + δ ν (cid:19) k f ( s ) k L (Ω per ) + 1 ν k g ( s ) k H − (Ω per ) ds (4.5) for t ≥ . Proof.
Since v p ∈ ∩ j =0 C j ( T ; H − j (Ω per )), one can prove the existence of asolution to (4.2) and (4.4) with u ∈ L ∗ (Ω per ) × H (Ω per ) in a standard way bycombining the method of characteristics and the parabolic theory.We prove the estimate (4.5). We employ the energy method by Heywood-Padula [9]. We compute Re(( ∂ t u + L ( t ) u, u )) t = Re(( F, u )) t . In a similar way tothe proof of [7, Lemma 4.3], we see from Lemma 2.1 and Proposition 3.1 thatthere exist positive constants ν , γ , ε and a such that if δ = C min { ν +˜ ν , νγ , γ } , ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o , then the follow-ing estimate holds:Re(( L ( t ) u, u )) t ≥ ν k∇ w k L (Ω per ) + ˜ ν k div w k L (Ω per ) + δ k φ k L (Ω per ) . On the other hand, we haveRe((
F, u )) t ≤ δ k φ k L (Ω per ) + ν k∇ w k L (Ω per ) + C (cid:18) δγ + δ ν (cid:19) k f k L (Ω per ) + Cν k g k H − (Ω per ) . It then follows12 ddt (( u ( t ) , u ( t ))) t + ν k∇ w k L (Ω per ) + ˜ ν k div w k L (Ω per ) + δ k φ k L (Ω per ) ≤ C (cid:18) δγ + δ ν (cid:19) k f k L (Ω per ) + Cν k g k H − (Ω per ) . Multiplying this by e β t and integrating the resulting inequality over [0 , t ], we13ave k u ( t ) k L (Ω per ) ,γ + Z t e − β ( t − s ) δ k φ ( s ) k L (Ω per ) ds + Z t e − β ( t − s ) (cid:16) ν k∇ w ( s ) k L (Ω per ) + ˜ ν k div w ( s ) k L (Ω per ) (cid:17) ds ≤ e − β t k u k L (Ω per ) ,γ + C Z t e − β ( t − s ) (cid:18) δγ + δ ν (cid:19) k f ( s ) k L (Ω per ) ds + C Z t e − β ( t − s ) ν k g ( s ) k H − (Ω per ) ds. (4.6)We apply Lemma 4.3 below to the right hand side of (4.6) and obtain k u ( t ) k L (Ω per ) ,γ + Z t e − β ( t − s ) δ k φ ( s ) k L (Ω per ) ds + Z t e − β ( t − s ) (cid:16) ν k∇ w ( s ) k L (Ω per ) + ˜ ν k div w ( s ) k L (Ω per ) (cid:17) ds ≤ e − β t k u k L (Ω per ) ,γ + C − e − β Z (cid:18) δγ + δ ν (cid:19) k f ( s ) k L (Ω per ) ds + C − e − β Z ν k g ( s ) k H − (Ω per ) ds. This completes the proof.
Lemma 4.3. If f ∈ L ( T ; L (Ω per )) , then Z t e − β ( t − s ) k f ( s ) k L (Ω per ) ds ≤ − e − β Z k f ( s ) k L (Ω per ) ds. Proof.
We set N = [ t ], then Z t e − β ( t − s ) k f ( s ) k L (Ω per ) ds = N − X k =0 Z k +1 k e − β ( t − s ) k f ( s ) k L (Ω per ) ds + Z tN e − β ( t − s ) k f ( s ) k L (Ω per ) ds =: I + I Since f ∈ L ( T ; L (Ω per )), we have I = N − X k =0 Z e − β ( t − ˜ s − k ) k f ( k + ˜ s ) k L (Ω per ) d ˜ s ≤ N − X k =0 e − β ( t − k − Z k f (˜ s ) k L (Ω per ) d ˜ s e β e β − Z k f (˜ s ) k L (Ω per ) d ˜ s and I = Z t − N e − β ( t − ˜ s − N ) k f (˜ s + N ) k L (Ω per ) d ˜ s ≤ Z t − N e − β ( t − ˜ s − N ) k f (˜ s ) k L (Ω per ) d ˜ s ≤ Z k f (˜ s ) k L (Ω per ) d ˜ s. Therefore, we obtain Z t e − β ( t − s ) k f ( s ) k L (Ω per ) ds ≤ e β e β − Z k f (˜ s ) k L (Ω per ) d ˜ s. This completes the proof.We are in a position to prove Proposition 4.1.
Proof of Proposition 4.1.
We denote by u ♯ the solution of (4.2) and (4.4)with u = 0. We then see from Lemma 4.2 that u ♯ satisfies k u ♯ ( t ) k L (Ω per ) ,γ ≤ C − e − β Z (cid:18) δγ + δ ν (cid:19) k f ( s ) k L (Ω per ) + 1 ν k g ( s ) k H − (Ω per ) ds (4.7)for t ≥
0. Let m, n ∈ N with m > n . Since F is periodic in t of period1, the function u ♯ ( t + ( m − n )) − u ♯ ( t ) is the solution of (4.2) and (4.4) with F = 0 and u = u ♯ ( m − n ). Hence, it follows from (4.5) with F = 0 that u ♯ ( t + ( m − n )) − u ♯ ( t ) satisfies k u ♯ ( t + ( m − n )) − u ♯ ( t ) k L (Ω per ) ,γ ≤ e − β t k u ♯ ( m − n ) k L (Ω per ) ,γ . We set t = n in this inequality. It then follows from (4.7) that k u ♯ ( m ) − u ♯ ( n ) k L (Ω per ) ,γ ≤ e − β n k u ♯ ( m − n ) k L (Ω per ) ,γ ≤ e − β n C − e − β Z (cid:18) δγ + δ ν (cid:19) k f ( s ) k L (Ω per ) + 1 ν k g ( s ) k H − (Ω per ) ds. We thus obtain k u ♯ ( m ) − u ♯ ( n ) k L (Ω per ) ,γ → n → ∞ ) . { u ♯ ( m ) } is a Cauchy sequence in L ∗ (Ω per ) × L (Ω per ). It then followsthat there exists ˜ u ♯ ∈ L ∗ (Ω per ) × L (Ω per ) such that u ♯ ( m ) converges to ˜ u ♯ strongly in L ∗ (Ω per ) × L (Ω per ), and ˜ u ♯ satisfies k ˜ u ♯ k L (Ω per ) ,γ ≤ C − e − β Z (cid:18) δγ + δ ν (cid:19) k f ( s ) k L (Ω per ) + 1 ν k g ( s ) k H − (Ω per ) ds. (4.8)We then see from the argument by Valli that the solution u of (4.2) and (4.4)with u = ˜ u ♯ is a time periodic solution of (4.2). Furthermore, applying (4.5)and (4.8), we obtain k u ( t ) k L (Ω per ) ,γ + Z t e − β ( t − s ) δ k φ ( s ) k L (Ω per ) ds + Z t e − β ( t − s ) (cid:16) ν k∇ w ( s ) k L (Ω per ) + ˜ ν k div w ( s ) k L (Ω per ) (cid:17) ds ≤ C − e − β Z (cid:18) δγ + δ ν (cid:19) k f ( s ) k L (Ω per ) + 1 ν k g ( s ) k H − (Ω per ) ds. This completes the proof.The following proposition shows that 0 is an eigenvalue of − B . We alsogive the estimates of an eigenfunction for the eigenvalue 0. Proposition 4.4.
There exist positive constants ν , γ , ε and a such that thefollowing assertions hold. If ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o ,then there exists a solution u (0) = ⊤ ( φ (0) , w (0) ) ∈ D ( B ) of ( B u (0) = 0 , hh φ (0) ii = 1 . (4.9) Furthermore, ˜ u (0) = ⊤ ( ˜ φ (0) , w (0) ) = ( φ (0) − , w (0) ) satisfies γ [[ ˜ φ (0) ( t )]] + [[ w (0) ( t )]] ≤ Cνγ , (4.10) Z ν + ˜ ν k ˜ φ (0) k H (Ω per ) + ν + ˜ νγ k ∂ t ˜ φ (0) k L (Ω per ) + ν ν + ˜ ν [[ w (0) ]] ds ≤ Cνγ . (4.11)Before proving Proposition 4.4, we state one proposition on the spectrum of − B which immediately follows from Proposition 4.4. Proposition 4.5.
Under the assumption of Proposition . for each k ∈ Z , πik is an eigenvalue of − B with eigenfunction e πikt u (0) . To prove Proposition 4.4, we decompose φ into φ = 1 + ˜ φ and rewrite (4.9)for u = ⊤ ( φ, w ) as (4.2) for ˜ u = ⊤ ( ˜ φ, w ). We thus consider the time periodic16roblem for (4.2) with F = − div v p ∇ (cid:16) P ′ ( ρ p ) γ ρ p (cid:17) + γ ρ p ( ν ∆ v p + ˜ ν ∇ div v p ) ! . (4.12)As in the proof of Proposition 4.1, we first consider the initial value problemfor (4.2) with F given in (4.12) under the initial condition (4.4) with u ∈ H ∗ (Ω per ) × ( H (Ω per ) ∩ H (Ω per )). Lemma 4.6.
There exist positive constants ν , γ , ε and a such that if ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o , then there exists aunique solution u = ⊤ ( φ, w ) ∈ ∩ j =0 C j ([0 , ∞ ); H − j ∗ (Ω per ) × H − j (Ω per )) ∩ H j ([0 , ∞ ); H − j ∗ (Ω per ) × ( H − j (Ω per ) ∩ H (Ω per ))) to (4 . and (4 . with u ∈ H ∗ (Ω per ) × ( H (Ω per ) ∩ H (Ω per )) and F given by (4 . . Furthermore, u satisfies γ [[ φ ( t )]] + [[ w ( t )]] + Z t e − a ν +˜ νγ ( t − s ) ν ν + ˜ ν [[ w ( s )]] ds + Z t e − a ν +˜ νγ ( t − s ) (cid:18) ν + ˜ ν k φ ( s ) k H (Ω per ) + ν + ˜ νγ k ∂ t φ ( s ) k L (Ω per ) (cid:19) ds ≤ e − a ν +˜ νγ t (cid:18) γ [[ φ ]] + [[ w ]] (cid:19) + C − e − a ν +˜ νγ γ . Proof.
Since v p ∈ ∩ j =0 C j ( T ; H − j (Ω per )), one can prove the existence ofsolution to (4.2) and (4.4) with u ∈ H ∗ (Ω per ) × ( H (Ω per ) ∩ H (Ω per )) ina standard way by combining the method of characteristics and the parabolictheory. We rewrite L ( t ) as L ( t ) = A + M ( t ) , where A = (cid:18) γ div ∇ − ν ∆ − ˜ ν ∇ div (cid:19) ,M ( t ) = v p ( t ) · ∇ + div v p ( t ) γ div( φ p ( t ) · ) ∇ ( p (1) ( φ p ( t )) φ p ( t ) · ) φ p ( t )1+ φ p ( t ) ( ν ∆ + ˜ ν ∇ div) ! + γ ρ p ( t ) ( ν ∆ v p ( t ) + ˜ ν ∇ div v p ( t )) v p ( t ) · ∇ + ⊤ ( ∇ v p ( t )) ! . Here p (1) ( φ ) = R p ′′ (1+ θφ ) γ (1+ θφ ) dθ . Furthermore, we rewrite (4.2) as ∂ t u + Au = F − M u.
17e apply Proposition 5.6 below with m = 2 for (cid:18) fg (cid:19) = F − M u, ˜ f = φ div v p + γ div( φ p w ) . (4.13)As in the proof of Proposition 5.7, we can show that there exists a positiveconstant ν such that if ν ν +˜ ν ≥ ν , then ddt E ( u ) + D ( u ) ≤ N ( u ) . In a similar manner to Proposition 5.12 below by using Proposition 3.1, we canestimate N ( u ) with (4.13) as N ( u ) ≤ C γ ν + ˜ ν [[ v p ]] D ( u ) + 14 D ( u ) + C ν + ˜ ν [[ v p ]] + C ν [[ ∇ φ p ]] . It then follows that there exist positive constants ν , γ , ε and a such that if ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o , we have ddt E ( u ) + D ( u ) ≤ C ν + ˜ ν [[ v p ]] + C ν [[ ∇ φ p ]] . (4.14)By the Poincar´e inequality, we also have D ( u ) ≥ a ν + ˜ νγ E ( u ) . This, together with (4.14), implies E ( u ( t )) + Z t e − a ν +˜ νγ ( t − s ) D ( u ( s )) ds ≤ e − a ν +˜ νγ t E ( u ) + C Z e − a ν +˜ νγ ( t − s ) (cid:18) ν + ˜ ν [[ v p ]] + 1 ν [[ ∇ φ p ]] (cid:19) ds. By using Lemma 4.3, we arrive at E ( u ( t )) + Z t e − a ν +˜ νγ ( t − s ) D ( u ( s )) ds ≤ e − a ν +˜ νγ t E ( u ) + C − e − a ν +˜ νγ Z (cid:18) ν + ˜ ν [[ v p ]] + 1 ν [[ ∇ φ p ]] (cid:19) ds. It then follows from Proposition 3.1 that there exist positive constants ν , γ , ε and a such that if ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o ,then E ( u ( t )) + Z t e − a ν +˜ νγ ( t − s ) D ( u ( s )) ds ≤ e − a ν +˜ νγ t E ( u ) + C − e − a ν +˜ νγ γ . Proof of Proposition 4.4.
Decomposing φ into φ = 1 + ˜ φ , we rewrite theproblem (4.9) for u = ⊤ ( φ, w ) as the problem (4.2) with F given in (4.12)for ˜ u = ⊤ ( ˜ φ, w ). Based on Lemma 4.6, in a similar manner to the proof ofProposition 4.1, by using the argument of Valli [27], we can obtain a timeperiodic solution ˜ u = ⊤ ( ˜ φ, w ) to (4.2) with F given in (4.12) satisfying1 γ [[ ˜ φ ( t )]] + [[ w ( t )]] ≤ C − e − a ν +˜ νγ γ and Z (cid:18) ν + ˜ ν k ˜ φ k H (Ω per ) + ν + ˜ νγ k ∂ t ˜ φ k L (Ω per ) + ν ν + ˜ ν [[ w ]] (cid:19) ds ≤ C − e − a ν +˜ νγ γ . Since 1 − e − a ν +˜ νγ ≥ a ν +˜ νγ if γ ν +˜ ν ≥ γ for some positive constant γ , we have1 γ [[ ˜ φ ( t )]] + [[ w ( t )]] ≤ Cνγ and Z (cid:18) ν + ˜ ν k ˜ φ k H (Ω per ) + ν + ˜ νγ k ∂ t ˜ φ k L (Ω per ) + ν ν + ˜ ν [[ w ]] (cid:19) ds ≤ Cνγ . This completes the proof.To prove that 0 is a simple eigenvalue of − B , we prepare the followinglemma. Lemma 4.7.
Let Π (0) be defined by Π (0) u = hh u, u (0) ∗ ii u (0) = hh φ ii u (0) for u = ⊤ ( φ, w ) , where u (0) ∗ = γ γ ρ p P ′ ( ρ p ) ! . Then the following assertions hold. (i) u (0) ∗ satisfies u (0) ∗ ∈ D ( B ∗ ) , B ∗ u (0) ∗ = 0 and h u (0) ( t ) , u (0) ∗ ( t ) i = 1 . (ii) Π (0) is a bounded projection on X satisfying Π (0) B ⊂ B Π (0) = 0 , Π (0) u (0) = u (0) and Π (0) X = span { u (0) } . − B . Let X and X bedefined by X = Π (0) X and X = ( I − Π (0) ) X. Observe that u = ⊤ ( φ, w ) ∈ X if and only if hh φ ii = 0.As for X and X , it holds the following assertions. Proposition 4.8. If ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o ,then (i) X = Ker( B ) and X = Ran( B ); X is closed. (ii) X = X ⊕ X . (iii) − B . Proof.
Let us show X = Ran( B ). We first assume that F = ⊤ ( f, g ) ∈ Ran( B ). There exists a function u = ⊤ ( φ, w ) ∈ D ( B ) such that B u = F .Applying Π (0) to B u = F , we have0 = Π (0) B u = Π (0) F = hh f ii u (0) . This implies that hh f ii = 0 and hence F ∈ X . We thus obtain Ran( B ) ⊂ X .We next prove X ⊂ Ran( B ). Let F = ⊤ ( f, g ) ∈ X . We will show thatthere exists a unique solution u ∈ D ( B ) ∩ X to B u = F. (4.15)We define ˜Π (0) by ˜Π (0) u = ⊤ ( h φ i ,
0) for u = ⊤ ( φ, w ). We decompose u = ⊤ ( φ, w )as u = ˜Π (0) u + ( I − ˜Π (0) ) u = ⊤ ( h φ i ,
0) + u , where u = ⊤ ( φ , w ) ∈ L ( T ; L ∗ (Ω per ) × L (Ω per )). Applying ˜Π (0) and ( I − ˜Π (0) ) to B u = F , we have ∂ t h φ i = h f i , (4.16) ∂ t u + L ( t ) u = F − L ( t ) ⊤ ( h φ i ,
0) (4.17)since ˜Π (0) L ( t ) u = 0, where F = ⊤ ( f , g ) := ( I − ˜Π (0) ) F . Integrating (4.16) in[0 , t ], we have h φ ( t ) i = h φ (0) i + Z t h f ( s ) i ds. Furthermore, we determine h φ (0) i = R s h f ( s ) i ds so that R h φ ( t ) i dt = 0. Con-sequently, we obtain h φ ( t ) i = Z s h f ( s ) i ds + Z t h f ( s ) i ds. (4.18)20s for (4.17), it follows from Proposition 4.1 that there exist positive constants ν , γ , ε and a such that if δ = C min { ν +˜ ν , νγ , γ } , ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o , there exists a time periodic solution u = ⊤ ( φ , w ) ∈ L ( T ; L ∗ (Ω per ) × H (Ω per )) to (4.17) satisfying k u ( t ) k L (Ω per ) ,γ + Z t e − β ( t − s ) δ k φ ( s ) k L (Ω per ) ds + Z t e − β ( t − s ) (cid:16) ν k∇ w ( s ) k L (Ω per ) + ˜ ν k div w ( s ) k L (Ω per ) (cid:17) ds ≤ C − e − β Z (cid:18) δγ + δ ν (cid:19) k f ( s ) k L (Ω per ) + 1 ν k g ( s ) k H − (Ω per ) ds. (4.19)Hence, there exists a unique solution u = ⊤ ( φ, w ) ∈ D ( B ) ∩ X to (4.15). Thisshows X ⊂ Ran( B ). Therefore X = Ran( B ).Let us show X = Ker( B ). We assume that u is the solution of B u = 0.Decomposing u into u = u + u with u j ∈ X j for j = 0 ,
1, we have B u j = 0.It follows from the previous argument that u is a unique solution to B u = 0,and we see from (4.19) with F = 0 that u = 0. Consequently, it holds that u = u ∈ X and Ker( B ) ⊂ X . Therefore, Ker( B ) = X . This completes theproof. Remark 4.9.
One can show that, for each k ∈ Z , πik is a simple eigenvalueof − B . We next establish the resolvent estimate for η ′ = 0. We consider λu + ∂ t u + L ( t ) u = F, (4.20)where F = ⊤ ( f, g ) ∈ L ( T ; L (Ω per ) × H − (Ω per )). Proposition 4.10.
There exist positive constants ν , γ , ε and a such that if ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o , then there existsa unique time periodic solution u = ⊤ ( φ, w ) of (4 . for λ ∈ C with Re λ > − β and λ = 2 πik ( k ∈ Z ) . Furthermore, u satisfies k u ( t ) k L (Ω per ) ,γ + Z t e − (2Re λ + β )( t − s ) k ( φ − h φ i )( s ) k L (Ω per ) ds + Z t e − (2Re λ + β )( t − s ) (cid:16) k∇ w ( s ) k L (Ω per ) + k div w ( s ) k L (Ω per ) (cid:17) ds ≤ C (2Re λ + β ) | − e − λ | Z k f ( s ) k L (Ω per ) ds + C − e − (2Re λ + β ) Z k F ( s ) k L (Ω per ) × H − (Ω per ) ds for t ∈ T . roof. As in the proof of Proposition 4.8, we apply ˜Π (0) and ( I − ˜Π (0) ) to (4.20).Then we have, λ h φ i + ∂ t h φ i = h f i (4.21) λu + ∂ t u + L ( t ) u = G , (4.22)where G = − L ( t ) ⊤ ( h φ i ,
0) + F . Since we look for a time periodic solution, h φ i must satisfy ( h φ i ( t ) = e − λt h φ i (0) + R t e − λ ( t − s ) h f ( s ) i ds, h φ i (1) = h φ i (0) . (4.23)In (4.23), set t = 1. We then obtain h φ i (0) = e − λ h φ i (0) + Z e − λ (1 − s ) h f ( s ) i ds. Therefore, if 1 − e − λ = 0, namely, if λ = 2 πik for k ∈ Z , then h φ i (0) = 11 − e − λ Z e − λ (1 − s ) h f ( s ) i ds. (4.24)Substituting (4.24) to the first equation of (4.23), we have h φ i ( t ) = e − λt − e − λ Z e − λ (1 − s ) h f ( s ) i ds + Z t e − λ ( t − s ) h f ( s ) i ds. Therefore, if λ ∈ Σ, then we obtain |h φ i ( t ) | ≤ C | − e − λ | Z k f k L (Ω per ) dt. (4.25)We next consider (4.22). We set v ( t ) = e λt u ( t ). Since v satisfies (4.2) with F = e λt G and the estimate (4.5), there exist positive constants ν , γ , ε and a such that if ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o , wehave k u ( t ) k L (Ω per ) ,γ + Z t e − (2Re λ + β )( t − s ) k φ ( s ) k L (Ω per ) ds + Z t e − (2Re λ + β )( t − s ) (cid:16) k∇ w ( s ) k L (Ω per ) + k div w ( s ) k L (Ω per ) (cid:17) ds ≤ e − (2Re λ + β ) t k u (0) k L (Ω per ) + C Z t e − (2Re λ + β )( t − s ) k G ( s ) k L (Ω per ) × H − (Ω per ) ds.
22s in the proofs of Proposition 4.1 and Lemma 4.2, one can see that if Re λ + β >
0, then there exists a time periodic solution u = ⊤ ( φ , w ) to (4.22) satisfying k u ( t ) k L (Ω per ) ,γ + Z t e − (2Re λ + β )( t − s ) k φ ( s ) k L (Ω per ) ds + Z t e − (2Re λ + β )( t − s ) (cid:16) k∇ w ( s ) k L (Ω per ) + k div w ( s ) k L (Ω per ) (cid:17) ds ≤ C Z t e − (2Re λ + β )( t − s ) k G ( s ) k L (Ω per ) × H − (Ω per ) ds ≤ C (2Re λ + β ) | − e − λ | Z k f ( s ) k L (Ω per ) ds + C Z t e − (2Re λ + β )( t − s ) k F ( s ) k L (Ω per ) × H − (Ω per ) ds. This completes the proof.
Proposition 4.11.
There exist positive constants ν , γ , ε and a such that if ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o , then Σ := { λ : Re λ > − β λ = 2 πik, ∀ k ∈ Z } ⊂ ρ ( − B ) . Furthermore, u = ⊤ ( φ, w ) = ( λ + B ) − F satisfies the estimates k u k X ≤ C (2Re λ + β ) | − e − λ | Z |h f ( s ) i| ds + C (1 − e − (2Re λ + β ) ) k F k L ( T ; L (Ω per ) × H − (Ω per )) , k∇ w k L ( T ; L (Ω per )) ≤ C (2Re λ + β ) | − e − λ | Z |h f ( s ) i| ds + C (1 − e − (2Re λ + β ) ) k F k L ( T ; L (Ω per ) × H − (Ω per )) We are now in a position to prove Theorem 3.3.
Proof of Theorem 3.3.
We first observe that k B (1) j u k L ([0 , L (Ω per ) × L (Ω per )) ≤ C (cid:0) k u k L ([0 , L (Ω per ) × L (Ω per )) + k∇ w k L ([0 , L (Ω per )) (cid:1) (4.26)and k B (2) j,k u k L ([0 , L (Ω per ) × L (Ω per )) ≤ C k w k L ([0 , L (Ω per )) . (4.27)23et Σ be the set given in Theorem 3.3. We see from Proposition 4.11 thatif λ ∈ Σ, then k ( λ + B ) − F k L ( T ; L (Ω per ) × H (Ω per )) ≤ C k F k X . This, together with (4.26) and (4.27), implies that k M η ′ ( λ + B ) − F k X ≤ C ( | η ′ | + | η ′ | ) k F k X . It then follows that there exists positive constant r = r ( ν, ˜ ν, γ, ν , γ ) suchthat if | η ′ | ≤ r , then k M η ′ ( λ + B ) − F k X ≤ k F k X . We thus find that if | η ′ | ≤ r , then Σ ⊂ ρ ( − B η ′ ) and for λ ∈ Σ( λ + B η ′ ) − = ( λ + B ) − ∞ X N =0 ( − N [ M η ′ ( λ + B ) − ] N and k ( λ + B η ′ ) − F k L ( T ; L (Ω per ) × H (Ω per )) ≤ C k F k X . This proves the assertion (i).As for the assertion (ii), it suffices to show that if | η ′ | ≤ r , then σ ( − B η ′ ) ∩ (cid:26) λ ∈ C ; Re λ ≥ − β , | λ | ≤ π (cid:27) = { λ η ′ , } ,λ η ′ , = − i n − X j =1 a j η j − n − X j,k =1 a jk η j η k + O ( | η ′ | ) ( η ′ → a j , a jk ∈ R and λ η ′ , satisfiesRe λ η ′ , ≤ − κ γ ν | η ′ | . In view of Proposition 4.8, Proposition 4.11, (4.26) and (4.27), we can applythe analytic perturbation theory ([18]) to see that the set σ ( − B η ′ ) ∩ (cid:26) λ ∈ C ; Re λ ≥ − β , | λ | ≤ π (cid:27) consists of a simple eigenvalue, say λ η ′ , , for sufficiently small η ′ , and that λ η ′ , is expanded as λ η ′ , = λ + n − X j =1 η j λ (1) j + n − X j,k =1 η j η k λ (2) jk + O ( | η ′ | ) ( η ′ → , λ = 0 ,λ (1) j = −hh B (1) j u (0) , u (0) ∗ ii ,λ (2) jk = − hh ( B (2) j,k + B (2) k,j ) u (0) , u (0) ∗ ii + 12 hh ( B (1) j SB (1) k + B (1) k SB (1) j ) u (0) , u (0) ∗ ii . Here S = [( I − Π (0) ) B ( I − Π (0) )] − . By definition of B (1) j , u (0) and u (0) ∗ , wehave λ (1) j = − i hh v jp φ (0) + γ ρ p w (0) ,j ii . As for λ (2) jk , since hh ( B (2) j,k + B (2) k,j ) u (0) , u (0) ∗ ii = 0, we obtain λ (2) jk = 12 hh ( B (1) j SB (1) k + B (1) k SB (1) j ) u (0) , u (0) ∗ ii . We set u ( k )1 = ⊤ ( iφ ( k )1 , iν w ( k )1 ) = SB (1) k u (0) . Then u ( k )1 is a solution of ( B u = ( I − Π (0) ) B (1) k u (0) , hh φ ii = 0 , (4.28)where ( I − Π (0) ) B (1) k u (0) = i v kp φ (0) + γ ρ p w (0) · e k ( P ′ ( ρ p ) γ ρ p φ (0) ) e k − ρ p (2 ν∂ x k w (0) + e k ˜ ν div w (0) − ˜ ν ∇ w (0) ,k ) ! − i ≪ v kp φ (0) + γ ρ p w (0) · e k ≫ u (0) . In fact, there exists a solution u ( k )1 = ⊤ ( φ ( k )1 , w ( k )1 ) to (4.28), and λ (2) jk is writtenas λ (2) jk = 12 hh B (1) j u ( k )1 + B (1) k u ( j )1 , u (0) ∗ ii = 12 hh v jp φ ( k )1 + γ ν ρ p w ( k )1 · e j ii + 12 hh v kp φ ( j )1 + γ ν ρ p w ( j )1 · e k ii . We thus estimate u ( k )1 to prove the estimate (3.6). Lemma 4.12.
Assume that ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ , S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o and | η ′ | ≤ r . Then the following estimate holds : Z (cid:18) δ k φ ( k )1 ( s ) k L (Ω per ) + 1 ν k∇ w ( k )1 ( s ) k L (Ω per ) (cid:19) ds ≤ Cν . (4.29)25 roof.
As in Proposition 4.1, we have Z (cid:18) δ k φ ( k )1 ( s ) k L (Ω per ) + 1 ν k∇ w ( k )1 ( s ) k L (Ω per ) (cid:19) ds ≤ C (cid:18) δγ + δ ν (cid:19) Z (cid:13)(cid:13)(cid:13)(cid:16) ( I − Π (0) ) B (1) k u (0) (cid:17) (cid:13)(cid:13)(cid:13) L (Ω per ) ds + Cν Z (cid:13)(cid:13)(cid:13)(cid:16) ( I − Π (0) ) B (1) k u (0) (cid:17) (cid:13)(cid:13)(cid:13) H − (Ω per ) ds, where ( u ) = φ and ( u ) = w for u = ⊤ ( φ, w ). Since δγ + δ ν ≤ γ (cid:16) ν + γ + ν +˜ νγ (cid:17) ,it holds that (cid:18) δγ + δ ν (cid:19) Z (cid:13)(cid:13)(cid:13)(cid:16) ( I − Π (0) ) B (1) k u (0) (cid:17) (cid:13)(cid:13)(cid:13) L (Ω per ) ds ≤ Cν (cid:18) ν + 1 νγ + 1 γ (cid:19) , (4.30)1 ν Z (cid:13)(cid:13)(cid:13)(cid:16) ( I − Π (0) ) B (1) k u (0) (cid:17) (cid:13)(cid:13)(cid:13) H − (Ω per ) ds ≤ Cν , and we have the desired estimate.To derive the estimate (3.6), we next introduce ˜ u ( k )1 = ⊤ ( i ˜ φ ( k )1 , iν ˜ w ( k )1 ), whichis a unique stationary solution of the Stokes system A ˜ u ( k )1 = F ( k ) , h ˜ φ ( k )1 i = 0 , (4.31)where F ( k ) = (cid:18) e k (cid:19) . We use the following lemma ([17, Theorem 4.7]).
Lemma 4.13 ([17]) . Let ˜ κ ( η ′ ) be defined by ˜ κ ( η ′ ) = n − X j,k =1 ˜ a jk η j η k for η ′ ∈ R n − , where ˜ a jk = γ ν ( ∇ ˜ w ( j )1 , ∇ ˜ w ( k )1 ) = γ ν h e j · ˜ w ( k )1 i . Then there exists a constant κ > independent of ν , ˜ ν and γ such that ˜ κ ( η ′ ) ≥ κ γ ν | η ′ | for all η ′ ∈ R n − .
26y using Lemma 4.13, we have the following estimate.
Lemma 4.14.
Assume that ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ , S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o and | η ′ | ≤ r , then the following estimate holds : Z k∇ ( w ( k )1 − ˜ w ( k )1 ) k L (Ω per ) ds ≤ C (cid:18) ν + 1 νγ + 1 γ (cid:19) . (4.32) Proof.
We consider( ∂ t + L ( t ))( u ( k )1 − ˜ u ( k )1 ) = ( I − Π (0) ) B (1) k u (0) − F ( k ) − M ˜ u ( k )1 . It follows from the estimate for the Stokes problem (see, e.g., [8]) that k ∂ x ˜ φ ( k )1 k L (Ω per ) + k ∂ x ˜ w ( k )1 k L (Ω per ) ≤ C , and we have k M ˜ u ( k )1 k L ( T ; L (Ω per ) × H − (Ω per )) ≤ C ν + ˜ νγ . (4.33)By using (4.30), (4.33) and Z (cid:13)(cid:13)(cid:13)(cid:16) ( I − Π (0) ) B (1) k u (0) − F ( k ) (cid:17) (cid:13)(cid:13)(cid:13) H − (Ω per ) ds ≤ C (cid:18) γ + 1 ν (cid:19) , (4.34)we obtain 1 ν Z k∇ ( w ( k )1 − ˜ w ( k )1 ) k L (Ω per ) ds ≤ Cν (cid:18) ν + 1 νγ + 1 γ (cid:19) . This completes the proof.
Proof of (3.6).
By Lemmas 4.12, 4.13 and 4.14, if ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o , we have n − X j,k =1 η j η k λ (2) jk = − n − X j,k =1 η j η k hh v jp φ ( k )1 ( t ) + γ ν ρ p w ( k )1 ( t ) · e j ii = − n − X j,k =1 η j η k hh v jp φ ( k )1 ( t ) + γ ν φ p w ( k )1 ( t ) · e j ii− n − X j,k =1 η j η k ( hh ( w ( k )1 ( t ) − ˜ w ( k )1 ( t )) · e j ii + γ ν hh ˜ w ( k )1 ( t ) · e j ii ) ≤ − γ ν (cid:18) κ − νγ (cid:19) | η ′ | ≤ − κ γ ν | η ′ | . λ + B ) − F in a higher order Sobolev space. Lemma 4.15.
There exist positive constants ν , γ , ε and a such that if ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ , and S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o for λ satis-fying uniformly π ≤ | λ | ≤ π , then u = ⊤ ( φ, w ) = ( λ + B ) − F with F = ⊤ ( f, g ) satisfies [[ u ]] + Z t (cid:0) [[ φ ( s )]] + [[ w ( s )]] (cid:1) ds ≤ C Z (cid:0) [[ f ( s )]] + [[ g ( s )]] (cid:1) ds. (4.35) Proof.
We consider ( λ + B ) u = F, u ∈ D ( B ) . (4.36)We set Π = I − Π (0) . Applying Π (0) and Π , (4.36) is decomposed into λ Π (0) u = Π (0) F, (4.37) λu + B u = G , (4.38)where u = Π u and G = ⊤ ( g , ˜ g ) := Π F . If λ = 0, then we rewrite (4.37)as Π (0) u = 1 λ Π (0) F. (4.39)We next consider (4.38). We see from the proofs of Propositions 4.8 and 4.10that there exists a unique solution u ∈ D ( B ) ∩ X to (4.38) if λ ∈ Σ . Fur-thermore, it follows from the proofs of Lemma 4.6 and Proposition 4.10 that[[ u ]] + Z t e − (2Re λ + a ν +˜ νγ )( t − s ) (cid:0) [[ φ ( s )]] + [[ w ( s )]] (cid:1) ds ≤ C − e − (2Re λ + a ν +˜ νγ ) Z (cid:0) [[ g ( s )]] + [[˜ g ( s )]] (cid:1) ds. Let π ≤ | λ | ≤ π . We set u = 1 λ Π (0) F + u . Then u is a solution of ( λ + B ) u = G , where G = ⊤ ( g , ˜ g ) = Π (0) F + G . Thenit holds that[[ u ]] + Z t (cid:0) [[ φ ( s )]] + [[ w ( s )]] (cid:1) ds ≤ C | λ | + 11 − e − (2Re λ + a ν +˜ νγ ) ! Z (cid:0) [[ g ( s )]] + [[˜ g ( s )]] (cid:1) ds. (4.40)28his implies that ( λ + B ) − is bounded and u = ( λ + B ) − G satisfies (4.40).Since Z (cid:0) [[ g ( s )]] + [[˜ g ( s )]] (cid:1) ds ≤ C Z (cid:0) [[ f ( s )]] + [[ g ( s )]] (cid:1) ds, there exist positive constants ν , γ and ε such that if ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o , then λ + B has a bounded inverse for λ satisfying π ≤ | λ | ≤ π and u = ⊤ ( φ, w ) = ( λ + B ) − F satisfies[[ u ]] + Z t (cid:0) [[ φ ( s )]] + [[ w ( s )]] (cid:1) ds ≤ C Z (cid:0) [[ f ( s )]] + [[ g ( s )]] (cid:1) ds. This completes the proof.We are now in a position to prove Theorem 3.4.
Proof of Theorem 3.4. If λ ∈ Σ, then for | η ′ | ≤ r , ( λ + B η ′ ) − is given bythe Neumann series expansion( λ + B η ′ ) − = ( λ + B ) − ∞ X N =0 ( − N [ M η ′ ( λ + B η ′ ) − ] N . It then follows that u η ′ = u (0) + iη ′ · u (1) + u (2) ,u ∗ η ′ = u (0) ∗ + iη ′ · u (1) ∗ + u (2) ∗ , h ˜ u η ′ , ˜ u ∗ η ′ i t = 1 + O ( η ′ ) ≥ | η ′ | ≤ r . Here, u (1) = − πi Z | λ | = π ( λ + B ) − B (1) j ( λ + B ) − u (0) dλ,u (2) ( η ′ ) = 12 πi Z | λ | = π R (2) ( λ, η ) u (0) dλ, with R (2) ( λ, η ) = − ( λ + B ) − B (2) j,k ( λ + B ) − + ( λ + B ) − ∞ X N =2 ( − N η N − [( B (1) j + ηB (2) j,k )( λ + B ) − ] N . By Proposition 4.15 and the definition of B (1) j , B (2) j,k , we obtain k u (1) k C ([0 , H (Ω per )) ≤ C and k u (2) ( η ) k C ([0 , H (Ω per )) ≤ C. k u η ′ ( t ) k H (Ω per ) ≤ C, k u η ′ ( t ) − u (0) ( t ) k H (Ω per ) ≤ C | η ′ | . Similarly, we obtain k ˜ u ∗ η ′ ( t ) k H (Ω per ) ≤ C. This completes the proof.
In this section we give a proof of Proposition 3.1. We set ρ = 1 + 1 γ φ, v = w. The system (2.1)-(2.2) is then written as ∂ t φ + γ div w = f ( φ, w ) , (5.1) ∂ t w − ν ∆ w − ˜ ν ∇ div w + ∇ φ = g ( φ, w, G ) , (5.2)where f ( φ, w ) = − div( φw ) ,g ( φ, w, G ) = SG − w · ∇ w − φγ + φ { ν ∆ w + ˜ ν ∇ div w } + φγ + φ ∇ φ − γ + φ ∇ (cid:18) p (1) (cid:18) φγ (cid:19) φ (cid:19) . We consider (5.1)-(5.2) on Ω per under the conditions w | x n =0 , = 0 , (5.3) h φ i = 0 . (5.4)Under some smallness assumption on the size of S , we have the followingresult on the existence of a time periodic solution of (5.1)-(5.4). Proposition 5.1.
Let G ∈ ∩ j =0 H j ( T ; H − j (Ω per )) with [ G ] , , Ω per = 1 .There exist positive constants ν , γ , ε and a such that if ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ ε γ √ ν +˜ ν q − e − a ν +˜ νγ min n , ν γ o , then there exists a time periodic so-lution u = ( φ, w ) ∈ ∩ j =0 C j ( T ; H − j (Ω per ) × H − j (Ω per )) ∩ H j ( T ; H − j (Ω per ) × H − j (Ω per )) to problem (5 . - (5 . , and u satisfies sup t ∈ T (cid:26) γ [[ φ ( t )]] + [[ w ( t )]] (cid:27) ≤ C ν γ , T (cid:26) ν ν + ˜ ν [[ w ( t )]] + 1 ν + ˜ ν [[ ∇ φ ( t )]] + ν + ˜ νγ k ∂ t φ ( t ) k L (Ω per ) (cid:27) dt ≤ C ν γ , where C is independent of ν , ˜ ν , γ and S . Remark 5.2. If γ ν +˜ ν ≫ , then − e − a ν +˜ νγ ∼ a ν +˜ νγ , and so the condition of S in Proposition . implies S ≤ ε γ √ ν + ˜ ν q − e − a ν +˜ νγ min (cid:26) , ν γ (cid:27) ∼ ε √ a min (cid:26) , ν γ (cid:27) . Proposition 5.1 follows from Valli’s argument (see [27]). Since we look fora solution in a higher order Sobolev space than that considered in [27] and weneed to take care of the dependence of the estimates on the parameters, we heregive a proof of Proposition 5.1To prove Proposition 5.1 we first consider the initial boundary problem for(5.1)-(5.2) under the boundary condition (5.3) and the initial condition u | t =0 = u = ⊤ ( φ , w ) . (5.5)In what follows we assume that u ∈ H ∗ (Ω per ) × ( H (Ω per ) ∩ H (Ω per )) ,G ∈ ∩ j =0 H j ( T ; H − j (Ω per )) , [ G ] , , Ω per = 1 . (5.6)We will also impose the following compatibility conditions on u = ⊤ ( φ , w )and G : w ∈ H (Ω per ) , (5.7) ν ∆ w + ˜ ν ∇ div w − ∇ φ + g ( φ , w , G (0)) ∈ H (Ω per ) . (5.8)We note that if u ∈ H ∗ (Ω per ) × (cid:0) H (Ω per ) ∩ H (Ω per ) (cid:1) and G ∈ ∩ j =0 H j ( T ; H − j (Ω per )),then ν ∆ w + ˜ ν ∇ div w − ∇ φ + g ( φ , w , G (0)) ∈ H (Ω per ). In this subsection, we prove the global existence of solution to (5.1)-(5.5) withenergy estimate that is stated in the following proposition.
Proposition 5.3.
Let u and G satisfy (5.6) and the compatibility conditions (5 . and (5 . . There exist positive constants ν , γ , ε , ε , a and C such thatif ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ , S ≤ γ √ ν +˜ ν q − e − a ν +˜ νγ min n ε νγ , ε o and E ( u ) ≤ C S ν +˜ νν − e − a ν +˜ νγ , then there exists a unique global solution u to (5 . - (5 . in ∩ j =0 C j ([0 , ∞ ); H − j (Ω per ) × H − j (Ω per )) ∩ H j ([0 , ∞ ); H − j (Ω per ) × H − j (Ω per )) which satisfies E ( u ( t )) + 14 Z t e − a ν +˜ νγ ( t − s ) D ( u ( s )) ds ≤ e − a ν +˜ νγ t E ( u ) + 2 C ν S − e − a ν +˜ νγ , (5.9)31 here E m ( u ) and D m ( u ) ( m = 2 , are the quantities satisfying the followinginequalities with some positive constant C : C − (cid:18) γ [[ φ ]] m + [[ w ]] m (cid:19) ≤ E m ( u ) ≤ C (cid:18) γ [[ φ ]] m + [[ w ]] m (cid:19) C − (cid:18) ν ν + ˜ ν [[ w ]] m +1 + 1 ν + ˜ ν [[ ∇ φ ]] m − + ν + ˜ νγ k ∂ m t φ k L (Ω per ) (cid:19) ≤ D m ( u ) ≤ C (cid:18) ν ν + ˜ ν [[ w ]] m +1 + 1 ν + ˜ ν [[ ∇ φ ]] m − + ν + ˜ νγ k ∂ m t φ k L (Ω per ) (cid:19) . As in [6, 13], we can prove Proposition 5.3 by combining local existenceand the a priori estimates. The local existence is proved by applying the localsolvability result in [15, 27]. In fact, we can show that the following assertion.
Proposition 5.4.
Let u and G satisfy (5.6) and the compatibility conditions (5 . and (5 . . Then there exists a positive number T depending only on k u k H (Ω per ) , ν , ˜ ν , γ and S such that the problem (5 . - (5 . has a uniquesolution u ( t ) = ⊤ ( φ ( t ) , w ( t )) ∈ ∩ j =0 C j ([0 , T ]; H − j (Ω per ) × H − j (Ω per )) ∩ H j ([0 , T ]; H − j (Ω per ) × H − j (Ω per )) satisfying [[ u ( t )]] + Z T [[ φ ( s )]] + [[ w ( s )]] ds ≤ C k u k H (Ω per ) . The global existence is proved by combining Proposition 5.4 and the follow-ing a priori estimates.
Proposition 5.5.
Let T be a positive number and assume that u is a solution of (5 . - (5 . in ∩ j =0 C j ([0 , T ]; H − j (Ω per ) × H − j (Ω per )) ∩ H j ([0 , T ]; H − j (Ω per ) × H − j (Ω per )) . There exist positive constants ν , γ , ε , ε , a and C in-dependent of T , ν , ˜ ν , γ and S such that if ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ , S ≤ γ √ ν +˜ ν q − e − a ν +˜ νγ min n ε νγ , ε o and E ( u ( t )) ≤ C S ν +˜ νν − e − a ν +˜ νγ for t ∈ [0 , T ] , then E ( u ( t )) + Z t e − a ν +˜ νγ ( t − s ) D ( u ( s )) ds ≤ e − a ν +˜ νγ t E ( u ) + 2 C ν S − e − a ν +˜ νγ . (5.10) for t ∈ [0 , T ] . Proposition 5.5 is proved by the energy estimate and nonlinear estimates.In what follows, we denote T j,l = ∂ jt ∂ lx ′ .
32e also denote ˙ φ := ∂ t φ + w · ∇ φ . It then follows that˙ φ = − γ div w + ˜ f , ˜ f = − φ div w. (5.11)We have the following basic estimates in a similar manner to [7, Section 5]. Proposition 5.6.
The following estimates hold true :12 ddt k T j,l u k L (Ω per ) ,γ + ν k T j,l ∇ w k L (Ω per ) + ˜ ν k T j,l div w k L (Ω per ) + ν + ˜ νγ k T j,l ˙ φ k L (Ω per ) ≤ ( T j,l F, T j,l u ) γ + C ν + ˜ νγ k T j,l ˜ f k L (Ω per ) , (5.12) where j, l ∈ Z , j, l ≥ , j + l ≤ m , F = ⊤ ( f, g ) and ( u , u ) γ = 1 γ ( φ , φ ) + ( w , w ) for u j = ⊤ ( φ j , w j ) ( j = 1 , γ ddt k T j,l ∂ k +1 x n φ k L (Ω per ) + 12( ν + ˜ ν ) k T j,l ∂ k +1 x n φ k L (Ω per ) + ν + ˜ ν γ k T j,l ∂ k +1 x n ˙ φ k L (Ω per ) ≤ C n γ ( T j,l ∂ k +1 x n ( w · ∇ φ ) , T j,l ∂ k +1 x n φ ) + ν + ˜ νγ k T j,l ∂ k +1 x n ˜ f k L (Ω per ) + 1 ν + ˜ ν k T j,l ∂ kx n g n k L (Ω per ) + 1 ν + ˜ ν k T j +1 ,l ∂ kx n w n k L (Ω per ) + ν k T j,l +1 ∂ kx n ∇ w k L (Ω per ) o , (5.13) where j, l, k ∈ Z , j, l, k ≥ , j + l + k ≤ m − , ν ν + ˜ ν k T j,l ∂ k +2 x w k L (Ω per ) + 1 ν + ˜ ν k T j,l ∂ k +1 x φ k L (Ω per ) ≤ C n ν + ˜ νγ k T j,l ˜ f k H k +1 (Ω per ) + ν + ˜ νγ k T j,l ˙ φ k H k +1 (Ω per ) + 1 ν + ˜ ν k T j,l g k H k (Ω per ) + 1 ν + ˜ ν k T j,l ∂ t w k H k (Ω per ) o , (5.14) where j, l, k ∈ Z , j, l, k ≥ , j + l + k ≤ m − , ν + ˜ νγ k ∂ j +1 t φ k L (Ω per ) ≤ C n ( ν + ˜ ν ) k ∂ jt div w k + ν + ˜ νγ k ∂ jt ˜ f k L (Ω per ) + ν + ˜ νγ ( ∂ jt ( w · ∇ φ ) , ∂ j +1 t φ ) o , (5.15) where j = 0 , . H m -energy estimate ( m = 2 ,
4) in a similar argument to that in [7, Section 5].
Proposition 5.7.
There exists positive constant ν such that if ν ν +˜ ν ≥ ν , then ddt E m ( u ) + D m ( u ) ≤ N m ( u ) (5.16) for m = 2 , . Here N m ( u ) = C n ν + ˜ νγ [[ ˜ f ]] m + 1 ν + ˜ ν [[ g ]] m − + 1 γ X j + k ≤ m ( ∂ jt ∂ kx ( w · ∇ φ ) , ∂ jt ∂ kx φ )+ 1 γ X j + l ≤ m ( T j,l ˜ f , T j,l φ ) + X j + l ≤ m ( T j,l g, T j,l w ) + ν + ˜ νγ ( ∂ m − t ( w · ∇ φ ) , ∂ m t φ ) o , where C is independent of ν , ˜ ν , γ and S . Proof.
We prove (5.16) in the case of m = 4. Let b j ( j = 1 , · · · ,
9) be positivenumbers independent of ν , ˜ ν and γ and consider the following equality: X j + l ≤ (5 .
12) + X j + l ≤ { b × (5 . k =0 + b × (5 . k =0 } + X j + l ≤ { b × (5 . k =1 + b × (5 . k =1 } + X j + l ≤ { b × (5 . k =2 + b × (5 . k =2 } + b × (5 . k =3 + b × (5 . k =3 + b × (5 . . As in [7, Section 5], taking b j ( j = 1 , · · · ,
9) suitably small, if ν ≥
1, we canobtain ddt ˜ E ( u ) + D ( u ) ≤ N ( u ) , (5.17)where there exists a constant C such that C − γ [[ φ ]] + X j + l ≤ k T j,l w k L (Ω per ) ≤ ˜ E ( u ) ≤ C γ [[ φ ]] + X j + l ≤ k T j,l w k L (Ω per ) . Furthermore, we also have ddt k ∂ t ∂ x w k L (Ω per ) = 2( ∂ t ∂ x w, ∂ t ∂ x w ) ≤ C k ∂ t w k L (Ω per ) k ∂ t ∂ x k L (Ω per ) ≤ ν D ( u ) (5.18)and ddt k ∂ x w k L (Ω per ) = 2( ∂ t ∂ x w, ∂ x w ) ≤ C k ∂ t ∂ x w k L (Ω per ) k ∂ x w k L (Ω per ) ≤ ν D ( u ) . (5.19)34t then follows from (5.17)-(5.19) that there exists a positive constant ν such that if ν ν +˜ ν ≥ ν , we have ddt E ( u ) + D ( u ) ≤ N ( u ) . (5.20)In the case of m = 2, we can prove (5.16) in a similar manner. This completesthe proof.We next estimate N ( u ). Proposition 5.8.
There exist positive constants ν , γ , a , C and C such thatif ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ √ C ν √ ν +˜ ν q − e − a ν +˜ νγ , then the followingestimate holds. N ( u ) ≤ D ( u ) + C ν S X j =0 k ∂ jt G k H − j (Ω per ) + C νγ p E ( u ) D ( u ) . Proposition 5.8 is an immediate consequence of the following Propositions5.9 and 5.10.
Proposition 5.9. If ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ √ C ν √ ν +˜ ν q − e − a ν +˜ νγ ,then the following estimate holds. (i) ν + ˜ νγ [[ ˜ f ]] ≤ C ν + ˜ ννγ E ( u ) D ( u )(ii) ν + ˜ νγ k ∂ t ( w · ∇ φ ) k L (Ω per ) ≤ ( ν + ˜ ν ) γ E ( u ) D ( u )(iii) 1 ν + ˜ ν [[ g ]] ≤ ν + ˜ ν S + C (cid:18) ν + 1 γ (cid:19) E ( u ) D ( u ) Proposition 5.10.
Let j and k be non-negative integer with j + k ≤ . If ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S ≤ √ C ν √ ν +˜ ν q − e − a ν +˜ νγ , then the followingestimates hold. (i) 1 γ | ( ∂ jt ∂ kx ( w · ∇ φ ) , ∂ jt ∂ kx φ ) | ≤ (cid:18) γ + γν (cid:19) p E ( u ) D ( u )(ii) 1 γ | ( ∂ jt ∂ kx ˜ f , ∂ jt ∂ kx φ ) | ≤ (cid:18) γ + γν (cid:19) p E ( u ) D ( u )(iii) | ( ∂ jt ∂ kx g, ∂ jt ∂ kx w ) | ≤ D ( u )+ Cν S X j =0 k ∂ jt G k H − j (Ω per ) + Cγ p E ( u ) D ( u ) Here C is some constant independent of ν , ˜ ν , γ and S .
35o prove Propositions 5.9 and 5.10, we use the following Sobolev inequalities.
Lemma 5.11.
Let ≤ p ≤ . Then the inequality k f k L p (Ω per ) ≤ C k f k H (Ω per ) holds for f ∈ H (Ω per ) . Furthermore, the inequality k f k L ∞ (Ω per ) ≤ C k f k H (Ω per ) holds for f ∈ H (Ω per ) . Proof of Propositions 5.9 and 5.10.
The estimates (i) and (ii) can beobtained by the integration by parts and straightforward computations basedon Lemma 5.11.As for the estimate (iii), we first make the following observation. We set F (cid:18) φγ (cid:19) = φγ + φ , − γ + φ (cid:18) γ p (1) ′ ( γ − φ ) φ + 2 p (1) ( γ − φ ) φ − φ (cid:19) . It follows from Lemma 5.11 that if γ ≥ | φ | ≤ γ , then (cid:13)(cid:13)(cid:13)(cid:13) F (cid:18) φγ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ (Ω per ) ≤ Cγ k φ k L ∞ (Ω per ) , (cid:13)(cid:13)(cid:13)(cid:13) F ( k ) (cid:18) φγ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ (Ω per ) ≤ C. (5.21)Using Lemma 5.11 and (5.21), we can obtain the desired estimate (iii) in a sim-ilar manner to [6, Section 7] by Lemma 5.11 and (5.21) by direct computations.We omit the details. We can similarly obtain the estimates in Proposition 5.10by using Lemma 5.11, Proposition 5.9 and integration by parts. This completesthe proof.We are now in a position to prove Proposition 5.5. Proof of Proposition 5.5.
From Propositions 5.7 and 5.8, we have ddt E ( u ) + 34 D ( u ) ≤ C ν S X j =0 k ∂ jt G k H − j (Ω per ) + C νγ p E ( u ) D ( u ) . Since D ( u ) ≥ a ν + ˜ νγ E ( u ) , it holds that ddt E ( u )+ a ν + ˜ νγ E ( u )+ 12 D ( u ) ≤ C ν S X j =0 k ∂ jt G k H − j (Ω per ) + C νγ p E ( u ) D ( u ) .
36y the assumption, E ( u ( t )) ≤ C ν S − e − a ν +˜ νγ for t ∈ [0 , T ], therefore if ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S satisfies S ≤ √ C γ √ ν + ˜ ν q − e − a ν +˜ νγ min (cid:26) νγ , C (cid:27) , (5.22)we have ddt E ( u ) + a ν + ˜ νγ E ( u ) + 14 D ( u ) ≤ C ν S X j =0 k ∂ jt G k H − j (Ω per ) . By integrating (4.14) in [0 , t ], it holds that E ( u ( t )) + Z t e − a ν +˜ νγ ( t − s ) D ( u ( s )) ds ≤ e − a ν +˜ νγ t E ( u ) + C ν S Z t e − a ν +˜ νγ ( t − s ) 2 X j =0 k ∂ jt G ( s ) k H − j (Ω per ) ds. It follows from (5.6) and Lemma 4.3 that E ( u ( t )) + Z t e − a ν +˜ νγ ( t − s ) D ( u ( s )) ds ≤ e − a ν +˜ νγ t E ( u ) + 2 C ν S − e − a ν +˜ νγ . This completes the proof.
We first consider the H -energy estimate for the difference of the solution of(5.1)-(5.5).Let u j ( j = 1 ,
2) and G satisfy (5.6) and the compatibility conditions (5.7)and (5.8). Assume that u j satisfy E ( u j ) ≤ C ν S − e − a ν +˜ νγ ( j = 1 , . (5.23)Let u j ( j = 1 ,
2) be the solutions of (5.1)-(5.5) with u = u j obtained byProposition 5.3 with u and u replaced by u j and u j ( j = 1 , u = u − u = ⊤ ( φ − φ , w − w ). Then ˜ u = ⊤ ( ˜ φ, ˜ w ) satisfies ∂ t ˜ φ + γ div ˜ w = f, (5.24) ∂ t ˜ w − ν ∆ ˜ w − ˜ ν ∇ div ˜ w + ∇ ˜ φ = g, (5.25)where f = − w · ∇ ˜ φ − ˜ f , = −{ w · ∇ ˜ w + ˜ w · ∇ w + φ γ + φ ∆ ˜ w + 1 γ + φ γ γ + φ ∆ w ˜ φ + F (cid:18) φ γ (cid:19) ∇ ˜ φ + F (1) (cid:18) φ γ , φ γ (cid:19) ∇ φ γ ˜ φ } . Here ˜ f = ˜ w · ∇ φ + φ div ˜ w + φ div w ,F ˜ φγ ! = − γ + ˜ φ γ p (1) ′ ˜ φγ ! ˜ φ + 2 p (1) ˜ φγ ! ˜ φ − ˜ φ ! ,F (1) (cid:18) φ γ , φ γ (cid:19) = Z F ′ (cid:18) θ φ γ + (1 − θ ) φ γ (cid:19) dθ. Similarly to the previous section one can obtain ddt E (˜ u ) + D (˜ u ) ≤ N (˜ u ) . (5.26)One can also obtain the following estimate for N (˜ u ) in a similar manner tothe previous section. Proposition 5.12. If ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S satisfies (5 . , then thereexists positive constant C independent of ν , ˜ ν , γ and S such that N (˜ u ) ≤ C γ ν + ˜ ν (cid:16)p E ( u ) + p E ( u ) (cid:17) D (˜ u ) . In fact, Proposition 5.12 is an immediate consequence of the following Propo-sition 5.13.
Proposition 5.13. If ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S satisfies (5 . , then (i) ν + ˜ νγ [[ ˜ f ]] ≤ C ( E ( u ) + E ( u )) D (˜ u ) , (ii) 1 ν + ˜ ν [[ g ]] ≤ C ( E ( u ) + E ( u )) D (˜ u ) , (iii) ν + ˜ νγ k w · ∇ ˜ φ k L (Ω per ) ≤ ( ν + ˜ ν ) γ E ( u ) D (˜ u ) , (iv) 1 γ | ( ∂ jt ∂ kx ( w · ∇ ˜ φ ) , ∂ jt ∂ kx ˜ φ ) | ≤ C γ ν + ˜ ν (cid:16)p E ( u ) + p E ( u ) (cid:17) D (˜ u ) , (v) 1 γ | ( ∂ jt ∂ kx ˜ f , ∂ jt ∂ kx ˜ φ ) | ≤ C γ ν + ˜ ν (cid:16)p E ( u ) + p E ( u ) (cid:17) D (˜ u ) , (vi) | ( ∂ jt ∂ kx g, ∂ jt ∂ kx ˜ w ) | ≤ C (cid:16) γν (cid:17) (cid:16)p E ( u ) + p E ( u ) (cid:17) D (˜ u ) , here C is some constant independent of ν , ˜ ν , γ and S . Proposition 5.13 can be obtained in a similar manner to Propositions 5.9and 5.10.We now establish the H -energy estimate for ˜ u . Proposition 5.14.
Let u j ( j = 1 , be the solutions of (5 . - (5 . with u = u j ( j = 1 , . If ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ , then the following estimate holds. Thereexists a constant ε > such that if S ≤ γ √ ν +˜ ν q − e − a ν +˜ νγ min n ε , ε ν γ o ,then E (˜ u ( t )) + 14 Z t e − a ν +˜ νγ ( t − s ) D (˜ u ( s )) ds ≤ e − a ν +˜ νγ t E ( u − u ) . (5.27) Proof.
By (5.26) and Proposition 5.12, we have ddt E (˜ u ) + ν + ˜ νγ E (˜ u ) + D (˜ u ) ≤ C γ ν + ˜ ν (cid:16)p E ( u ) + p E ( u ) (cid:17) D (˜ u ) . It then follows that if ν ν +˜ ν ≥ ν , γ ν +˜ ν ≥ γ and S satisfies S ≤ √ C γ √ ν + ˜ ν q − e − a ν +˜ νγ min (cid:26) ε , C ν γ (cid:27) , (5.28)then ddt E (˜ u ) + ν + ˜ νγ E (˜ u ) + 14 D (˜ u ) ≤ . Proposition 5.14 now follows from this inequality. This completes the proof.We now show the existence of a time-periodic solution of (5.1)-(5.4).
Proof of Proposition 5.1.
Let φ ♭ = 0 and w ♭ ∈ H (Ω per ) ∩ H (Ω per ) be thesolution to − ν ∆ w − ˜ ν ∇ div w = g (0 , w, G (0)) . The existence of the stationary solution was shown in [7]. Furthermore, we canprove that the stationary solution satisfies E ( u ♭ ) ≤ C ν S − e − a ν +˜ νγ . (5.29)This can be seen by the same energy method as that in Section 5.1 without timederivatives. By Proposition 5.3, we have the global solutions u ♭ ( t ) of (5.1)-(5.5)with u = u ♭ and u ♭ ( t ) satisfies E ( u ♭ ( t )) ≤ C ν S − e − a ν +˜ νγ ( t ≥ .
39e next consider the functions u and u defined by u ( t ) = u ♭ ( t ) , u ( t ) = u ♭ ( t + ( m − n )) , where m, n ∈ N with m > n . As in the proof of Proposition 4.1, we can showthat E ( u ♭ ( n ) − u ♭ ( m )) → n → ∞ ) , and it holds that there exists ˜ u ♭ ∈ H ∗ (Ω per ) × ( H (Ω per ) ∩ H (Ω per )) such that u ♭ ( m ) converges to ˜ u ♭ strongly in H ∗ (Ω per ) × ( H (Ω per ) ∩ H (Ω per )) and weaklyin H ∗ (Ω per ) × ( H (Ω per ) ∩ H (Ω per )), and ˜ u ♭ satisfies E (˜ u ♭ ) ≤ C ν S − e − a ν +˜ νγ . We therefore see from Proposition 5.3 that there exists a unique global solution u ∈ ∩ j =0 C j ([0 , ∞ ); H − j (Ω per ) × H − j (Ω per )) ∩ H j ([0 , ∞ ); H − j (Ω per ) × H − j (Ω per )) of (5.1)-(5.5) in with u = ˜ u ♭ . It then follows from the argumentby Valli [27] that u is a time-periodic solution of (5.1)-(5.4) satisfying E ( u ( t )) ≤ Cν S − e − a ν +˜ νγ , Z D ( u ( s )) ds ≤ Cν S − e − a ν +˜ νγ . Using the condition (5.28), we have1 γ [[ φ ]] + [[ w ]] ≤ C ν γ , Z (cid:26) ν ν + ˜ ν [[ w ( s )]] + 1 ν + ˜ ν [[ ∇ φ ( s )]] + ν + ˜ νγ k ∂ t φ ( s ) k L (Ω per ) (cid:27) ds ≤ C ν γ . This completes the proof.
Acknowledgements.
Y. Kagei was partly supported by JSPS KAKENHIGrant Numbers 16H03947, 16H06339 and 20H00118. S. Enomoto was partlysupported by JSPS KAKENHI Grant Numbers 18J01068.
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