Li and Gu Boundary Value Problems (2016) 2016:39 DOI 10.1186/s13661-016-0552-4

RESEARCH

Open Access

Existence of periodic solutions of Boussinesq system Hengyan Li* and Liuxin Gu *

Correspondence: [email protected] School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou, 450011, China

Abstract This paper is devoted to the study of the dynamical behavior of a Boussinesq system, which is a basic model in describing the flame propagation in a gravitationally stratified medium. This system consists of an incompressible Navier-Stokes equation coupled with a reaction-advection-diffusion equation under the Boussinesq approximation. We prove that this system possesses time dependent periodic solutions, bifurcating from a steady solution. Keywords: Boussinesq system; periodic solution; Hopf bifurcation

1 Introduction and main results The Boussinesq-type equation of reactive flows is a basic model in describing the flame propagation in a gravitationally stratified medium, and its non-dimensional form is given by →

Ut + (U · ∇)U – νU + ∇P = T ρ , ∇U = ,

(.)

Tt + (U · ∇)T – T = g(T), where (t, x) ∈ R+ × R , U ∈ R is the velocity field, T is the temperature function, ν >  denotes the Prandtl number, which is the ratio of the kinematic and thermal diffusivities (inverse proportional to the Reynolds number); P(x, t) ∈ R denotes the pressure; the vector → → → ρ = ρ g corresponds to the non-dimensional gravity g scaled by the Rayleigh number ρ > . The reaction term of Kolmogorov-Petrovskii-Piskunov (KPP) type is of the form g(T) =

αT( – T) . 

Here, α is the reaction rate. See [] for the derivation of this model and the related parameters. When the initial temperature T is identically zero (or constant), the above system reduces to the classical incompressible Navier-Stokes equation: Ut + (U · ∇)U – νU + ∇P = , ∇U = . © 2016 Li and Gu. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Since the work of Sattinger [], Iudovich [], and Iooss [] in , the bifurcation of stationary solutions into time-periodic solutions (i.e. Hopf bifurcation) of incompressible Navier-Stokes equation has attracted much attention, see [–], etc. When the linearized operator possesses a continuous spectrum up to the imaginary axis and that a pair of imaginary eigenvalues crosses the imaginary axis, Melcher, et al. [] proved Hopf bifurcation for the vorticity formulation of the incompressible Navier-Stokes equations in R . Their work is mainly motivated by the work of Brand et al. [] who studied the Hopf-bifurcation problem and its exchange of stability for a coupled reaction diffusion model in Ra . Inspired by the work of [, ], this paper is to establish the corresponding Hopf-bifurcation result for the three-dimensional Boussinesq system. The Boussinesq system is a very important model in fluid mechanics, which exhibits extremely rich phenomena, for example, Rayleigh-Bénard convection [–], geophysical fluid dynamics [, ] etc. A key problem in the study of the dynamic behavior of Boussinesq system is how to understand the time-periodic solutions, quasi-periodic solutions and traveling waves, etc. There were several papers on the existence of time-periodic solutions [] and traveling waves [–] for the Boussinesq system (.). To our knowledge, there is no theoretical result on bifurcation analysis for the Boussinesq system on R . In the present paper, we consider the reactive Boussinesq system with external timeindependent force in R →

Ut + (U · ∇)U – νU + ∇P = T ρ + f (x, ),

(.)

∇U = ,

(.)

Tt + (U · ∇)T – T = g(T) + h(x, ),

(.)

with initial conditions U(x, ) = U (x),

T(x, ) = T (x),

(.)

where f (x, ) and h(x, ) ∈ R × R are external time independent forces, which depend smoothly on some parameter , g(T) = T( – T). Meanwhile, external forces f (x, ) and h(x, ) can be chosen suitably so that (U (x), T (x), P (x)) is the solution of the steady Boussinesq system →

–νU + (U · ∇)U + ∇P = T ρ + f (x, ), ∇U = , –T + (U · ∇)T = g(T) + h(x, ), with the condition lim U (x) = ,

|x|→∞

lim T (x) = .

|x|→∞

Furthermore, assume that the steady solution (U (x), T (x), P (x)) satisfies the following certain decay properties:

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(A) For p ∈ (, ) and s > ,       U (x) p , T (x) p , P (x) p ≤ C, L L L s

s

s

p

where C and Ls denote a positive constant and the weighted Lebesgue space to be specified blow. We also assume that the solution of system (.)-(.) has the form U(x, t) = u(x, t) + U (x),

T(x, t) = v(x, t) + T (x),

P (x, t) = p(x, t) + P (x),

where 

   U (x), T (x), P (x) = u (x) + c, T˜  (x) + , P (x) ,

and (u (x), T˜  (x), P (x)) is the solution of the following steady problem: →

–νU + (U · ∇)U + ∇P = T ρ + f (x), ∇U = , –T + (U · ∇)T = g(T) + h (x), with the condition lim U (x) = ,

|x|→∞

lim T (x) = .

|x|→∞

Then the deviation (u(x, t), v(x, t), p(x, t)) from the stationary (U (x), T (x), P (x)) satisfies →

ut – νu + c∂x u + (u · ∇)u + (u · ∇)u + (u · ∇)u + ∇p = v ρ ,

(.)

∇u = ,

(.)

vt – v – v + ∂x v + (u · ∇)v + (u · ∇)T˜  + (u · ∇)v + vT˜  + v = .

(.)

Here, for general matrices u = (uij )i,j=,, ,  ∇ ·u=

  j=

∂x uj ,

  j=

∂x uj ,

 

T ∂x uj

.

j=

In fact, by the incompressible condition (.), it follows that   ∇ · uvT = u · ∇u + u∇ · u = u · ∇u. So, system (.)-(.) can be rewritten as       → ut – νu + c∂x u + ∇ · u uT + ∇ · uuT + ∇ · uuT + ∇p = v ρ ,       vt – v – v + ∂x v + ∇ · u vT + ∇ · uT˜ T + ∇ · uvT – u ∇ · v – u∇ · T˜  – u∇ · v + vT˜  + v = ,

(.)

(.)

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with the incompressible condition ∇ · u = . It is convenient to rewrite system (.)-(.) in the stream function and vorticity formulation in dimensionless form. The vorticity associated with the velocity field u of the fluid is defined by ω = ∇ × u. Then, using     ∇ × ∇ · uuT = ∇ · ωuT – uωT , we can rewrite system (.)-(.) in the stream function and the vorticity formulation in dimensionless form →

ωt – ω + c∂x ω – ∇ · M(ω , ω) – ∇ · M(ω, ω) = ρ ∇v,

(.)

vt – v – v + ∂x v + ∇ · N(u, u , v, T˜  ) – B(u, v, T˜  ) = ,

(.)

where M(ω , ω ) = ω uT + ω uT – u ωT – u ωT , N(u, u , v, T˜  ) = u vT + uT˜ T + uvT , B(u, v, T˜  ) = u ∇ · v + u∇ · T˜  + u∇ · v – vT˜  – v . Note that we can assume that ∇ · ω = . This is because the space of divergence free vector fields is invariant under the evolution of (.). Denote ϕ = (ω, v)T . Then we can write system (.)-(.) as the evolution equation of the form dϕ + N ϕ = F(ϕ), dt

(.)

where 

– + c∂x N= 

→

ρ · ∇v – –  + ∂x



and  ∇ · M(ω , ω) + ∇ · M(ω, ω) . F(ϕ) = B(u, v, T˜  ) – ∇ · N(u, u , v, T˜  ) 

For y ∈ R , the Fourier transform F and the inverse Fourier transform F – are given by ˆ = F (u)(y) = u(y)

 (π)

 u(x)e–ix·y dx, R

 ˆ F (u)(x) = u(x) = (π)



–

R

ix·y ˆ u(y)e dy.

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For s ≥ ,  ≤ p ≤ , and p + q = , the Fourier transform is a continuous mapping from p q p Ls into Wκ . Especially, when p = , the Fourier transform is an isomorphism between Ls q and Wκ . Since we deal with the problem in the whole space R , it is advantageous to apply the Fourier transform to the evolution equation (.). Denote ϕˆ = (ω, ˆ vˆ )T . Then dϕˆ ˆ ϕ), + Nˆ ϕˆ = F( ˆ dt

(.)

where   → iρ · y |y| + icy ˆ N= ,  |y| +  + iy   ˆ ω, ˆ ωˆ  , ω) ˆ + iy · M( ˆ ω) ˆ iy · M( ˆ ϕ) , F( ˆ = ˆ u, ˆ uˆ  , vˆ , Tˆ˜  ) + B( ˆ vˆ , Tˆ˜  ) P( and ˆ ωˆ  , ωˆ  ) = ωˆ  ∗ uˆ T + ωˆ  ∗ uˆ T – uˆ  ∗ ωˆ T – uˆ  ∗ ωˆ T , M(   ˆ uˆ  , vˆ , Tˆ˜  ) = –iy · uˆ  ∗ vˆ T + iuˆ  ∗ (y · vˆ ) – ˆv ∗ Tˆ˜  , P(   ˆ u, ˆ vˆ , Tˆ˜  ) = iy · uˆ ∗ Tˆ˜ T + uˆ ∗ vˆ T + iuˆ ∗ (y · Tˆ˜  ) + iuˆ ∗ (y · vˆ ) – vˆ ∗ vˆ . B( Here, ∗ denotes the convolution. That is,  uˆ ∗ vˆ (y) =

R

ˆ – x)ˆv(x) dx, u(y

and for general matrices u = (ukj )k,j=,, ,  iy · u = i

  j=

∂x yj uj ,

  j=

∂x yj uj ,

 

T ∂x yj uj

.

j=

To overcome the essential spectrum of operator Jˆ (defined in (.)) up to the imaginary axis, for  < p <  and s > ( – p ), we need the following assumption: ˆ (A) For any  ∈ [c –  , c +  ],  is not an eigenvalue of J. + ˆ (A) For  = c , the operator J has two pair eigenvalues (λ , μ+ ) and (λ– , μ– ) satisfying ± for c > , λ±  (c ) = μ (c ) = ±i c = ,  ±   ±  d d Re λ () Re μ () > , > . d d =c =c

(.) (.)

(A) The remaining eigenvalue of Jˆ is strictly bounded away from the imaginary axis in the left half plane for all  ∈ [c –  , c +  ]. Here is our main result in this paper.

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Theorem . Assume that (A)-(A) hold. Then for p ∈ (, ) and s > ( – p ), system (.) has a time-periodic solution 

    ˆ t), vˆ (y, t) = uˆ n (y)ein t , vˆ n (y)ein t , ϕ(y, ˆ t) = u(y, n∈Z

n∈Z

ˆ vˆ )Lps = O( ), – c = O( ). with  = c +  ,  ∈ (, β), (u, This paper is organized as follows. In Section , we give the basic setting of the problem and derive some priori estimates needed in the proof in next section. The proof of the main result occupies the Section .

2 Preliminary and some estimates We start this section by introducing some notations. Consider the following standard Sobolev space, a spatially weighted Lebesgue space:

  Dα uq q < ∞ , Wκq := u : uqκ := L |α|≤κ

 p p Ls := u : us :=

ρ (x)u (x) dx < ∞ , sp

R

p

 where the weighted function ρ(x) =  + |x| . To investigate periodic solutions of system (.)-(.), we also introduce the space

 ˆ Xps := Xps := uˆ = (uˆ n )n∈Z : u uˆ n Lps < ∞ n∈Z

and the weighted space p

p

Lps = Lps × Ls+ ,

Xsp = Xps × Xs+ ,

with norms ϕLps := uLps + vLp , s+

p

ϕXsp := uXps + vXp , s+

p

for ϕ = (u, v)T ∈ Ls or Xs , respectively. As we known, the vorticity ω = ∇ × u, where u is the velocity field. By the Biot-Savart law, u is recovered from ω as u(x) = –

 π

 R

(x – y) × ω(y) dy. |x – y|

The following estimates are taken from [], which show the norm relationship uˆ with ω. ˆ Lemma . Let p ∈ [, +∞]. For k = , ,  and ωˆ ∈ (Lp (R )) , there exists a constant C such that ˆ Lp ≤ Cω iyk u ˆ Lp .

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Furthermore, for every p ∈ [, ), p , p ∈ [, ∞] with

 p

+

 p

= p ,

  ˆ Lp ≤ C ω ˆ Lp + ω ˆ Lp . u Meanwhile, if ωˆ ∈ Lp (R ) ∩ Lp (R ), then uˆ ∈ Lp (R ) and the above estimate also holds. p

Then, for the weighted space Ls , the following Sobolev embedding holds. Lemma . For p ≥ p and s > holds.

a p

–

a , p

p

the continuous embedding Ls  (Ra ) ⊂ Lp (Ra )



Proof Note that ρ(y) = ( + |y| )  , y ∈ Ra . By direct computation, we have  –s p ρ (y) p = L 

 

=  ≤

dy Ra

( + |y| )

sp 



dy sp |y|≤ ( + |y| ) 

dy |y|≤

sp  

( + |y| )

+

dy

|y|>



( + |y| )

∞

+C 

sp 

dx , xsp –a+

∀p > . p

Hence, for sp > a, the above inequality implies that ρ –s (y)Lp is bounded. Let p = p + p . By Hölder’s inequality and the above inequality, it follows that for ∀ϕ ∈ Lp (Ra ),       ϕLp (Ra ) = ρ s ϕρ –s Lp (Ra ) ≤ ρ s ϕ Lp (Ra ) ρ –s Lp (Ra )   = ϕLps  (Ra ) ρ –s Lp (Ra ) ≤ CϕLps  (Ra ) . 

This completes the proof. From Corollary . in [], the following result holds. p

Lemma . Let p ∈ (  , +∞]. For any ωˆ  , ωˆ  ∈ Ls and s > ( – p ), there exists a positive constant C such that   M( ˆ ωˆ  , ωˆ  ) p ≤ Cωˆ   p ωˆ   p . Ls Ls L s

Furthermore, let p ∈ (, ). Then, for s > ,   M( ˆ ωˆ  , ωˆ  )

L∞

≤ Cωˆ  Lps ωˆ  Lps .

Lemma . Let p ∈ (  , +∞] and s > ( – p ). Then there exists a constant C >  such that   iuˆ ∗ (y · vˆ ) p ≤ Cω ˆ Lps ˆvLp , L s

s+

ˆ vˆ ) ∈ Lps . for (u,

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Proof By Young’s inequality and Lemma ., it follows that   iuˆ ∗ (y · vˆ ) p Ls   ˆ Lp + u ˆ Lps yˆvL + u ˆ L ˆvLp ≤ C u s+     ˆ L + ω ≤ C ω ˆ Lp + ω ˆ Lp + ω ˆ Lp yˆvL + ˆvLp ω ˆ Lp , s–

s+

(.)

where p + p =  and p ∈ [, ). p For s > ( – p ) and p > p , applying the Sobolev embedding Ls ⊂ L ∩ Lp to (.) yields   iuˆ ∗ (y · vˆ ) p ≤ Cω ˆ Lps ˆvLp . L s

s+



This completes the proof. Lemma . Let p ∈ (, ) and s > . Then there exists a constant C >  such that   iuˆ ∗ (y · vˆ ) ∞ ≤ Cω ˆ Lps ˆvLp , L s+

ˆ vˆ ) ∈ Lps . for (u,

Proof By Young’s inequality and Lemma ., we have    iuˆ ∗ (y · vˆ ) ∞ ≤ u ˆ Lp yˆvLp ≤ C ω ˆ L



p L 

where

 p

+

 p

= ,

For p ≥ p and s p

p

 = p , and p ∈ [, ). p > ( p – p ), p ≥ p and s > p

 p

 + ω ˆ Lp yˆvLp ,

(.)

+

Ls ⊂ Lp and Ls ⊂ L to (.), we derive

( p – p ), applying the Sobolev embedding 

  iuˆ ∗ (y · vˆ ) ∞ ≤ Cω ˆ Lps ˆvLp . L s+

This completes the proof.



Consider the linearized operator of (.)   J (ϕ) ˆ = Nˆ + DF(ϕ ) ϕ, ˆ where  ˆ ωˆ  , ω) ˆ iy · M( , DF(ϕ )ϕˆ = ˆ uˆ  , vˆ , Tˆ˜  ) P( 

and   ˆ uˆ  , vˆ , T˜ˆ  ) = iuˆ  ∗ (y · vˆ ) – ˆv ∗ T˜ˆ  – iy · uˆ  ∗ vˆ T . P( Then we can rewrite system (.) as dϕˆ ˆ = G(ϕ), ˆ + J (ϕ) dt

(.)

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where 

ˆ ω, iy · M( ˆ ω) ˆ G(ϕ) ˆ = ˆ ˆB(u, ˜ ˆ vˆ , T )



and ˆ ωˆ  , ωˆ  ) = ωˆ  ∗ uˆ T + ωˆ  ∗ uˆ T – uˆ  ∗ ωˆ T – uˆ  ∗ ωˆ T , M(   ˆ u, ˆ vˆ , Tˆ˜  ) = iy · uˆ ∗ Tˆ˜ T + uˆ ∗ vˆ T + iuˆ ∗ (y · Tˆ˜  ) + iuˆ ∗ (y · vˆ ) – vˆ ∗ vˆ . B( Remark . By applying the theorem of Riesz, it is easy to see that the operators J and p Nˆ differ by a relatively compact perturbation in Ls , for p ∈ (, ) and s > ( – p ). Hence, the essential spectrum of the operator J equals the essential spectrum of the operator Nˆ (see [], p. ). p

Lemma . Let p ∈ (  , +∞] and s > ( – p ). Then, for (ω, ˆ vˆ ) ∈ Ls , there exists a positive constant C such that   B( ˆ u, ˆ vˆ , Tˆ˜  )

p

Ls

  ≤ C ω ˆ Lps Tˆ˜  Lp + ω ˆ Lp + ˆvLp . s+

s

s+

Moreover, for p ∈ (, ) and s > , there exists a positive constant C such that   B( ˆ u, ˆ vˆ , Tˆ˜  )

L∞

  ≤ C ω ˆ Lps Tˆ˜  Lp + ω ˆ Lp + ˆvLp . s+

s

s+

Proof Applying Lemma . and Young’s inequality for convolution, it is easy to derive this result.  Lemma . Let p, p ≥ . Then, for s > ,

 p

=

 p

+

 , p

and fˆ ∈ Ls (R ), the equation p

Nˆ ϕˆ = fˆ p has a unique solution ϕ = Nˆ – fˆ ∈ Ls (R ).

Proof Let η(y) ∈ C∞ (R, [, ]) be a cut-off function satisfying η(y) = ,

for |y| ≤ ;

η(y) = ,

for |y| ≥ .

Consider   ϕˆ = Nˆ – fˆ = η(y)Nˆ – fˆ +  – η(y) Nˆ – fˆ , where   |y| + icy Nˆ – = (|y| + icy )(|y| + iy + ) 

 –iρy  . |y| +  + iy

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Denote  = ϒ|y|≤  = ϒ|y|≤  = ϒ|y|≤

|y|

 , + iy + 

|y|

 , + icy

 = ϒ˜ |y|>  ϒ˜ |y|> =

|y|

|y|

–iρy  , (|y| + icy )(|y| + iy + )

 , + iy + 

 , + icy  ϒ˜ |y|> =

–iρy  . (|y| + icy )(|y| + iy + )

By Minkowski’s inequality, Hölder’s inequality, and Lemma ., for s > ( p – p ), we have      ϕ ˆ Lps ≤ η(y)Nˆ – fˆ Lp +   – η(y) Nˆ – fˆ 

 p

=

 p

+ p , p > p , and

p

Ls

s

      ≤ C η(y)Nˆ – Lp fˆ Lps  +   – η(y) Nˆ – L∞ fˆ Lps       ≤ C η(y)Nˆ –  p fˆ Lps  +   – η(y) Nˆ – L∞ fˆ Lps .

(.)

Ls

j It is easy to check that ϒ˜ |y|> L∞ < +∞ for j = , , . For

  p  ϒ |y|≤ p =



Ls

|y|≤

s

( + |y| )  dy ≤ C ||y| + iy + |p



 

 

 xp –s–

+

s 

< p <

 

+ s , we have

dx < +∞.

In the same way, we get   p ϒ  |y|≤ Lp < +∞,   p ϒ  |y|≤ Lp < +∞,

 s  s + < p < + ,      s s for + < p <  + .    for

Therefore, by (.) and the above estimates, for  + s < p <  + s and   η(y)Nˆ – 

 p

– p ≥  , we obtain

< +∞.

(.)

This completes the proof.



p

Ls

ˆ Lemma . Let p, p ∈ (  , +∞]. For s ≥  and p = p + p . Then the operator Nˆ – · DF(ϕ) p is a compact operator on Ls . Furthermore, the operator ˆ : Lps → Lps  := I + Nˆ – · DF(ϕ) is a Fredholm operator with index . Proof Denote the set   S = ϕˆ ∈ Lps : ϕ ˆ Lps ≤  ˆ ϕˆ n for any sequence ϕˆ n = (ωˆ n , vˆ n ) ∈ S . and χn = (Nˆ – · DF(ϕ))

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By Minkowski’s inequality, Hölder’s inequality, and Lemma ., for and s > ( p – p ), we have

 p

=

 p

+

 , p

p > p ,

   χn Lps =  Nˆ – · DF(ϕ) ˆ ϕˆn Lp s    –      ≤ η(y) Nˆ · DF(ϕ) ˆ ϕˆn Lp +   – η(y) Nˆ – · DF(ϕ) ˆ ϕˆ n Lp s s         –   –        ˆ ˆ ≤ C η(y)N Lp DF(ϕ) ˆ ϕˆ n Lp +  – η(y) N L∞ DF(ϕ) ˆ ϕˆ n Lp s s        –    – ˆ ϕˆ n Lp +   – η(y) Nˆ L∞ DF(ϕ) ˆ ϕˆ n Lp . (.) ≤ C η(y)Nˆ  p DF(ϕ) Ls

s

s

As proved in Lemma ., by Young’s inequality for convolutions, for p, p ∈ (  , +∞] and s > ( – p ), we can get   DF(ϕ) ˆ ϕˆ n Lp s

≤C



ωˆ n Lp s

  DF(ϕ) ˆ ϕˆ n 

 + ˆvn Lp + uˆ  Lp + ωˆ  Lp + Tˆ˜  Lp < +∞, s

s+

s

(.)

s+

p

Ls

  ≤ C ωˆ n  p + ˆvn  p + uˆ   p + ωˆ   p + Tˆ˜   p < +∞. Ls

From (.), for

 

  η(y)Nˆ – 

Ls+

< p <  +

+

s 

p

< +∞.

Ls

Ls

s 

and

 p

Ls

–

 p

Ls+

(.)

≥  , we derive (.)

By (.)-(.), it follows that    ˆ ϕˆ n Lp < +∞. χn Lps =  Nˆ – · DF(ϕ) s

p Therefore, Nˆ – · DF(ϕ) ˆ S is a precompact set in Ls . This completes the proof.

Lemma . Let p, p ∈ (  , +∞]. For s ≥ ,

 p

=

 p

+

 p



p and fˆ ∈ Ls (R ), the equation

J ϕˆ = fˆ has a unique solution ϕˆ = J– fˆ =  – Nˆ – fˆ ∈ Ls (R ), where the operator p

  ˆ : Lps → Lps .  – = I + Nˆ – · DF(ϕ) Proof This is a direct result from Lemma ..



3 Proof of the main result This section is devoted to proving the main result. Since the linear operator which we get in solving equation (.) is not invertible for  =  , the implicit function theorem cannot be applied directly. The Lyapunov-Schmidt reduction is a powerful method to deal with

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this case. Assume that | –  | and | –  | are suitable small. We find the π/ -timep periodic solution ϕˆ = (ω, ˆ vˆ ) ∈ Xs , which can be made the ansatz as ϕ(y, ˆ t) =



ωˆ n (y)e

in t

,

n∈Z



vˆ n (y)e

in t

 ,

n∈Z

and satisfying dϕˆ + J ϕˆ = G(ϕ). ˆ dt

(.)

Introduce the projection Sn onto the nth Fourier mode, i.e.,

(Sn ϕ)(y) ˆ =

π



π

ω(y, ˆ t)e

in t



dt, π



π

 vˆ (y, t)e

in t

dt



and the J -invariant orthogonal projection Pn,c onto the subspace spanned by the eigenvector associated with the eigenvalue (in , in ). We denote Pn,s =  – Pn,c . Applying the projection Sn to equation (.) we get lattice systems for the Fourier modes Sn ϕˆ in ϕˆ n + J ϕˆn = Gn (ϕ), ˆ

n ∈ Z,

(.)

where Gn (ϕ) ˆ =



G(ϕˆ n–m , ϕˆm ).

n∈Z

Denote  in in ∗ = 

  . in

Then we rewrite lattice system (.) as in ∗ ϕˆ n + J ϕˆ n = Gn (ϕ), ˆ

for n = ±, ±, . . . ,

±i ∗ ϕˆ n,s + J ϕˆ n,s = Pn,s G± (ϕ), ˆ

J ϕˆ = G (ϕ), ˆ

for n = ±,

for n = ,

±i ∗ ϕˆ n,c + J ϕˆ n,c = Pn,c G± (ϕ), ˆ

(.) (.) (.)

for n = ±.

In the following, we want to prove that if ϕˆ ±,c = P±,c ϕˆ± = (P±,c ωˆ ± , P±,c vˆ ± ) ∈ Lps is given, then the above lattice systems are solvable.

(.)

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By assumptions (A)-(A), (.)-(.), and Lemma ., we obtain –  ϕˆ n = in ∗ + J Gn (ϕ), ˆ for n = ±, ±, . . . ,   – ϕˆ n,s = ±i ∗ + J Pn,s G± (ϕ), ˆ for n = ±,

(.)

ˆ ϕˆ  = J– G (ϕ),

(.)

(.)

for n = .

So we rewrite (.)-(.) as

F (ϕˆc , ϕˆs ) = ,

(.)

where ϕˆ c = (. . . , , ϕˆ –,c , , ϕˆ ,c , , . . . ), ϕˆ s = (. . . , ϕˆ – , ϕˆ –,s , ϕˆ  , ϕˆ ,s , ϕˆ – , . . . ). p

p

ˆ n∈Z = (n ωˆ n , n vˆ n ). Then Lemma . Define  = (n , n )n∈Z : Ls → Ls and (ϕ) ϕ ˆ Xsp ≤ sup Lps →Lps ϕ ˆ Xsp . n∈Z

Proof ϕ ˆ Xsp ≤

  n ωˆ n Lps + n vˆ n Lp s+

n∈Z

 ≤ sup n Lps →Lps + n Lp

p

s+ →Ls+

n∈Z

   ωˆ n Lps + ˆvn Lp s+

n∈Z

≤ sup Lps →Lps ϕ ˆ Xsp .



n∈Z

 

Lemma . Let p >    Bˆ n (ϕ) ˆ

 

p

n∈Z Xs

p

and s > ( – p ). Then, for any ϕˆ ∈ Xs ,

  ≤ Cϕ ˆ Xsp Tˆ˜  Xp + ϕ ˆ Xsp . s+

Proof By Lemma ., we obtain    Bˆ n (ϕ) ˆ

 

p n∈Z Xs

=

    B( ˆ u, ˆ vˆ , Tˆ˜  ) n Lp s

n∈Z

≤C



 ωˆ n Lps Tˆ˜ n Lp + ωˆ n Lp + ˆvn Lp s+

s

n∈Z

  ≤ C ϕ ˆ Xsp Tˆ˜  Xp + ϕ ˆ X p . s+

s

Lemma . There exists a constant C >  such that     in ∗ – J – (iy, )·

p

p

Ls →Ls

    ±i ∗ – J – Pn,s (iy, )·

p

≤ C, p

Ls →Ls

n ∈ Z \ {±, },

≤ C,

n = ±.

s+



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Proof This result is directly derived from assumption (A), the property of the sectorial operators J and Nˆ , Lemma ., and Lemma ..  p

p

Now, we return to equation (.). From Lemmas .-., F : Xs → Xs is well defined p p and smooth for p ∈ (, ) and s ≥ . It is obvious that F (, ) = , Dϕˆ F : Xs → Xs is invertible and Dϕˆ F (, ) = I. Therefore, by the implicit function theorem, there exists a unique smooth solution ϕˆ s = ϕˆ s (ϕˆ c ) satisfying ϕˆ s (ϕˆ c )Xsp ≤ Cϕˆc Xsp . Finally, we give the proof of our main result. This proof is based on the classical Hopf bifurcation (see []) applied to solve the equation (.) by the implicit function theorem. p Let ψn+ ∈ Xs denote the eigenfunctions associated with the eigenvalues (±i  , ±i  ). χ+ () is the eigenvalues of operator J under the basis (ψn+ , ψn+ ). Introduce pn,c by Pn,c ϕ = pn,c (ϕ)ψn+ . Then, for ξ ∈ C \ , it follows (.) that     –i ∗ ξ ψn+ + χ+ ()ψn+ – pn,c G+ ϕˆ s ξ ψn+ ψn+ = , which implies that     –i ∗ ξ + χ+ () – pn,c G+ ϕˆ s ξ ψn+ = . Define the complex-valued smooth function ⎧ ⎨–i( ∗ + ) + χ + ( + β) – α – ( + β, α), α = , c  c c ϒ(α; , β) := ⎩–i( ∗ + ) + χ + (c + β), α = ,  c where (c + β, α) := pn,c (G+ (ϕˆ s (ξ ψn+ ))). Denote    i c ∗ i c =  i c and  χ+ () =

λ+ 

  . μ+

By (.)-(.) and Lemma ., we know that ϒ(; , ) =  and the determinant of the Jacobi matrix  det D,β ϒ(α; , β)|α==β= = det

 –

d d d d

Re χ+ ()|=c Im χ+ ()|=c

 ×

d d + + Re λ () Re μ () = + > . d d =c =c Therefore, there exists a function α → ((α), β(α)) with () = β() =  satisfying       –iα c∗ + (α) + αχ+ c + β(α) –  c + β(α), α = , for |α| sufficient small.

(.)

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Due to the degree of nonlinearity term in (.), it is easy to see that there exists a function α(β) such that ϕˆ n,c = α(β)ψn+ is the solution of (.) for =  + (α(β)) and  =  + β. This completes the proof. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Acknowledgements This work is supported by the project of Education department of Henan Province 14A110013, and the project of Tianyuan 11426106. Received: 19 August 2015 Accepted: 2 February 2016 References 1. Zeldovich, YB, Barenblatt, GI, Librovich, VB, Makhviladze, GM: The Mathematical Theory of Combustion and Explosions. Consultants Bureau, New York (1985) 2. Sattinger, DH: Bifurcation of periodic solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 41, 66-80 (1971) 3. Iudovich, VI: Appearance of auto-oscillations in a fluid. Prikl. Mat. Meh. 35, 638-655 (1971) 4. Iooss, G: Bifurcation des solutions périodiques de certains problèmes d’évolution. C. R. Acad. Sci. Paris, Sér. A 273 624-627 (1971) 5. Chen, ZM, Price, WG: Time dependent periodic Navier-Stokes flows on a two-dimensional torus. Commun. Math. Phys. 179, 577-597 (1996) 6. Chen, ZM: Bifurcations of a steady-state solution to the two-dimensional Navier-Stokes equations. Commun. Math. Phys. 201, 117-138 (1999) 7. Chen, ZM, Price, WG: Remarks on the time dependent periodic Navier-Stokes flows on a two-dimensional torus. Commun. Math. Phys. 207, 81-106 (1999) 8. Glowinski, R, Guidoboni, G: Hopf bifurcation in viscous incompressible flow down an inclined plane: a numerical approach. J. Math. Fluid Mech. 10, 434-454 (2008) 9. Iooss, G: Bifurcation of a periodic solution of the Navier-Stokes equations into an invariant torus. Arch. Ration. Mech. Anal. 58, 35-56 (1975) 10. Iooss, G, Mielke, A: Bifurcating time-periodic solutions of Navier-Stokes equations in infinite cylinders. J. Nonlinear Sci. 1, 107-146 (1991) 11. Melcher, A, Schneider, G, Uecker, H: A Hopf-bifurcation theorem for the vorticity formulation of the Navier-Stokes equations in R3 . Commun. Partial Differ. Equ. 33, 772-783 (2008) 12. Brand, T, Kunze, M, Schneider, G, Seelbach, T: Hopf bifurcation and exchange of stability in diffusive media. Arch. Ration. Mech. Anal. 171, 263-296 (2004) 13. Bodenschatz, E, Pesch, W, Ahlers, G: Recent developments in Rayleigh-Bénard convection. Annu. Rev. Fluid Mech. 32, 709-778 (2000) 14. Getling, AV: Rayleigh-Bénard Convection. Structures and Dynamics. Advanced Series in Nonlinear Dynamics, vol. 11. World Scientific, River Edge (1998) 15. Tritton, D: Physical Fluid Dynamics, 2nd edn. Oxford Science Publications. Oxford University Press, New York (1988) 16. Lions, JL, Temam, R, Wang, S: On the equations of large-scale ocean. Nonlinearity 5, 1007-1053 (1992) 17. Pedlosky, J: Geophysical Fluid Dynamics, 2nd edn. Springer, New York (1987) 18. Climent-Ezquerra, B, Guillén-González, F, Rojas-Medar, MA: Time-periodic solutions for a generalized Boussinesq model with Neumann boundary conditions for temperature. Proc. R. Soc. A 463, 2153-2164 (2007) 19. Constantin, P, Kiselev, A, Ryzhik, L: Fronts in reactive convection: bounds, stability and instability. Commun. Pure Appl. Math. 56, 1781-1803 (2003) 20. Constantin, P, Kiselev, A, Ryzhik, L: Traveling waves in 2D reactive Boussinesq systems with no-slip boundary conditions. Nonlinearity 19(11), 2605-2615 (2006) 21. Malham, S, Xin, J: Global solutions to a reactive Boussinesq system with front data on an infinite domain. Commun. Math. Phys. 193, 287-316 (1998) 22. Lewicka, M: Existence of traveling waves in the Stokes-Boussinesq system for reactive flows. J. Differ. Equ. 237, 343-371 (2007) 23. Lewicka, M, Mucha, PB: On the existence of traveling waves in the 3D Boussinesq system. Commun. Math. Phys. 292, 417-429 (2009) 24. Winn, B: Traveling fronts in a reactive Boussinesq system: bounds and stability. Commun. Math. Phys. 259, 451-474 (2005) 25. Henry, D: Geometric Theory of Semilinear Parabolic Equations. Springer, New York (1981) 26. Marsden, JE, Mccracken, M: The Hopf Bifurcation and Its Applications. Springer, Berlin (1976)