Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 103, pp. 1–34. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu TIME PERIODIC SOLUTIONS FOR THE NON-ISENTROPIC COMPRESSIBLE QUANTUM HYDRODYNAMIC EQUATIONS WITH VISCOSITY IN R3 MIN LI Communicated by Hongjie Dong Abstract. This article concerns the existence and uniqueness of a time pe- riodic solution for the non-isentropic quantum hydrodynamic equations with viscosity. By applying the Leray-Schauder theory, subtle energy estimates and a limiting method, we obtain the existence of time periodic solutions under some smallness assumptions on the time periodic external force in R3. The uniqueness can be proved by similar energy estimates. In particular, the quan- tum effects and the energy equation are taken into account in this paper which play a significant role in the uniform (in the domain R and the positive con- stant ε) estimates, especially in the selection of the norm. 1. Introduction In this article, we consider the three-dimensional compressible quantum hydro- dynamic equations effected by a time periodic external force, which can be used widely in fluid models of nucleus, superconductivity, superfluidity and ultra small electronic devices [1, 9, 16], ∂n ∂t + div(nu) = 0, (1.1a) m[ ∂(nu) ∂t + div(nu⊗ u)] + divP = divS + n∇ψ + nf − mnu τm , (1.1b) ∂W ∂t + div(uW + uP ) + div q = div(uS) + nu · ∇ψ − W − 3 2n τe , (1.1c) ∆ψ = n− b(x), (1.1d) where n, u, ψ, P = (Pij)3×3, W denote the electron density, the electron velocity, the electric potential, the stress tensor and the energy density respectively. m, b(x), τm, τe are the electron mass, the prescribed background ion density, the momentum and the energy relaxation coefficients. Moreover, f is the given time periodic external force with the period T ∗ > 0 and q = −κ∇T− ~2n 24m (∆u+2∇ div u) is the dispersive heat flux, where the vorticity R in [15] is assumed to be “small” 2010 Mathematics Subject Classification. 47H11, 35B10, 76Y05, 35Q35, 35G25, 76N10. Key words and phrases. Time periodic solutions; uniform energy estimates; full quantum hydrodynamic equations with viscosity; Leray-Schauder degree theory. c©2020 Texas State University. Submitted March 29, 2020. Published October 2, 2020. 1 2 M. LI EJDE-2020/103 i.e. R = O(~2), T and κ are the temperature and the heat conductivity coefficient respectively. To close the moment expansion at the third order, we define the above quantities by Pij = nTδij − ~2n 12m ∂2 ∂xi∂xj log n+O(~4), W = 3 2 nT + 1 2 nm|u|2 − ~2n 24m ∆ log n+O(~4), respectively, where the quantum correction is involved and ~ > 0 is the Planck constant divided by 2π, much smaller than the macro quantities [9, 14]. In addition, we denote the viscous stress tensor S as S = µ(∇u+ (∇u)>)− 2µ 3 (div u)I, where µ > 0. It follows from [29, 2] that the quantum stress tensor is closely related to the quantum Bohm potential. Moreover, by direct calculation, we obtain the following relation −∇Q(n) = ~2 4nm div(n(∇⊗∇) log n) = ~2 4m (∆∇n n −∇n(∆n+∇2n) (n)2 + |∇n|2∇n (n)3 ) . Recently, a great deal of research has been devoted to many topics of the time periodic solutions, specially on the existence, stability and convergence of solutions for isentropic or non-isentropic models in bounded domain or in the whole space. See [7, 17, 18, 20, 27, 28] for Navier-Stokes equations. See [5, 13, 25, 26] and the references therein for other studies of the compressible Navier-Stokes-Korteweg equations, and [3, 4, 21, 24] for the magnetohydrodynamic equations. In the direc- tion of a bounded domain of R3, one can see [7] for the strong and weak periodic solutions under the inhomogeneous boundary data of Dirichlet type for example. Without the quantum effects, the above system (1.1) is well known as the classical Navier-Stokes equation and has been extensively studied by [3, 5, 17, 20, 24, 26, 31]. However, the dimension of the time periodic solutions in [5, 20, 24] need to satisfy n > 3, since the convergence of the integral with respect to the time depends closely on the space dimension. It is worthy of noting that the whole space case in two or three dimension seems more physically meaningful and mathematically difficult, and thus comparatively less studies were obtained to the best of our knowledge. By means of the spectral properties of the time-T-map in a hybrid type function space, Tsuda [26] confined the case to n = 3 and estimated the nonlinear terms by the contraction mapping argument. As for the isentropic Naver-Stokes equation in R3, Jin and Yang [17] studied the time periodic solutions by the symmetry condition and the topological degree. In this work, with the aid of the conservation law of mass and the fact that u is an odd function for the space variables, they obtain the “integration”= 0 conditions, which are necessary to construct a closed solution op- erator. Motivated by this, Cai and Tan considered the existence and uniqueness of the time periodic strong solutions to the isentropic magnetohydrodynamic model in the whole space [4] and the periodic domain [3]. Almost all of the above mentioned results are the isentropic case, the full Navier-Stokes equation is more interesting which brings new difficulty to the energy estimates. The aim of our paper is to present the existence and uniqueness of time periodic solutions for the three dimensional full quantum hydrodynamic equations with the viscosity and the damping near the constant stationary solution (n, u, T ) = (1, 0, 1). EJDE-2020/103 TIME PERIODIC SOLUTIONS 3 The main differences from the previous work [17] are in the following sense: we need neither the oddness assumptions for the space variable on the external force nor the “integration equal zero” conditions here since we introduce the damping terms and the Poisson equation, from which we can obtain the L2-norm of the variables by using such a structure of system (1.1) directly. Precisely, we utilize carefully a function of entropy to derive the closed L2 energy estimates. In addition, we take the quantum effects into account, which needs more effort for our purpose than those known results of classical hydrodynamic equations due to the quantum higher order terms. The main results of this paper are stated in Theorems 1.1 and 1.2. To this end, we use the delicate energy approach and the Leray-Schauder degree theory to obtain the existence of time periodic solutions for the approximated system (2.1) in a periodic domain ΩR, where we can see clearly how the quantum corrections and the energy equation affect the energy estimates, and then by means of a limiting process, we derive the sequence of solutions in the bounded domain with periodic boundary converges to that in the whole space. Furthermore, we not only need to construct suitable energy norms in coming estimates, but also to control the electric potential due to the special structure of system (2.1). We solve the difficult by employing the continuity equation and the curl-div decomposition of the gradient. The Sobolev space we finally adopt is a set of H5×H4×H3 defined in (2.5) which includes the quantum parameter ~. On account of the third order quantum terms in (1.2c), obtaining the uniform estimate requires some elaborated treatments thereof, and thus we can not obtain the same Sobolev space for (ρ, u, θ). Below we briefly review some results for the quantum fluid dynamics equations, and we only mention some results related to our paper. The derivation of the full compressible quantum hydrodynamic model for semiconductor devices is from Gardner [9] by the Wigner-Boltzmann equation. Moreover, in [6], the author for- mally derived nonlocal quantum hydrodynamic models. For recent works, Jüngel and Milîsić [16] introduced the full compressible quantum hydrodynamic model with viscosity which is just the system (1.1). Moreover, Pu and Guo [22] studied the global existence of smooth solutions with small initial data and established the semiclassical limit to the classical hydrodynamic system. The global existence of smooth solutions to the quantum hydrodynamic equations without viscosity in a three-dimensional domain with the insulating boundary conditions was investigated recently by [23]. The interested readers may also refer to [2, 8, 9, 10, 29] for more results. For related monograph, one can see [12]. In this article, we only put emphasis on the two quantum terms in the non- isothermal model of quantum compressible fluids system (1.1). Letting (ρ, u, θ) = (n− 1, u, T − 1), system (1.1) transforms into ∂tρ+ div u = −div(ρu), (1.2a) (1 + ρ)∂tu+ (1 + ρ)u− µ∆u− µ 3 ∇ div u+∇θ +∇ρ− ~2 12 ∇∆ρ− (1 + ρ)∇ψ = −ρ∇θ − θ∇ρ− (1 + ρ)u · ∇u− ~2 12 ∇ρ∆ρ+∇ρ · ∇2ρ 1 + ρ + ~2 12 (∇ρ · ∇ρ)∇ρ (1 + ρ)2 + (1 + ρ)f, (1.2b) 4 M. LI EJDE-2020/103 ∂tθ + θ + 2 3 div u− 2κ 3 ∆θ 1 + ρ − ~2 18 div ∆u = −u · ∇θ − 2 3 θ div u+ ~2 18 ∇ρ · ∇ div u 1 + ρ + 2 3(1 + ρ) (µ 2 |∇u+∇u>|2 − 2µ 3 (div u)2 ) + |u|2 3 + ~2 36 ∆ lnn, (1.2c) ∆ψ = ρ, (1.2d) where we let m, τm, τe, b(x) = 1 for the sake of brevity. The existence of the time periodic solutions in R3 is stated as follows. Theorem 1.1. Assume the time periodic function f ∈ L2(0, T ∗;H3(R3)). Then there are some sufficiently small constants λ and d0, independent of ε, R, such that if ∫ T∗ 0 ‖(f, ~∇f)‖2H2(R3)dt < λ, then problem (1.2) admits a time periodic solution (ρ, u, θ) ∈ Xd0 , where Xd0 is first defined in Definition 2.2. Theorem 1.2. Under the assumption of Theorem 1.1, provided that d0 sufficiently small, there exists a unique solution (ρ, u, θ) ∈ Xd0 . Remark 1.3. Studies pertaining to the unbounded domain over two or three di- mensions is also open for the case of time periodic problem of the full Naver-Stokes equations without the damping. More precisely, the symmetry condition, which has been applied successfully to get the closed solution space for the isentropic compressible Navier-Stokes system in [17], is not applicable for the reason that the temperature is an even function of the no-conserved quantity. This article is arranged as follows. In Section 2, we introduce the approximated system (2.1) and prove the completely continuous operator (2.12) to the linear system (2.6) is well defined. Then we derive some basic lemmas which will be used several times in the coming estimates. In Section 3, to obtain the existence of time periodic solutions to (2.1), we give several uniform estimates of system (3.1) with respect to the domain R and the positive constant ε. Moreover, with the help of the unform estimates and the Leray-Schauder degree theory, we complete the proof of Theorem 3.1. Finally, the main theorems are proved rigorously in Sections 4 by passing the limit of corresponding sequence of solutions in a bounded domain ΩR to the whole space R3 and the similar energy estimates as Section 3. Throughout this article, we denote α by the multi-index, and ∂α = ∂α1 x1 ∂α2 x2 ∂α3 x3 by the partial differential derivatives. We sometimes abuse the notation ∂α±1 to take the place of ∂α±β , |β| = 1 for some multi-index β. Here and in the subsequent, Hs denote the usual Sobolev space with norm ‖f‖Hs = ∑ |α|≤s ‖∂αx f‖L2 , and write W k,p(ΩR) as W k,p to simplify the symbol, but still denote W k,p(R3) by the Sobolev norm in the whole space to avoid confusion. Moreover, we use [A,B] = AB−BA to denote the commutator of A and B. The commutator estimate appears on Lemma 2.5. EJDE-2020/103 TIME PERIODIC SOLUTIONS 5 2. Preliminaries 2.1. Regularized system. To study the time periodic solutions for the quantum hydrodynamic system (1.2), we introduce the following regularized problem in a bounded domain ΩR = (−R,R)3 ⊆ R3 with periodic boundary conditions ∂tρ+ div u− ε∆ρ = −div(ρu), (2.1a) (1 + ρ)∂tu+ (1 + ρ)u− µ∆u− µ 3 ∇ div u+∇θ +∇ρ− ~2 12 ∇∆ρ − (1 + ρ)∇ψ = −ρ∇θ − θ∇ρ− (1 + ρ)u · ∇u− ~2 12 ∇ρ∆ρ+∇ρ · ∇2ρ 1 + ρ + ~2 12 (∇ρ · ∇ρ)∇ρ (1 + ρ)2 + (1 + ρ)fR, (2.1b) ∂tθ + 2 3 div u− 2κ 3 ∆θ 1 + τρ + θ − ~2 18 div ∆u = −u · ∇θ − 2 3 θ div u+ ~2 18 ∇ρ · ∇ div u 1 + ρ + 2 3(1 + ρ) (µ 2 |∇u+∇u>|2 − 2µ 3 (div u)2 ) + 1 3 |u|2 + ~2 36 ∆ ln(1 + ρ), (2.1c) ∆ψ = ρ, (2.1d) where fR is a smooth time periodic function with the period T ∗ that satisfies fR → f in L2(0, T ∗;H3(R3)), (2.2) as R tends to ∞. Definition 2.1. Let XR and X be the solution spaces in the bounded domain ΩR and in the whole space R3 respectively XR = { (ρ, u, θ) :ρ ∈ L∞(0, T ∗;H5(ΩR)) ∩ L2(0, T ∗;H5(ΩR)) u ∈ L∞(0, T ∗;H4(ΩR)) ∩ L2(0, T ∗;H5(ΩR)) θ ∈ L∞(0, T ∗;H3(ΩR)) ∩ L2(0, T ∗;H4(ΩR)) } (2.3) and X = { (ρ, u, θ) :ρ ∈ L∞(0, T ∗;H5(R3)) ∩ L2(0, T ∗;H5(R3)) u ∈ L∞(0, T ∗;H4(R3)) ∩ L2(0, T ∗;H5(R3)) θ ∈ L∞(0, T ∗;H3(R3)) ∩ L2(0, T ∗;H4(R3)) } , (2.4) where (ρ, u, θ) is periodic in time with the same period as the external force f . 6 M. LI EJDE-2020/103 Definition 2.2. We say that the solution (ρ, u, θ) ∈ Xd(or XR d ), if (ρ, u, θ) ∈ X(or XR) and satisfies |||(ρ, u, θ)|||2 = sup t∈[0,T∗] ( ‖(ρ, u, θ)‖2H3 + ‖(~∇ρ, ~∇u, ~2∆ρ)‖2H3 ) + ∫ T∗ 0 ( ‖(ρ, u, θ)‖2H3 + ‖(∇u,∇θ, ~∆u)‖2H3 + ‖(~∇ρ, ~2∆ρ)‖2H3 ) dt < d2, (2.5) with d being suitably small. 2.2. Introduction of an operator χ. For any given (ρ̃, ũ, θ̃) ∈ XR d , we consider the linear system ∂tρ+ div u− ε∆ρ = −τ div(ρ̃ũ), (2.6a) ∂tu+ u− µ∆u− µ 3 ∇ div u+∇θ +∇ρ− ~2 12 ∇∆ρ−∇ψ = −τ ρ̃ũ− τ ρ̃∂tũ− τ ρ̃∇θ̃ − τ θ̃∇ρ̃− τ(1 + τ ρ̃)ũ · ∇ũ − ~2τ 12 ∇ρ̃∆ρ̃+∇ρ̃ · ∇2ρ̃ 1 + τ ρ̃ + ~2τ 12 (∇ρ̃ · ∇ρ̃)∇ρ̃ (1 + τ ρ̃)2 + τ(1 + τ ρ̃)fR, (2.6b) ∂tθ + 2 3 div u− 2κ 3 ∆θ 1 + τ ρ̃ + θ − ~2 18 div ∆u− ~2 36 ∆ρ 1 + τ ρ̃ = −τ ũ · ∇θ̃ − 2 3 θ̃ div ũ+ ~2 18 ∇ρ̃ · ∇ div ũ 1 + τ ρ̃ + 2 3(1 + τ ρ̃) (µ 2 |∇ũ+∇ũ>|2 − 2µ 3 (div ũ)2 ) + τ 3 |ũ|2 + ~2τ 36 |∇ρ̃|2 (1 + τ ρ̃)2 , (2.6c) ∆ψ = ρ, (2.6d) for any τ ∈ [0, 1]. Provided that (ρ, u, θ) ∈ XR d , 0 < d < 1/2, we have ‖(ρ, θ)‖L∞ ≤ C‖(∇ρ,∇θ)‖H1 ≤ Cd. (2.7) This implies 1 2 ≤ 1 + τρ ≤ 3 2 , 1 2 ≤ 1 + τθ ≤ 3 2 , (2.8) which can be also applied to 1 + τ ρ̃, 1 + τ θ̃ when (ρ̃, θ̃) ∈ XR d . Letting U = (ρ, u, θ), Ũ = (ρ̃, ũ, θ̃), system (2.6) takes the form Ut +AU +QU = W (Ũ) + F, (2.9) where W (Ũ) only depends on the known functions (ρ̃, ũ, θ̃), and A = −ε∆ div 0 ∇ −µ∆− µ 3∇ div +I ∇ 0 2 3 div − 2κ 3 ∆ 1+τρ̃ + 1  , (2.10) EJDE-2020/103 TIME PERIODIC SOLUTIONS 7 where I is the unit operator, Q =  0 0 0 −~2 12∇∆ 0 0 0 −~2 18 ∆ div 0  , F = ( 0, τ(1 + τ ρ̃)fR, 0 )> . (2.11) We define the operator χ for system (2.6) in ΩR as χ : XR d × [0, 1]→ XR by ((ρ̃, ũ, θ̃), τ)→ (ρ, u, θ). (2.12) In what follows, we focus on the properties of the operator χ. Lemma 2.3. For any τ ∈ [0, 1], the operator χ is well defined. Proof. First of all, we show that U is the time periodic solution to (2.9) with the period T ∗. Consider the liner system ∂tρ+ div u− ε∆ρ = 0, (2.13a) ∂tu+ u− µ∆u− µ 3 ∇ div u+∇θ +∇ρ− ~2 12 ∇∆ρ−∇ψ = 0, (2.13b) ∂tθ + 2 3 div u+ θ − 2κ 3 ∆θ 1 + τ ρ̃ − ~2 18 div ∆u+ ~2 36 ∆ρ 1 + τ ρ̃ = 0, (2.13c) ∆ψ = ρ. (2.13d) Multiplying (2.13b) by u, we obtain 〈T u, u〉 =− ∫ ΩR ∇θ · u− ∫ ΩR ∇ρ · u+ ~2 12 ∫ ΩR ∇∆ρ · u+ ∫ ΩR ∇ψ · u , 4∑ i=1 R1,i. (2.14) Here, we define the abbreviated operator T as 〈T u, u〉 = 1 2 d dt ‖u‖2L2 + ‖u‖2L2 + Cµ‖∇u‖2L2 , where Cµ is a constant dependent on µ, and the strong elliptic operator µ∆u + µ 3∇div u is equivalent to ∇2u because of the assumption µ > 0. Now, we deal with the right-hand side of (2.14) one by one. Integration by parts and using (2.13c), we have R1,1 =− 3 2 ∫ ΩR θ(∂tθ + θ − 2κ 3 ∆θ 1 + τ ρ̃ − ~2 18 div ∆u+ ~2 36 ∆ρ 1 + τ ρ̃ ) =− 3 4 d dt ∫ ΩR |θ|2 − 3 2 ∫ ΩR |θ|2 − κ ∫ ΩR |∇θ|2 1 + τ ρ̃ − ~2 12 ∫ ΩR ∇θ ·∆u − κ ∫ ΩR ∇( 1 1 + τ ρ̃ ) · ∇θθ + ~2 24 ∫ ΩR ∇ρ · ∇θ 1 + τ ρ̃ + ~2 24 ∫ ΩR ∇( 1 1 + τ ρ̃ ) · ∇ρθ ≤− 3 4 d dt ∫ ΩR |θ|2 − 3 2 ∫ ΩR |θ|2 − κ ∫ ΩR |∇θ|2 1 + τ ρ̃ − ~2 12 ∫ ΩR ∇θ ·∆u+ δ‖∇θ‖2L2 + Cτ‖∇ρ̃‖L∞‖θ‖2H1 + Cτ‖∇ρ̃‖L∞‖(θ, ~∇ρ)‖2L2 + C~4‖∇ρ‖2L2 , 8 M. LI EJDE-2020/103 where δ is a sufficiently small positive constant. Again integrating by parts and using (2.13a), we derive R1,2 = ∫ ΩR ρdiv u = − ∫ ΩR ρ(∂tρ− ε∆ρ) = −1 2 d dt ∫ ΩR |ρ|2 − ε ∫ ΩR |∇ρ|2. Similarly, R1,3 = −~2 12 ∫ ΩR ∆ρdiv u = ~2 12 ∫ ΩR ∆ρ(∂tρ− ε∆ρ) = −~2 24 d dt ∫ ΩR |∇ρ|2 − ~2ε 12 ∫ ΩR |∆ρ|2. Using (2.13d) and the expression of div u again, we have R1,4 =− ∫ ΩR ψ div u = ∫ ΩR ψ(∂t∆ψ − ε∆2ψ) =− 1 2 d dt ∫ ΩR |∇ψ|2 − ε ∫ ΩR |ρ|2. Putting the above estimates together, and using the bounds (2.8), we conclude that d dt ∫ ΩR |(u, ρ, θ,∇ψ, ~∇ρ)|2 + ‖θ‖2H1 + ‖u‖2H1 + ε‖ρ‖2H1 + ~2ε‖∆ρ‖2L2 + ~2 12 ∫ ΩR ∇θ ·∆u ≤ δ‖∇θ‖2L2 + Cτ(d0 + ~2)‖(θ,∇θ, ~∇ρ)‖2L2 . (2.15) On the other hand, multiplying (2.13b) by −~2∆u, we obtain ~2 2 d dt ‖∇u‖2L2 + ~2Cµ‖∇u‖2H1 − ~2 ∫ ΩR ∇θ ·∆u = ~2 ∫ ΩR ∇ρ ·∆u− ~2 ∫ ΩR ∇ψ ·∆u− ~4 12 ∫ ΩR ∇∆ρ ·∆u , 7∑ i=5 R1,i, (2.16) thanks to (2.8) and the fact µ∆u+µ/3∇ div u is a strong elliptic operator. Almost exactly as the estimate for R1,3, R1,5 can be bounded by R1,5 = −~2 ∫ ΩR ρ∆ div u = −~2 2 d dt ∫ ΩR |∇ρ|2 − ~2ε ∫ ΩR |∆ρ|2. (2.17) EJDE-2020/103 TIME PERIODIC SOLUTIONS 9 It follows from (2.13a) and (2.13d) that R1,6 =~2 ∫ ΩR ψ∆ div u =− ~2 ∫ ΩR ψ(∂t∆ 2ψ − ε∆3ψ) =− ~2 d dt ∫ ΩR |∆ψ|2 − ~2ε ∫ ΩR |∇∆ψ|2 =− ~2 d dt ∫ ΩR |ρ|2 − ~2ε ∫ ΩR |∇ρ|2. (2.18) With the aid of integration by parts and (2.13a), the last term R1,7 can be estimated by R1,7 =− ~4 12 ∫ ΩR ∆ρ(∂t∆ρ− ε∆∆ρ) =− ~4 24 d dt ∫ ΩR |∆ρ|2 − ~4ε 12 ∫ ΩR |∇∆ρ|2. Multiplying (2.16) by 1/12 on both sides, taking δ, ~ to be sufficiently small and combining (2.15), we arrive at d dt ∫ ΩR |(ρ, u, θ,∇ψ, ~∇u, ~∇ρ, ~2∆ρ)|2 + ε‖ρ‖2H1 + ‖θ‖2H1 + ‖u‖2H1 + ~2‖∇2u‖2L2 + ~2ε‖∆ρ‖2L2 + ~4ε‖∇∆ρ‖2L2 ≤ 0. (2.19) Therefore, ρ ≡ u ≡ θ ≡ 0. From system (2.9), we derive ‖U(t)‖L2 = ‖S~(t)U0‖L2 ≤ C‖U0‖L2e−Cε,Rt, where S~(t) = e−t(A+Q) is the solution operator to system (2.9) and Cε,R is a constant dependent on ε, R. Thanks to Duhamel’s principle and the time periodic property of (ρ̃, ũ, θ̃, fR), we derive solutions to system (2.9) can be estimated as ‖U(t)‖L2 = ∫ t −∞ ‖S~(t− s)(W (Ũ(s)) + F (s))‖L2ds = ∫ t −∞ e−Cε,R(t−s)‖W (Ũ(s)) + F (s)‖L2ds = ∞∑ i=0 ∫ t−iT∗ t−(i+1)T∗ e−Cε,R(t−s)‖W (Ũ(s)) + F (s)‖L2ds ≤ ∞∑ i=0 (∫ T∗ 0 e−2Cε,R((i+1)T∗−s)ds )1/2(∫ t+T∗ t ‖W (Ũ(s)) + F (s)‖2L2ds )1/2 ≤ Cε,R sup t∈R ‖W (Ũ(t)) + F (t)‖L2 . (2.20) 10 M. LI EJDE-2020/103 Since fR and (ρ̃, ũ, θ̃) are periodic in time with period T ∗, we obtain U(t+ T ∗) = ∫ t+T∗ −∞ S~(t+ T ∗ − s)(W (Ũ(s)) + F (s))ds = ∫ t+T∗ −∞ S~(t− (s− T ∗))(W (Ũ(s− T ∗)) + F (s− T ∗))ds = ∫ t −∞ S~(t− s′)(W (Ũ(s′) + F (s′))ds′ = U(t). (2.21) Therefore, by (2.20) and (2.21), we have that U(t) =∈ L∞(0, T ∗;L2(ΩR)) is the de- sired periodic solution of system (2.9) with the time period T ∗. Moreover, following the energy estimates similar but much easier to those in Section 3, we derive, for any (ρ̃, ũ, θ̃) ∈ XR d , τ ∈ [0, 1], there exists a time periodic solution (ρ, u, θ) ∈ XR. In fact, we do not need the uniform (in ε, R) estimates here. Finally, we will prove the uniqueness of the time periodic solutions to system (2.6). Assume that for some Ũ ∈ XR d , τ ∈ [0, 1], there exists two different solutions U1 and U2 to system (2.9). Then we can deduce U1 − U2 is the solution to the homogeneous system ∂t(U1 − U2) + (A + Q)(U1 − U2) = 0. Recalling (2.19) and integrating the resultant inequality from 0 to T ∗, we obtain U1 = U2. Therefore, it follows from (2.12) that χ((ρ̃, ũ, θ̃), 0) = 0, which implies we need only to consider the condition τ ∈ (0, 1] in the following section. This completes the proof. � Lemma 2.4. Assume that d is suitably small, then the operator χ is compact and continuous. A proof of the above can be found in [3, Lemmas 2.4 and 2.5]. 2.3. Basic lemmas. The following lemmas will be used later. Lemma 2.5. Let α be any multi-index with |α| = k, k ≥ 1 and p ∈ (1,∞). Then ‖∂α(fg)‖Lp ≤ C‖f‖Lp1‖g‖Ẇk,p2 + C‖f‖Ẇk,p3 ‖g‖Lp4 , ‖[∂α, f ]g‖Lp ≤ C‖∇f‖Lp1 ‖g‖Ẇk−1,p2 + C‖f‖Ẇk,p3 ‖g‖Lp4 , where f, g ∈ S, the Schwartz class, Ẇ is the homogeneous Sobolev space, and p2, p3 ∈ (1,∞) such that 1 p = 1 p1 + 1 p2 = 1 p3 + 1 p4 . Lemma 2.6. Let p, q, r be integers with p, q, r ∈ [1,∞], and 0 ≤ a,m ≤ l. Then ‖∇af‖Lp ≤ C‖∇lf‖cLq‖∇mf‖1−cLr , (2.22) for some generic constant c ∈ [0, 1], where a 3 − 1 p = ( l 3 − 1 q ) c+ (m 3 − 1 r ) (1− c); when c = 1, it holds l − a 6= 3/q. EJDE-2020/103 TIME PERIODIC SOLUTIONS 11 3. Existence of time periodic solution in a bounded domain This section is devoted to the proof of the existence of time periodic solutions to system (2.1) in the bounded domain ΩR by combining the topological degree theory with the delicate energy estimates. The main theorem is stated as follows. Theorem 3.1. Let the time periodic force fR ∈ L2(0, T ∗;H3). If there exist some sufficiently small constants λ and d0 such that Cλ2 + Cd3 0 ≤ d2 0/2, where C is the generic positive constant, and∫ T∗ 0 ‖(fR, ~∇fR)‖2H2dt < λ, then(ρR, uR, θR) ∈ XR d0 is the strong time periodic solution of the approximate sys- tem (2.1) in the bounded domain ΩR with periodic boundary. To do this, we introduce the nonlinear approximated equations ∂tρ+ div u− ε∆ρ = −τ div(ρu), (3.1a) (1 + τρ)∂tu+ (1 + τρ)u− µ∆u− µ 3 ∇ div u+ (1 + τρ)∇θ + (1 + τθ)∇ρ− ~2 12 ∇∆ρ− (1 + τρ)∇ψ = −τ(1 + τρ)u · ∇u− ~2τ 12 ∇ρ∆ρ+∇ρ · ∇2ρ 1 + τρ + ~2τ 12 |∇ρ|2∇ρ (1 + τρ)2 + τ(1 + τρ)fR, (3.1b) ∂tθ + θ + 2 3 (1 + τθ) div u− 2κ 3 ∆θ 1 + τρ − ~2 18 div ∆u− ~2 36 ∆ρ 1 + τρ = −τu · ∇θ + ~2τ 18 ∇ρ · ∇ div u 1 + τρ + 2τ 3(1 + τρ) (µ 2 |∇u+∇u>|2 − 2µ 3 (div u)2 ) + τ |u|2 3 − ~2τ 36 |∇ρ|2 (1 + τρ)2 , (3.1c) ∆ψ = ρ, (3.1d) where τ ∈ (0, 1]. Provided τ = 1, we naturally derive the existence of time periodic solutions to system (2.1). Letting (ρ, u, θ) ∈ XR d , we will utilize the Kronecker’s existence theorem (see [30, Theorem 11.1.6]) to show that the approximated system (2.1) admits a solution (ρ, u, θ) ∈ XR, which is equivalent to prove χ(U, 1) = U, U = (ρ, u, θ) ∈ XR d . To this end, we need to choose d0 > 0, such that, for any τ ∈ (0, 1], (I − χ(·, τ))(∂B̂d0(0)) 6= 0, (3.2) where B̂d0(0) is a ball of radius d0 centered at the origin in XR. Next, our plan is to derive inequality (3.2) and some suitable energy estimates by the deep analysis for the special structure of the approximated system (3.1). For this, we first give the basic L2 estimate. 12 M. LI EJDE-2020/103 3.1. Basic entropy estimate. Motivated by [19], we give the zero order estimates for the approximated system (3.1) with the aid of the transform τs = (1 + τθ)/(1 + τρ)2/3 − 1, (3.3) and we define the energy function E by E(ρ, u, s) = 3(1 + τs) 2τ2 ( (1 + τρ)5/3 − 1− 5τρ 3 ) + (1 + τρ) 2 |u|2 + sρ+ 3(1 + τρ)s2 4 . (3.4) Lemma 3.2. There exists a constant 0 < ρ2 ≤ 1/2, such that E is positive definite, and satisfies E ∼= ρ2 + |u|2 + θ2. The above lemma can be proved by modifying [19, Lemma 2.5]. Under this notation (ρ, u, s), system (3.1) transforms into ∂tρ+ div u− ε∆ρ = −τ div(ρu), (3.5a) (1 + τρ)∂tu+ (1 + τρ)u− µ∆u− µ 3 ∇ div u+ 1 τ ∇ ( (1 + τρ)5/3(1 + τs) ) − ~2 12 ∇∆ρ− (1 + τρ)∇ψ = −τ(1 + τρ)u · ∇u− ~2τ 12 ∇ρ∆ρ+∇ρ · ∇2ρ 1 + τρ + ~2τ 12 (∇ρ · ∇ρ)∇ρ (1 + τρ)2 + τ(1 + τρ)fR, (3.5b) ∂ts+ s (1 + τρ)2/3 + 2 3 (1 + τs)ρ (1 + τρ)2/3 − 2κ 3 div ( ∇s 1 + τρ + 2(1 + τs)∇ρ 3(1 + τρ)2 ) − 10 9 κτ ( ∇s (1 + τρ)2 + 2(1 + τs)∇ρ 3(1 + τρ)3 ) ∇ρ− ~2 18 div ∆u (1 + τρ)2/3 − 2ε 3 (1 + τs)∆ρ 1 + τρ − ~2 36 ∆ρ (1 + τρ)5/3 = −τu · ∇s+ ~2τ 18 ∇ρ · ∇ div u (1 + τρ)5/3 + τ 3 |u|2 (1 + τρ)2/3 − ~2τ 36 |∇ρ|2 (1 + τρ)8/3 + 2τ 3(1 + τρ)5/3 (µ 2 |∇u+∇u>|2 − 2µ 3 (div u)2 ) , (3.5c) ∆ψ = ρ. (3.5d) Lemma 3.3. Let (ρ, u, θ) ∈ ∂B̂d0(0) be a solution to system (3.1). Then there exists a sufficiently small positive constant d0, independent ε, R, such that d dt ‖(ρ, u, θ,∇ρ,∇ψ, ~∇ρ)‖2L2 + ‖u‖2H1 + ‖θ‖2H1 + ‖ρ‖2H1 + ε‖ρ‖2H1 + ε‖∆ρ‖2L2 + ~2‖∆ρ‖2L2 + ~2ε‖∆ρ‖2L2 ≤ Cτd0‖(u,∇u,∇θ,∇ρ, ~∆ρ)‖2L2 + C~4‖∆u‖2L2 + Cτ‖fR‖2L2 . (3.6) Proof. Recalling the definition of E(ρ, u, s) in (3.4), we have d dt ∫ ΩR E(ρ, u, s) = ∫ ΩR (1 + τρ)uut + 3 2τ ∫ ΩR (1 + τρ) ( (1 + τρ)2/3 − 1 + τs ) st EJDE-2020/103 TIME PERIODIC SOLUTIONS 13 + ∫ ΩR ( τ |u|2 2 + 5 2τ (1 + τs) ( (1 + τρ)2/3 − 1 ) + s+ 3τ 4 s2 ) ρt. Integrating by parts, using (3.5a) and doing careful computations, we have 1 τ ∫ ΩR ∇ ( (1 + τρ)5/3(1 + τs) ) · u + ∫ ΩR ( 5 2τ (1 + τs) ( (1 + τρ)2/3 − 1 ) + s+ 3τ 4 s2 ) div((1 + τρ)u) + 3 2 ∫ ΩR (1 + τρ) ( (1 + τρ)2/3 − 1 + τs ) (u · ∇s) = 0, and − τ ∫ ΩR (1 + τρ)u · ∇u · u+ τ ∫ ΩR |u|2 2 ρt = τ 2 ∫ ΩR div ( (1 + τρ)u ) |u|2 + τ 2 ∫ ΩR ρt|u|2 = τε 2 ∫ ΩR |u|2∆ρ = −τε 2 ∫ ΩR ∇(|u|2) · ∇ρ ≤ Cτ‖u‖L∞‖(∇u,∇ρ)‖2L2 . Combining the above estimates with Young’s inequality and Lemma 2.6, we deduce that d dt ∫ ΩR E(ρ, u, s) ≤ ~2 12 ∫ ΩR ∇∆ρ · u+ ∫ ΩR (1 + τρ)∇ψ · u+ ~2 24τ ∫ ΩR ( (1 + τρ)2/3 − 1 + τs ) ∆ρ (1 + τρ)2/3 + ε τ ∫ ΩR (1 + τs) ( (1 + τρ)2/3 − 1 + τs ) ∆ρ − ~2τ 12 ∫ ΩR (∇ρ∆ρ+∇ρ · ∇2ρ 1 + τρ − (∇ρ · ∇ρ)∇ρ (1 + τρ)2 ) · u + ~2 12τ ∫ ΩR (1 + τρ)1/3 ( (1 + τρ)2/3 − 1 + τs ) div ∆u− 4κ 3 ∫ ΩR ∇s · ∇ρ − 2 ∫ ΩR ρs− 4κ 9 ‖∇ρ‖2L2 − ‖u‖2L2 − 2 3 ‖ρ‖2L2 − 3 2 ‖s‖2L2 − 5ε 3 ‖∇ρ‖2L2 − µ‖∇u‖2L2 − κ‖∇s‖2L2 − µ 3 ‖div u‖2L2 + δ‖u‖2L2 + Cτ ( ‖(u, ρ, s)‖L∞ + ‖(ρ, s,∇ρ)‖2L∞ ) ‖(u,∇u,∇s,∇ρ)‖2L2 + Cτ‖fR‖2L2 + Cε‖∇s‖2L2 + C~4‖∇ div u‖2L2 ≤ 6∑ i=1 R2,i − 4κ 3 ∫ ΩR ∇s · ∇ρ− 2 ∫ ΩR ρs− 4κ 9 ‖∇ρ‖2L2 − ‖u‖2L2 − 2 3 ‖ρ‖2L2 − 3 2 ‖s‖2L2 − 5ε 3 ‖∇ρ‖2L2 − µ‖∇u‖2L2 − κ‖∇s‖2L2 − µ 3 ‖div u‖2L2 + Cτd0‖(u,∇u,∇s,∇ρ)‖2L2 + δ‖u‖2L2 + Cτ‖fR‖2L2 + Cε‖∇s‖2L2 14 M. LI EJDE-2020/103 + C~4‖∇ div u‖2L2 , (3.7) where δ is a sufficiently small positive constant. Estimates for the right-hand side of (3.7). For the first term R2,1, by the equation (3.5a) and integration by parts, we obtain R2,1 = ~2 12 ∫ ΩR ∆ρ(∂tρ− ε∆ρ+ τu · ∇ρ) 1 + τρ =− ~2 24 d dt ∫ ΩR |∇ρ|2 1 + τρ − ~2ε 12 ∫ ΩR |∆ρ|2 1 + τρ + ~2τ 24 ∫ ΩR ∂tρ|∇ρ|2 (1 + τρ)2 + ~2τ 12 ∫ ΩR ∆ρu · ∇ρ 1 + τρ ≤− ~2 24 d dt ∫ ΩR |∇ρ|2 1 + τρ − ~2ε 12 ∫ ΩR |∆ρ|2 1 + τρ − ~2τ 12 ∫ ΩR ∇ρ>∇u∇ρ 1 + τρ + ~2τ 24 ∫ ΩR div( u 1 + τρ )|∇ρ|2 + Cτ‖∇ρ‖L∞‖~∇ρ‖L2‖~∂tρ‖L2 ≤− ~2 24 d dt ∫ ΩR |∇ρ|2 1 + τρ − ~2ε 12 ∫ ΩR |∆ρ|2 1 + τρ + δ‖~ε1/2∆ρ‖2L2 + Cτd0‖(~∇u, ~∇ρ)‖2L2 , where δ is a sufficiently small positive constant. For the second term R2,2, by (3.5a), (3.5d) and integration by parts, we obtain R2,2 =− ∫ ΩR ψ div ( (1 + τρ)u ) = ∫ ΩR ψ(∂t∆ψ − ε∆2ψ) =− 1 2 d dt ∫ ΩR |∇ψ|2 − ε ∫ ΩR |ρ|2. For the third term R2,3, by integration by parts again, we obtain R2,3 =− ~2 24τ ∫ ΩR ∇ ( (1 + τρ)2/3 − 1 + τs (1 + τρ)2/3 ) · ∇ρ ≤− ~2 36 ‖∇ρ‖2L2 + Cτ‖(ρ, s)‖L∞‖∇ρ‖2L2 + C~2‖(∇s,∇ρ)‖2L2 . Similarly, R2,4 ≤ − 2ε 3 ‖∇ρ‖2L2 + Cτ‖(ρ, s)‖L∞‖(∇s,∇ρ)‖2L2 + Cε‖(∇s,∇ρ)‖2L2 . For the fifth term R2,5, from integration by parts and Hölder’s inequality, we have R2,5 =− ~2τ 12 ∫ ΩR ∇ρ∆ρ 1 + τρ · u+ ~2τ 24 ∫ ΩR div ( u 1 + τρ ) |∇ρ|2 + ~2τ 12 ∫ ΩR |∇ρ|2∇ρ (1 + τρ)2 · u =− ~2τ 12 ∫ ΩR div(∇ρ⊗∇ρ)− 1 2∇(|∇ρ|2) 1 + τρ · u+ ~2τ 24 ∫ ΩR div ( u 1 + τρ ) |∇ρ|2 + ~2τ 12 ∫ ΩR (∇ρ · ∇ρ)∇ρ (1 + τρ)2 · u EJDE-2020/103 TIME PERIODIC SOLUTIONS 15 = ~2τ 12 ∫ ΩR (∇ρ)> div ( u 1 + τρ ) ∇ρ+ ~2τ 12 ∫ ΩR (∇ρ · ∇ρ)∇ρ (1 + τρ)2 · u ≤Cτ(‖∇ρ‖L∞ + ‖∇ρ‖2L∞)‖~∇ρ‖2L2 , where we have used the vector analysis formulation div ff = div(f ⊗ f)− 1 2 ∇(|f |2)− (∇× f)× f, (3.8) for any vector function f . Again, by integration by parts, we obtain R2,6 =− ~2 36 ∫ ΩR ∇ρ ( (1 + τρ)2/3 − 1 + τs ) ∆u (1 + τρ)2/3 − ~2 12τ ∫ ΩR (1 + τρ)1/3∇ ( (1 + τρ)2/3 − 1 + τs ) ∆u ≤C~4‖∆u‖2L2 + δ‖(∇ρ,∇s)‖2L2 + Cτ‖(s, ρ)‖2L∞‖∇ρ‖2L2 . Therefore, d dt ∫ ΩR E(ρ, u, s) + ~2 24 d dt ∫ ΩR |∇ρ|2 1 + τρ + 1 2 d dt ∫ ΩR |∇ψ|2 + 7ε 3 ‖∇ρ‖2L2 + ~2 36 ‖∇ρ‖2L2 + ε‖ρ‖2L2 + ‖u‖2L2 + µ‖∇u‖2L2 + 3 2 ‖s‖2L2 + κ‖∇s‖2L2 + 2 3 ‖ρ‖2L2 + 4κ 9 ‖∇ρ‖2L2 + µ 3 ‖ div u‖2L2 + ~2ε 12 ‖∆ρ‖2L2 ≤ −4κ 3 ∫ ΩR ∇s · ∇ρ− 2 ∫ ΩR sρ+ Cτ‖fR‖2L2 + Cε‖∇s‖2L2 + δ‖(u,∇ρ,∇s, ~ε1/2∆ρ)‖2L2 + C~2‖(∇ρ,∇s)‖2L2 + C~4‖∆u‖2L2 + Cτd0‖(u,∇u,∇s,∇ρ)‖2L2 , (3.9) where δ is a sufficiently small positive constant. Here, we have used the the Riesz operator Rj , (̂Rjf) = iξj |ξ| f̂ that ‖∇2f‖L2 = ‖∇∆−1∇∆f‖L2 = ‖RiRj∆f‖L2 ≤ C‖∆f‖L2 , (3.10) where RiRj is bounded from Lp to Lp with 1 < p <∞. On the other hand, we observe that d dt ∫ ΩR (1 2 |∇ρ|2 + 3 4µ (1 + τρ)2∇ρ · u ) = ∫ ΩR ( ∇ρ+ 3 4µ (1 + τρ)2u ) · ∇ρt + 3 4µ ∫ ΩR (1 + τρ)2∇ρ · ut + 3τ 2µ ∫ ΩR (1 + τρ)u · ∇ρρt = ∫ ΩR ( ∆ρ+ 3 4µ (1 + τρ)2 div u )( (1 + τρ) div u− ε∆ρ+ τu · ∇ρ ) + 3 4µ ∫ ΩR (1 + τρ)2∇ρ · ut ≤ − 3 8µ d dt ∫ ΩR (1 + τρ)|ρ|2 + 3 4µ ‖ div u‖2L2 − 3 4µ ∫ ΩR ∇ρ · ∇s 16 M. LI EJDE-2020/103 − 5 4µ ‖∇ρ‖2L2 − ε‖∆ρ‖2L2 − ~2 16µ ‖∆ρ‖2L2 − 3 4µ ‖ρ‖2L2 − 3ε 4µ ‖∇ρ‖2L2 + Cε1/2‖(ε1/2∆ρ, div u)‖2L2 + δ‖(∇ρ, ε1/2∆ρ)‖2L2 + Cτ‖fR‖2L2 + Cτ(‖(u, ρ,∇ρ,∇u)‖L∞ + ‖∇ρ‖2L∞)‖(ρ,∇ρ,∇s,∇u, ~∆ρ)‖2L2 , (3.11) where we have used∫ ΩR ∆ρ(1 + τρ) div u+ 3 4µ ∫ ΩR (1 + τρ)∇ρ(µ∆u+ µ 3 ∇div u) = ∫ ΩR ∆ρ(1 + τρ) div u− ∫ ΩR ∆ρ(1 + τρ) div u− τ ∫ ΩR div u|∇ρ|2 ≤ Cτ‖ div u‖L∞‖∇ρ‖2L2 , − 3 4µ ∫ ΩR (1 + τρ)2∇ρ · u = 3 4µ ∫ ΩR (1 + τρ) div ( (1 + τρ)u ) ρ+ 3τ 4µ ∫ ΩR ∇ρ(1 + τρ)uρ = − 3 4µ ∫ ΩR (1 + τρ)(∂tρ− ε∆ρ)ρ+ 3τ 4µ ∫ ΩR ∇ρ(1 + τρ)uρ = − 3 8µ d dt ∫ ΩR (1 + τρ)|ρ|2 + 3τ 8µ ∫ ΩR ρt|ρ|2 − 3ε 4µ ∫ ΩR (1 + τρ)|∇ρ|2 − 3τε 4µ ∫ ΩR ρ|∇ρ|2 + 3τ 4µ ∫ ΩR ∇ρ(1 + τρ)uρ ≤ − 3 8µ d dt ∫ ΩR (1 + τρ)|ρ|2 − 3ε 4µ ∫ ΩR (1 + τρ)|∇ρ|2 + Cτ‖ρt‖L2‖ρ‖L3‖ρ‖L6 + Cτ‖ρ‖L∞‖∇ρ‖2L2 + Cτ‖∇ρ‖L2‖u‖L3‖ρ‖L6 ≤ − 3 8µ d dt ∫ ΩR (1 + τρ)|ρ|2 − 3ε 4µ ∫ ΩR (1 + τρ)|∇ρ|2 + δε‖∆ρ‖2L2 + Cτd0‖ρ‖2H1 , 3 4µ ∫ ΩR (1 + τρ)2∇ρ · ∇ψ = − 3 4µ ∫ ΩR (1 + τρ)2ρ∆ψ − 3τ 2µ ∫ ΩR (1 + τρ)ρ∇ρ · ∇ψ = − 3 4µ ∫ ΩR (1 + τρ)2|ρ|2 − 3τ 2µ ∫ ΩR (1 + τρ)ρ∇ρ · ∇ψ ≤ − 3 4µ ‖ρ‖2L2 + Cτ‖∇ψ‖L6‖ρ‖L3‖∇ρ‖L2 + Cτ‖ρ‖L∞‖ρ‖2L2 ≤ − 3 4µ ‖ρ‖2L2 + Cτ‖∇2ψ‖L2‖ρ‖H1‖∇ρ‖L2 + Cτ‖ρ‖L∞‖ρ‖2L2 ≤ − 3 4µ ‖ρ‖2L2 + Cτd0‖ρ‖2H1 , and ~2 16µ ∫ ΩR (1 + τρ)∇∆ρ · ∇ρ EJDE-2020/103 TIME PERIODIC SOLUTIONS 17 = − ~2 16µ ∫ ΩR (1 + τρ)|∆ρ|2 − ~2τ 16µ ∫ ΩR |∇ρ|2∆ρ ≤ − ~2 16µ ∫ ΩR (1 + τρ)|∆ρ|2 + Cτ‖∇ρ‖L∞‖(∇ρ, ~∆ρ)‖2L2 ≤ − ~2 16µ ∫ ΩR (1 + τρ)|∆ρ|2 + Cτd0‖(∇ρ, ~∆ρ)‖2L2 , thanks to (3.1a), (3.1d), (3.10), integration by parts and Lemma 2.6. Multiplying (3.11) by a positive constant m, and taking δ and ε sufficiently small, we have d dt ∫ ΩR E(ρ, u, s) + ~2 24 d dt ∫ ΩR |∇ρ|2 (1 + τρ)2 + 1 2 d dt ∫ ΩR |∇ψ|2 +m d dt ∫ ΩR (1 2 |∇ρ|2 + 3 4µ (1 + τρ)2∇ρ · u+ 3 8µ (1 + τρ)|ρ|2 ) + ( 3m 4µ + 2 3 )‖ρ‖2L2 + ‖u‖2L2 + µ‖∇u‖2L2 + ( µ 3 − 3m 4µ )‖ div u‖2L2 + 3 2 ‖s‖2L2 + κ‖∇s‖2L2 + ~2ε 12 ‖∆ρ‖2L2 + ~2m 16µ ‖∆ρ‖2L2 + ( 7 3 + 3m 4µ )ε‖∇ρ‖2L2 + ε‖ρ‖2L2 +mε‖∆ρ‖2L2 + ( 4κ 9 + 5m 4µ )‖∇ρ‖2L2 + ~2 36 ‖∇ρ‖2L2 + ( 4κ 3 + 3m 4µ ) ∫ ΩR ∇s · ∇ρ+ 2 ∫ ΩR ρs ≤ Cτd0‖(u, ρ,∇u,∇s,∇ρ, ~∆ρ)‖2L2 + Cτ‖fR‖2L2 + C~4‖∆u‖2L2 . Moreover, we assume that 2µ(3− 4a) 9a < m < min{ 2µ2 9(1 + ρ̄)4 , 4µ2 9 , 16κµ 3 }, where 0 < ρ̄ ≤ 1/2, 0 < a < 3/4, then we obtain∫ ΩR E(ρ, u, s) +m ∫ ΩR (1 2 |∇ρ|2 + 3 4µ (1 + τρ)2∇ρ · u ) ≥ 1 8 (‖ρ‖2L2 + 7 9 ‖s‖2L2 + ‖u‖2L2) + m 4 ‖∇ρ‖2L2 ,(3m 4µ + 2 3 ) ‖ρ‖2L2 + 3 2 ‖s‖2L2 + 2 ∫ ΩR ρs ≥ (3m 4µ + 4a− 3 6a ) ‖ρ‖2L2 + (3 2 − 2a ) ‖s‖2L2 , and κ‖∇s‖2L2 + (µ 3 − 3m 4µ ) ‖div u‖2L2 + (4κ 3 + 3m 4µ ) ∫ ΩR ∇s · ∇ρ + (4κ 9 + 5m 4µ ) ∫ ΩR |∇ρ|2 ≥ 5κ 27 ‖∇s‖2L2 + 5κ 3 ‖∇ρ‖2L2 . 18 M. LI EJDE-2020/103 Combining the above estimates, we obtain d dt ‖(ρ, u, s,∇ρ,∇ψ, ~∇ρ)‖2L2 + ‖u‖2H1 + ‖s‖2H1 + ‖ρ‖2H1 + ε‖ρ‖2H1 + ~2ε‖∆ρ‖2L2 + ε‖∆ρ‖2L2 + ~2‖∆ρ‖2L2 ≤ Cτd0‖(∇u,∇s,∇ρ, ~∆ρ)‖2L2 + C~4‖∆u‖2L2 + Cτ‖fR‖2L2 . Equivalently, we have d dt ‖(ρ, u, θ,∇ρ,∇ψ, ~∇ρ)‖2L2 + ‖u‖2H1 + ‖θ‖2H1 + ‖ρ‖2H1 + ε‖ρ‖2H1 + ~2ε‖∆ρ‖2L2 + ε‖∆ρ‖2L2 + ~2‖∆ρ‖2L2 ≤ Cτd0‖(∇u,∇θ,∇ρ, ~∆ρ)‖2L2 + C~4‖∆u‖2L2 + Cτ‖fR‖2L2 , thanks to Lemma 3.2 and (3.3). The proof is complete. � It is worth mentioning that the derivative of the velocity u on the right of (3.6) is higher than that on the left which leads the following estimate. Lemma 3.4. Under the assumptions in Lemma 3.3, we arrive at d dt ‖(~∇u, ~∇ρ, ~2∆ρ)‖2L2 + ~2‖∆u‖2L2 + ε‖(~∆ρ, ~2∇∆ρ)‖2H3 ≤ δ‖(u, ε1/2∇ρ,∇θ, ~∆u)‖2H3 + Cτd2 0‖(∇u,∇θ,∇ρ, ~∆ρ, ~2∆ρ)‖L2 + Cτ‖fR‖2L2 . (3.12) Proof. Multiplying (3.1b) by 1/(1 + τρ), we obtain ∂tu+ u 1 + τρ − µ ∆u 1 + τρ − µ 3 ∇div u 1 + τρ +∇θ + 1 + τθ 1 + τρ ∇ρ− ~2 12 ∇∆ρ 1 + τρ −∇ψ = −τu · ∇u− ~2τ 12 ∇ρ∆ρ+∇ρ · ∇2ρ (1 + τρ)2 + ~2τ 12 |∇ρ|2∇ρ (1 + τρ)3 + τfR. (3.13) Taking inner product of (3.13) with −~2∆u, we obtain ~2 2 d dt ‖∇u‖2L2 + ~2Cµ‖∆u‖2L2 = −~4 12 ∫ ΩR ∇∆ρ ·∆u 1 + τρ + ~2 ∫ ΩR 1 + τθ 1 + τρ ∇ρ ·∆u+ ~2 ∫ ΩR ρdiv u + ~4τ 12 ∫ ΩR (∇ρ∆ρ+∇ρ · ∇2ρ (1 + τρ)2 − |∇ρ| 2∇ρ (1 + τρ)3 ) ∆u+ ~2 ∫ ΩR ∇θ ·∆u − µ~2τ 3 ∫ ΩR ∇ρ div u(∆u−∇ div u) (1 + τρ)2 + ~2τ 2 ∫ ΩR div u|∇u|2 + ~2τ ∫ ΩR ∂ku i∂iu j∂ku j + ~2 ∫ u∆u 1 + τρ − τ~2 ∫ ΩR fR∆u, (3.14) where we have used the estimate µ~2 3 ∫ ΩR ∇ div u∆u 1 + τρ = −µ~ 2 3 ∫ ΩR div u∆ div u 1 + τρ + µ~2τ 3 ∫ ΩR ∇ρdiv u∆u (1 + τρ)2 EJDE-2020/103 TIME PERIODIC SOLUTIONS 19 = µ~2 3 ∫ ΩR |∇div u|2 1 + τρ − µ~2τ 3 ∫ ΩR ∇ρdiv u∇ div u (1 + τρ)2 + µ~2τ 3 ∫ ΩR ∇ρ div u∆u (1 + τρ)2 , ~2 ∫ ΩR ∇ψ ·∆u = −~2 ∫ ΩR ψ∆ div u = −~2 ∫ ΩR ∆ψ div u = −~2 ∫ ΩR ρdiv u, (3.15) and τ~2 ∫ ΩR u · ∇u∆u = −τ~2 ∫ ΩR ∂ku i∂iu j∂ku j − ~2τ ∫ ΩR ui∂iku j∂ku j = ~2τ ∫ ΩR ∂ku i∂iu j∂ku j + ~2τ 2 ∫ ΩR div u|∇u|2. (3.16) We proceed to estimate the terms on the right-hand side of (3.14). For the first term, by integration by parts, (3.1a), Lemmas 2.5 and 2.6, we have − ~4 12 ∫ ΩR ∇∆ρ ·∆u 1 + τρ = ~4 12 ∫ ΩR ∆ρ∆ div u 1 + τρ + ~4 12 ∫ ΩR ∇ ( 1 1 + τρ ) ·∆u∆ρ = −~4 12 ∫ ΩR ∆ρ(∂t∆ρ− ε∆2ρ+ τu · ∇∆ρ) (1 + τρ)2 − ~4τ 12 ∫ ΩR ∆ρ([∆, u] · ∇ρ+ [∆, ρ] div u) (1 + τρ)2 + ~4 12 ∫ ΩR ∇ ( 1 1 + τρ ) ·∆u∆ρ = −~4 24 d dt ∫ ΩR |∆ρ|2 (1 + τρ)2 + ~4 24 ∫ ΩR ∂t ( 1 1 + τρ )2 |∆ρ|2 − ~4ε 12 ∫ ΩR |∇∆ρ|2 (1 + τρ)2 + ~4ετ 6 ∫ ΩR ∇ρ · ∇∆ρ∆ρ (1 + τρ)3 + ~4τ 12 ∫ ΩR div ( u (1 + τρ)2 ) |∆ρ|2 − ~4τ 12 ∫ ΩR ∆ρ[∆, u] · ∇ρ (1 + τρ)2 − ~4τ 12 ∫ ΩR ∆ρ[∆, ρ] div u (1 + τρ)2 + ~4 12 ∫ ΩR ∇ ( 1 1 + τρ ) ·∆u∆ρ ≤ −~4 24 d dt ∫ ΩR |∆ρ|2 (1 + τρ)2 − ~4ε 12 ∫ ΩR |∇∆ρ|2 (1 + τρ)2 + δ‖(ε1/2∇ρ, ~∆u, ~2ε1/2∇∆ρ)‖2H3 + Cτd2 0‖~2∆ρ‖L2 . By integration by parts and (3.10), we have − ~2 ∫ ΩR 1 + τθ 1 + τρ ∇ρ ·∆u = ~2 ∫ ΩR ρ∇ (1 + τθ 1 + τρ ) ·∆u− ~2 ∫ ΩR ρ∇ (1 + τθ 1 + τρ ) · ∇ div u − ~2 ∫ ΩR 1 + τθ 1 + τρ ∇ρ · ∇ div u 20 M. LI EJDE-2020/103 ≤ −~2 2 d dt ∫ ΩR (1 + τθ)|∇ρ|2 (1 + τρ)2 − ~2ε ∫ ΩR (1 + τθ)|∆ρ|2 (1 + τρ)2 + δ‖(ε1/2∇ρ,∇θ, ~ε1/2∆ρ)‖2H3 + Cτd2 0‖(∇θ, ~∇ρ)‖L2 + C~4‖∆u‖2L2 , (3.17) where δ is a sufficiently small positive constant. Here, we have used the following estimates, for any 1 ≤ p ≤ ∞, ‖ρt‖Lp ≤ C‖(div u, ε∆ρ)‖Lp + C‖u‖L∞‖∇ρ‖Lp , (3.18) and ‖θt‖Lp ≤C‖(θ,∆θ,div u, ~2∆ρ, ~2 div ∆u)‖Lp + Cτ‖u‖L∞‖(u,∇θ)‖Lp + Cτ‖∇ρ‖L∞‖(∇ρ, ~2∇ div u)‖Lp + Cτ‖∇u‖L∞‖∇u‖Lp ≤C‖(θ,∆θ,div u, ~2∆ρ, ~2 div ∆u)‖Lp + Cτ‖(∇u,∇ρ)‖H2‖(u,∇u,∇ρ,∇θ, ~∇ div u)‖Lp , (3.19) thanks to (3.1a) and (3.1c). By Hölder’s and Young’s inequality, the other terms on the right-hand side of (3.14) can be bounded by δ‖(u,∇θ)‖2L2 + C~4‖∆u‖2L2 + Cτd2 0‖(∇u,∇ρ, ~2∆ρ)‖L2 . Putting the above estimates together, and taking δ, ~ sufficiently small, we complete the proof. � 3.2. First, second and third order estimates for (ρ, u, θ). Lemma 3.5. Under the same assumptions in Lemma 3.3, it holds that d dt ‖(∇ρ,∇u,∇θ, ~∆ρ)‖2H2 + ‖∇u‖2H3 + ‖∇θ‖2H3 + ε‖∇ρ‖2H3 + ε~2‖∆ρ‖2H3 ≤ Cτd2 0‖(ρ, θ,∇u,∇θ, ~2∆ρ, ~2∆u)‖H3 + Cτd0‖(ρ, θ,∇u,∇θ)‖2H3 + Cτ‖∇fR‖2H1 + δ‖~∆u‖2H3 + C~4‖∆u‖2H3 + C~2‖~∇ρ‖2H3 . (3.20) Proof. Next we define α by any multi-index with |α| = k, k = 1, 2, 3, and write (∂αρ, ∂αu, ∂αθ) = (ρα, uα, θα) for simplicity. Applying the operator ∂α to (3.1a) and (3.1c), we derive ∂tρα + (1 + τρ) div uα − ε∆ρα = −τu · ∇ρα + g1, (3.21a) ∂tθα + θα + 2 3 (1 + τθ) div uα − 2κ 3 ∆θα 1 + τρ − ~2 18 div ∆uα = −τu · ∇θα + ~2 36 ∂α ( ∆ρ 1 + τρ ) + g2, (3.21b) where g1 = −τ [∂α, ρ] div u− τ [∂α, u] · ∇ρ, g2 = 2κ 3 [∂α, 1 1 + τρ ]∆θ − 2 3 τ [∂α, θ] div u− τ [∂α, u] · ∇θ + ~2τ 18 ∂α (∇ρ · ∇ div u 1 + τρ ) + 2τ 3 ∂α ( 1 1 + τρ (µ 2 |∇u+∇u>|2 − 2µ 3 (div u)2 )) + ∂α (τ 3 |u|2 − ~2τ 36 |∇ρ|2 (1 + τρ)2 ) . (3.22) EJDE-2020/103 TIME PERIODIC SOLUTIONS 21 Multiplying system (3.21) by 1+τθ 1+τρρα, 3 2 1+τρ 1+τθ θα, integrating over the periodic do- main ΩR, we obtain 1 2 d dt ∫ ΩR (1 + τθ)|ρα|2 1 + τρ + 3 4 d dt ∫ ΩR (1 + τρ)|θα|2 1 + τθ + ‖(θα,∇θα)‖2L2 + ε‖∇ρα‖2L2 + ∫ ΩR (1 + τθ) div uαρα + ∫ ΩR (1 + τρ) div uαθα = 1 2 ∫ ΩR ∂t (1 + τθ 1 + τρ ) |ρα|2 + 3 4 ∫ ΩR ∂t (1 + τρ 1 + τθ ) |θα|2 − τ ∫ ΩR 1 + τθ 1 + τρ u · ∇ραρα − 3τ 2 ∫ ΩR 1 + τρ 1 + τθ u · ∇θαθα + ∫ ΩR 1 + τθ 1 + τρ g1ρα + 3 2 ∫ ΩR 1 + τρ 1 + τθ g2θα + ~2 24 ∫ ΩR 1 + τρ 1 + τθ ∂α ( ∆ρ 1 + τρ ) θα − ~2 12 ∫ ΩR ∇ (1 + τρ 1 + τθ ) ·∆uαθα − κ ∫ ΩR ∇ ( 1 1 + τθ ) · ∇θαθα − ~2 12 ∫ ΩR (1 + τρ)∇θα ·∆uα 1 + τθ , 10∑ i=1 R3,i. (3.23) Estimates for the right-hand side of (3.23). It follows from (3.18) and (3.19) that R3,1 +R3,2 ≤C‖(∂tρ, ∂tθ)‖L∞‖(ρ, θ)‖2H3 ≤δε‖∇ρ‖2H3 + δ‖∇θ‖2H3 + C~4‖∆u‖2H3 + Cτ‖(ρ, θ)‖4H3 + Cτ‖(∇u,∇ρ,∇θ, ~∆ρ)‖H2‖(ρ, θ)‖2H3 + Cτ‖(∇u,∇ρ,∇θ, ~∇div u)‖2H2‖(ρ, θ)‖2H3 ≤δε‖∇ρ‖2H3 + δ‖∇θ‖2H3 + C~4‖∆u‖2H3 + Cτd0‖(ρ, θ)‖2H3 . By integration by parts and Hölder’s inequality, we have R3,3 +R3,4 = τ 2 ∫ ΩR div ( (1 + τθ)u 1 + τρ ) |ρα|2 + 3τ 4 ∫ ΩR div ( (1 + τρ)u 1 + τθ ) |θα|2 ≤ Cτ(1 + ‖u‖L∞)‖(∇u,∇ρ,∇θ)‖L∞‖(ρ, θ)‖2H3 ≤ Cτd0‖(ρ, θ)‖2H3 . For the fifth term, by Lemma 2.5, we have R3,5 ≤Cτ ( ‖(∇ρ,∇u)‖L∞‖(div u,∇ρ)‖H2 + ‖(∇ρ,∇u)‖H2‖(div u,∇ρ)‖L∞ ) ‖ρ‖H3 ≤Cτd2 0‖ρ‖H3 . By Lemmas 2.5 and 2.6, we have ‖ 1 1 + τρ ‖Ḣk ≤ Cτ(1 + ‖∇ρ‖kH2), (3.24) where k = 2, 3, and Ḣ is the homogeneous Sobolev spaces. For the sixth term, using (3.24) and Lemma 2.5 again, we have R3,6 ≤Cτ‖∇ ( 1 1 + τρ ) ‖H2‖∆θ‖H2‖θ‖H3 + Cτ‖(∇θ,∇u)‖H2‖(div u,∇θ)‖H2‖θ‖H3 + Cτ ( ‖∇ρ‖L∞‖~2∇ div u‖H3 + ‖~∇ρ‖H3‖~∇div u‖L∞ ) ‖θ‖H3 + Cτ(‖∇ρ‖H2 + ‖∇ρ‖4H2)‖~∇ div u‖L∞‖θ‖H3 22 M. LI EJDE-2020/103 + Cτ‖(u,∇u)‖L∞‖(∇u,∇2u)‖H2‖θ‖H3 + Cτ‖~∇ρ‖2H3‖θ‖H3 ≤Cτd2 0‖(θ,∇θ, ~2∆u)‖H3 . For the seventh term, with the aid of integration by parts and (3.24), we have R3,7 =− ~2 24 ∫ ΩR ∂ (1 + τρ 1 + τθ ) ∂α−1 ( ∆ρ 1 + τρ ) θα − ~2 24 ∫ ΩR 1 + τρ 1 + τθ ∂α−1 ( ∆ρ 1 + τρ ) θα+1 ≤Cτ‖(∇ρ,∇θ)‖L∞(1 + ‖∇ρ‖2H2)‖~∇ρ‖H3‖θ‖H3 + δ‖∇θ‖2H3 + C~2‖~∇ρ‖2H3 ≤δ‖∇θ‖2H3 + C~2‖~∇ρ‖2H3 + Cτd2 0‖θ‖H3 , where δ is a sufficiently small positive constant. Due to Hölder’s and Young’s inequalities, the other terms on the right-hand side of (3.23) can be bounded by δ‖~∆u‖2H3 + C~2‖∇θ‖2H3 + C~4‖∆u‖2H3 + Cτd2 0‖θ‖H4 . Adding the above estimates, and taking δ, ~, ε sufficiently small, we have d dt ‖(∇ρ,∇θ)‖2H2 + ‖∇θ‖2H3 + ε‖∇ρ‖2H3 + ∫ ΩR (1 + τθ) div uαρα + ∫ ΩR (1 + τρ) div uαθα ≤ δ‖~∆u‖2H3 + C~4‖∆u‖2H3 + C~2‖~∇ρ‖2H3 + Cτd2 0‖(ρ, θ,∇θ, ~2∆u)‖H3 + Cτd0‖(ρ, θ,∇θ)‖2H3 . (3.25) On the other hand, applying the operator div to (3.13), we have ∂t div u+ div u 1 + τρ − 4µ 3 ∆ div u 1 + τρ + ∆θ + 1 + τθ 1 + τρ ∆ρ− ~2 12 ∆2ρ 1 + τρ −∆ψ + τu · ∇ div u = ∇ ( 1 1 + τρ ) (µ∆u+ µ 3 ∇∆ div u)−∇ (1 + τθ 1 + τρ ) · ∇ρ + ~2 12 ∇ ( 1 1 + τρ ) · ∇∆ρ− τ∇u : ∇u− ~2τ 12 div (∇ρ∆ρ+∇ρ · ∇2ρ (1 + τρ)2 − |∇ρ| 2∇ρ (1 + τρ)3 ) + τ∇ρ · u+ τ div fR. (3.26) Applying the operator ∇× to (3.13), we have ∂t∇× u+ ∇× u 1 + τρ − µ 3 ∆∇× u 1 + τρ + τu · ∇∇ × u = ∇ ( 1 1 + τρ ) (µ∆u+ µ 3 ∇∆ div u)−∇ (1 + τθ 1 + τρ ) · ∇ρ + ~2 12 ∇ ( 1 1 + τρ ) · ∇∆ρ− τ(∇× udiv u−∇× u · ∇u) − ~2τ 12 ∇× (∇ρ∆ρ+∇ρ · ∇2ρ (1 + τρ)2 − |∇ρ| 2∇ρ (1 + τρ)3 ) + τ∇ρ · u+ τ∇× fR, (3.27) where we have used the vector analysis formula ∇× (f · ∇f) = ∇× (∇× f ⊗ f), (3.28) ∇× (f × g) = f div g − g div f + (g · ∇)f − (f · ∇)g, (3.29) for any vector functions f, g. EJDE-2020/103 TIME PERIODIC SOLUTIONS 23 Applying the operator ∂α−1 to systems (3.26) and (3.27), and multiplying the result by (1 + τρ)(∇ × uα−1,div uα−1), integrating over the periodic domain ΩR, we have 1 2 d dt ‖(1 + τρ)(∇× u,div u)‖2H2 + ‖(∇× u,div u)‖2H3 − τ 2 ∫ ΩR ∂tρ|div uα−1|2 + τ ∫ ΩR (1 + τρ)u · ∇ div uα−1 div uα−1 − τ 2 ∫ ΩR ∂tρ|∇ × uα−1|2 + τ ∫ ΩR (1 + τρ)u · ∇∇ × uα−1∇× uα−1 + ∫ ΩR (1 + τθ) div uα−1∆ρα−1 + ∫ ΩR (1 + τρ) div uα−1∆θα−1 − ∫ ΩR (1 + τρ)∆ψα−1 div uα−1 − ~2 12 ∫ ΩR ∆2ρα−1 div uα−1 = ∫ ΩR (1 + τρ)∂α−1g3(∇× uα−1,div uα−1) + ∫ ΩR (1 + τρ)g4(∇× uα−1,div uα−1) , R3,11 +R3,12, (3.30) where g3 =2τ∇ρ · u− τ∇× udiv u+ τ∇× u · ∇u− τ∇u : ∇u + 2∇ ( 1 1 + τρ ) (µ∆u+ µ 3 ∇∆ div u)− 2∇ (1 + τθ 1 + τρ ) · ∇ρ + ~2 6 ∇ ( 1 1 + τρ ) · ∇∆ρ+ τ∇× fR + τ div fR − ~2τ 12 ∇× (∇ρ∆ρ+∇ρ · ∇2ρ (1 + τρ)2 − (∇ρ · ∇ρ)∇ρ (1 + τρ)3 ) − ~2τ 12 div (∇ρ∆ρ+∇ρ · ∇2ρ (1 + τρ)2 − (∇ρ · ∇ρ)∇ρ (1 + τρ)3 ) , (3.31) and g4 = 4τ 3 [∂α−1, 1 1 + τρ ]∆ div u+ µ[∂α−1, 1 1 + τρ ]∆∇× u + ~2 12 [∂α−1, 1 1 + τρ ]∆2ρ− τ [∂α−1, u]∇ div u− τ [∂α−1, u]∇∇× u − [∂α−1, 1 + τθ 1 + τρ ]∆ρ− τ [∂α−1, ρ]∆θ − [∂α−1, 1 1 + τρ ] div u − [∂α−1, 1 1 + τρ ]∇× u. (3.32) Estimate for the left-hand side of (3.30). By integration by parts and (3.1a), we obtain τ ∫ ΩR (1 + τρ)u · ∇ div uα−1 div uα−1 − τ 2 ∫ ΩR ∂tρ|div uα−1|2 = −τ 2 ∫ ΩR ( ∂tρ+ div((1 + τρ)u) ) |div uα−1|2 24 M. LI EJDE-2020/103 = −τε 2 ∫ ΩR ∆ρ|div uα−1|2 ≥ −ε2‖∇ρ‖2H3 − Cτ‖∇u‖4H2 . Similarly, − τ 2 ∫ ΩR ∂tρ|∇ × uα−1|2 + τ ∫ ΩR (1 + τρ)u · ∇∇ × uα−1∇× uα−1 ≥ −ε2‖∇ρ‖2H3 − Cτ‖∇u‖4H2 . Integrating by parts, we have∫ ΩR (1 + τθ)∆ρα−1 div uα−1 + ∫ ΩR (1 + τρ) div uα−1∆θα−1 = −τ ∫ ΩR ∇θ · ∇ρα−1 div uα−1 − ∫ ΩR (1 + τθ)∇ρα−1 · ∇ div uα−1 − ∫ ΩR (1 + τρ)∇ div uα−1 · ∇θα−1 − τ ∫ ΩR ∇ρ · ∇θα−1 div uα−1 ≥ − ∫ ΩR (1 + τθ)ρα div uα − ∫ ΩR (1 + τρ) div uαθα − Cτd0‖(∇θ,∇ρ,∇u)‖2H2 . For the last but one term on the left-hand side of (3.30), by integration by parts, (3.1a), (3.1d), Lemmas 2.5 and 2.6, we arrive at − ∫ ΩR (1 + τρ)∆ψα−1 div uα−1 = − ∫ ΩR (1 + τρ)ρα−1 div uα−1 = ∫ ΩR ρα−1(∂tρα−1 − ε∆ρα−1 + τu · ∇ρα−1 + τ [∂α−1, ρ] div u+ τ [∂α−1, u]∇ρ) = 1 2 d dt ∫ ΩR |ρα−1|2 + ε ∫ ΩR |∇ρα−1|2 − τ 2 ∫ ΩR div u|ρα−1|2 + τ ∫ ΩR ρα−1[∂α−1, ρ] div u+ τ ∫ ΩR ρα−1[∂α−1, u] · ∇ρ ≥ 1 2 d dt ∫ ΩR |ρα−1|2 + ε ∫ ΩR |∇ρα−1|2 − Cτ‖(∇u,∇ρ)‖L∞‖(ρ,∇u)‖2H2 ≥ 1 2 d dt ∫ ΩR |ρα−1|2 + ε ∫ ΩR |∇ρα−1|2 − Cτd0‖(ρ,∇u)‖2H2 . For the last term on the left-hand side of (3.30), by (3.1a) and Lemma 2.5 again, we have − ~2 12 ∫ ΩR ∆2ρα−1 div uα−1 = ~2 12 ∫ ΩR ∇∆ρα−1 · ∇ div uα−1 = −~2 12 ∫ ΩR ∆ρα(∂tρα − ε∆ρα + τ∂α div(ρu)) = ~2 24 d dt ∫ ΩR |∇ρα|2 + ~2ε 12 ∫ ΩR |∆ρα|2 − ~2τ 12 ∫ ΩR ∆ρα div uρα EJDE-2020/103 TIME PERIODIC SOLUTIONS 25 − ~2τ 24 ∫ ΩR div u|∇ρα|2 − ~2τ 12 ∫ ΩR ∆ρα[∂α,div u]ρ + ~2τ 12 ∫ ΩR ∇ρα · [∂α∇, u] · ∇ρ ≥ ~2 24 d dt ∫ ΩR |∇ρα|2 + ~2ε 12 ∫ ΩR |∆ρα|2 − Cτ~2‖∆ρα‖L2(‖∇ div u‖L3‖ρ‖W 2,6 + ‖∇u‖H3‖ρ‖L∞)− Cτ~2‖∇ρα‖L2(‖∇u‖L∞‖∇ρ‖H3 + ‖∇u‖H3‖∇ρ‖L∞) − Cτ~2(‖∆ρ‖H3‖∇ρ‖H2 + ‖∇ρ‖2H3)‖∇u‖H3 ≥ ~2 24 d dt ∫ ΩR |∇ρα|2 + ~2ε 12 ∫ ΩR |∆ρα|2 − Cτd2 0‖∇u‖H3 . Estimates for the right-hand side of (3.30). For the first term R3,11, we consider here only the most difficult one, the other terms can be treated similarly. By the curl-div decomposition of the gradient, we have ‖∇u‖Hk ∼= ‖∇ × u‖Hk + ‖div u‖Hk , (3.33) where k = 0, 1, 2, 3. Invoking (3.10), (3.24), (3.33), Lemmas 2.5 and 2.6, we obtain, for α = 2, 3, − ~2τ 12 ∫ ΩR (1 + τρ)∂α−1 div (∇ρ∆ρ+∇ρ · ∇2ρ (1 + τρ)2 − (∇ρ · ∇ρ)∇ρ (1 + τρ)3 ) div uα−1 = ~2τ 12 ∫ ΩR (1 + τρ)∂α−2 div (∇ρ∆ρ+∇ρ · ∇2ρ (1 + τρ)2 − (∇ρ · ∇ρ)∇ρ (1 + τρ)3 ) div uα + ~2τ2 12 ∫ ΩR ∇ρ∂α−2 div (∇ρ∆ρ+∇ρ · ∇2ρ (1 + τρ)2 − (∇ρ · ∇ρ)∇ρ (1 + τρ)3 ) div uα−1 ≤ Cτ ( ‖∇ρ‖L∞‖~∆ρ‖H2 + ‖~∆ρ‖L∞(‖∇ρ‖H2 + ‖∇ρ‖L∞(1 + ‖∇ρ‖2H2)) + ‖∇ρ‖3H2(1 + ‖∇ρ‖2H2) ) ‖∇u‖H3 ≤ Cτ ( ‖∇ρ‖H2‖~∇ρ‖H3 + ‖∇ρ‖3H2‖~∇ρ‖H3 + ‖∇ρ‖3H2 + ‖∇ρ‖5H2 ) ‖∇u‖H3 ≤ Cτd2 0‖∇u‖H3 . For α = 1, we have −~2τ 12 ∫ ΩR div (∇ρ∆ρ+∇ρ · ∇2ρ (1 + τρ)2 − (∇ρ · ∇ρ)∇ρ (1 + τρ)3 ) div u ≤ Cτd2 0‖∇u‖L2 . Therefore, R3,11 ≤Cτ‖∇u‖H2‖∇u‖2H2 + Cτ‖∇u‖H2‖(∇θ,∇ρ)‖2H2 + Cτ‖∇u‖H2‖∇ ( 1 1 + τρ ) ‖H2(‖(∇u, ~2∆ρ)‖H3 + ‖∇ρ‖H2) + δ‖∇u‖2H3 + Cτ‖∇fR‖2H1 + Cτd2 0‖∇u‖H3 ≤δ‖∇u‖2H3 + Cτ‖∇fR‖2H1 + Cτd2 0(‖(∇θ,∇u, ~2∆ρ)‖H3 + ‖∇ρ‖H2). For the second term R3,12, with the aid of Hölder inequality, Lemmas 2.5 and 2.6, we have R3,12 ≤Cτ‖∇u‖H2(‖∇ 1 1 + τρ ‖L∞‖∇u‖H1 + ‖∇ 1 1 + τρ ‖H1‖∇u‖L∞) 26 M. LI EJDE-2020/103 + Cτ‖∇u‖H2(‖∇ 1 1 + τρ ‖L∞‖∇u‖H3 + ‖∇ 1 1 + τρ ‖H2‖∇3u‖L6) + Cτ‖∇u‖H2(‖∇ 1 1 + τρ ‖L∞‖~2∆ρ‖H3 + ‖∇ 1 1 + τρ ‖H2‖~2∆2ρ‖L6) + Cτ‖∇u‖H2‖(∇ρ,∇θ,∇u)‖L∞‖(∇ρ,∇θ,∇u)‖H2 + Cτ‖∇u‖H2‖(∇ρ,∇θ,∇u)‖H2‖(∇2u,∇2ρ,∇2θ)‖L6 ≤Cτd2 0‖∇u‖H3 . Putting these above estimates together, and recalling 1 ≤ |α| ≤ 3, we complete the proof of Lemma 3.5 by the curl-div decomposition of the gradient. � To close the uniform bound of solutions to system (3.1) with respect to ε, R, we must deal with the term ‖∆u‖H3 , which is caused by the quantum effect terms in the energy equation (3.21b). 3.3. Higher order estimates for (ρ, u, θ). Lemma 3.6. Under the assumptions in Lemma 3.3, we have d dt ‖(~∇u, ~∇ρ, ~2∆ρ)‖2H3 + ~2‖∇2u‖2H3 + ε~2‖∆ρ‖2H3 + ~4ε‖∇∆ρ‖2H3 ≤ Cτd2 0‖(ρ, u, θ,∇θ,∇u, ~∇ρ, ~2∆ρ)‖H3 + C~‖(ρ, ~div u)‖2H3 + C~2‖∇θ‖2H3 + δε‖∇ρ‖2H3 + Cτ‖~∇fR‖2H2 . (3.34) Proof. Applying the operator ∂α to (3.13), multiplying the result by −~2∆uα, we obtain ~2 2 d dt ‖∇uα‖2L2 + ~2Cµ‖∇uα‖2H1 = τ~2 ∫ ΩR u · ∇uα∆uα + ~2 ∫ ΩR 1 + τθ 1 + τρ ∇ρα ·∆uα − ~2 ∫ ΩR ∇ψα ·∆uα − ~4 12 ∫ ΩR ∇∆ρα ·∆uα 1 + τρ − ~2 ∫ ΩR g5∆uα + ~4τ 12 ∫ ΩR ∂α (∇ρ∆ρ+∇ρ · ∇2ρ (1 + τρ)2 − |∇ρ| 2∇ρ (1 + τρ)3 ) ∆uα + ~2 ∫ ΩR ∂α ( u 1 + τρ ) ∆uα − τ~2 ∫ ΩR ∂αfR∆uα − µ~2τ 3 ∫ ΩR ∇ρdiv uα(∆uα −∇ div uα) (1 + τρ)2 + ~2 ∫ ΩR ∇θα ·∆uα , 9∑ i=1 R4,i, (3.35) where g5 =− [∂α, u] · ∇u+ [∂α, 1 1 + τρ ](µ∆u+ µ 3 ∇div u) + [∂α, 1 + τθ 1 + τρ ]∇ρ − ~2 12 [∂α, 1 1 + τρ ]∇∆ρ. (3.36) EJDE-2020/103 TIME PERIODIC SOLUTIONS 27 Estimates for the right-hand side of (3.35). For the first three terms R4,1 ∼ R4,3, by a similar manner to the estimates (3.15), (3.16) and (3.17), we obtain R4,1 =− ~2τ ∫ ΩR ∂ku i∂iu j α∂ku j α − ~2τ ∫ ΩR ui∂iku j α∂ku j α =− ~2τ ∫ ΩR ∂ku i∂iu j α∂ku j α + ~2τ 2 ∫ ΩR div u|∇uα|2 ≤C‖∇u‖L∞‖~∇u‖2H3 , R4,2 ≤− ~2 2 d dt ∫ ΩR 1 + τθ (1 + τρ)2 |∇ρα|2 − ~2ε ∫ ΩR 1 + τθ (1 + τρ)2 |∆ρα|2 + δ‖(ε∇ρ, ~ε1/2∆ρ)‖2H3 + Cτd2 0‖(ρ, θ,∇u, ~∇ρ, ~∇u)‖H3 + C~4‖∆u‖2H3 , where δ is a sufficiently small positive constant, and R4,3 = ~2 ∫ ΩR ρα div uα ≤ C~‖(ρ, ~div u)‖2H3 . For the fourth term, by the equation (3.1a), we can infer that R4,4 = ~4 12 ∫ ΩR ∆ρα∆ div uα 1 + τρ + ~4 12 ∫ ΩR ∇ ( 1 1 + τρ ) ∆uα∆ρα =− ~4 12 ∫ ΩR ∆ρα(∂t∆ρα − ε∆∆ρα + τu · ∇∆ρα) (1 + τρ)2 − ~4τ 12 ∫ ΩR ∆ρα([∂α∆, u] · ∇ρ+ [∂α∆, ρ] div u) (1 + τρ)2 + ~4 12 ∫ ΩR ∇ ( 1 1 + τρ ) ∆uα∆ρα =− ~4 24 d dt ∫ ΩR |∆ρα|2 (1 + τρ)2 + ~4 24 ∫ ΩR ∂t ( 1 (1 + τρ)2 ) |∆ρα|2 − ~4ε 12 ∫ ΩR |∇∆ρα|2 (1 + τρ)2 + ~4ετ 6 ∫ ΩR ∇ρ · ∇∆ρα∆ρα (1 + τρ)3 + ~4τ 24 ∫ ΩR div ( u (1 + τρ)2 ) |∆ρα|2 − ~4τ 12 ∫ ΩR ∆ρα[∂α∆, u] · ∇ρ (1 + τρ)2 − ~4τ 12 ∫ ΩR ∆ρα[∂α∆, ρ] div u (1 + τρ)2 + ~4 12 ∫ ΩR ∇ ( 1 1 + τρ ) ∆uα∆ρα ≤− ~4 24 d dt ∫ ΩR |∆ρα|2 (1 + τρ)2 − ~4ε 12 ∫ ΩR |∇∆ρα|2 (1 + τρ)2 + δ‖(ε1/2∇ρ, ~2ε1/2∇∆ρ)‖2H3 + Cτd0‖~2∆ρ‖2H3 + Cτd2 0‖(∇u, ~∆u, ~2∆ρ)‖H3 + C~4‖∆u‖2H3 , thanks to Lemmas 2.5 and 2.6, Hölder’s and Young’s inequalities. Invoking the definition of g5, (3.24), Lemma 2.5 and Sobolev embedding, we arrive at R4,5 ≤C~2‖g5‖L2‖∆u‖H3 ≤C~4‖∆u‖2H3 + C‖g5‖2L2 ≤C~4‖∆u‖2H3 + C‖∇ ( 1 1 + τρ ) ‖2H2‖u‖2H3 + Cτ‖∇u‖2H2‖∇u‖2H3 28 M. LI EJDE-2020/103 + C‖∇ ( 1 1 + τρ ) ‖2H2‖∇u‖2H3 + C‖∇ (1 + τθ 1 + τρ ) ‖2H2‖∇ρ‖2H2 + C‖∇ ( 1 1 + τρ ) ‖2H2‖~2∆ρ‖2H3 ≤C~4‖∆u‖2H3 + Cτd2 0‖(ρ, u,∇u, ~2∆ρ)‖2H3 . For the sixth term R4,6, using lemma 2.5 and (3.24) again, we have R4,6 ≤Cτ‖~2∆u‖H3(‖∇ρ‖L∞‖~2∆ρ‖H3 + ‖~ ∇ρ (1 + τρ)2 ‖H3‖~2∆ρ‖L∞) ≤Cd2 0‖(~∇ρ, ~2∆ρ)‖H3 + C~4‖∆u‖2H2 . By Hölder’s and Young’s inequalities, the other terms on the right-hand side of (3.35) can be bounded by 9∑ i=7 R4,i ≤δ~2‖∆u‖2H3 + C~2‖∇θ‖2H3 + Cd2 0‖(ρ, u,∇u,∇θ, ~∇ρ, ~2∆ρ)‖H3 + Cτ‖~∇fR‖2H2 . Hence, choosing sufficiently small positive comstants δ, ~, we complete the proof. � 3.4. Estimates for ∇ρ. Lemma 3.7. Under the conditions in lemma 3.3, for any positive constant m < 1, we have m2‖(ρ,∇ρ, ~∆ρ)‖2 Ḣ2 +m2 d dt ∫ ΩR ∂2u∂2∇ρ ≤ Cm2(‖ε1/2∇ρ‖2H3 + ‖(∇θ,∇u)‖2H2) + Cτ‖∇fR‖2H1 + Cτ‖∇fR‖4H1 + Cτd0(‖~∇ρ‖2H3 + ‖(∇θ,∇u,∇ρ)‖2H2), (3.37) and m2‖(~ρ, ~∇ρ, ~2∆ρ)‖2 Ḣ3 +m2~2 d dt ∫ ΩR ∂3u∂3∇ρ ≤ Cm2‖(∇θ,∇u, ~∆u, ~ε1/2∆ρ)‖2H3 + Cτ‖∇fR‖2H1 + Cτd0‖(∇ρ,∇u, ~∇ρ, ~2∆ρ)‖2H3 . (3.38) Proof. Applying the operator ∂2 to (3.13), then taking inner product with ∂2∇ρ, we can obtain (3.37). The process is similar but much easier than the proof of the estimate (3.38). More precisely, applying the operator ∂3 to (3.13), multiplying the resultant equation by ~2∂3∇ρ, and using integration by parts, Sobolev embedding, Young’s inequality and Hölder’s inequality, we obtain ‖(~ρ, ~∇ρ, ~2∆ρ)‖2 Ḣ3 ≤ −~2 ∫ ΩR ∂3ut∂ 3∇ρ+ C‖(∇θ, ~∆u)‖2H3 + δ‖(~∇ρ, ~2∆ρ)‖2H3 + Cτd0‖(∇ρ,∇u, ~∇ρ, ~2∆ρ‖2H3 + Cτ‖∇fR‖2H1 , EJDE-2020/103 TIME PERIODIC SOLUTIONS 29 where δ is a sufficiently small positive constant. By integration by parts again, we have − ~2 ∫ ΩR ∂3ut∂ 3∇ρ = −~2 d dt ∫ ΩR ∂3u∂3∇ρ− ~2 ∫ ΩR ∂3 div u∂3ρt = −~2 d dt ∫ ΩR ∂3u∂3∇ρ+ ~2 ∫ ΩR ∂3 div u∂3 ( (1 + τρ) div u− ε∆ρ+ τu · ∇ρ ) ≤ −~2 d dt ∫ ΩR ∂3u∂3∇ρ+ C‖ div u‖2H3 + C‖~ε1/2∆ρ‖2H3 + Cτd0‖(∇u, ~∇ρ)‖2H3). Therefore, multiplying a suitably small positive constant m2, we derive (3.38). � 3.5. Proof of Theorem 3.1. In what follows, we will use the above uniform (in ε, R) estimates to show the condition (3.2) of the Leray-Schauder degree theory is satisfied. Then, we establish the existence of time periodic solutions to the approximated system (2.1) in the bounded domain ΩR by the topological degree theory. It is worth noting that the uniform (in ε, R) bounds of solutions to system (3.1) are needed especially for the proof of Theorem 1.1. Proof. Adding Lemmas 3.3-3.7 up, and integrating the resultant inequality from 0 to T ∗, we obtain, for some suitably small positive constant m, ∫ T∗ 0 ( ‖(ρ, u, θ)‖2H3 + ‖(∇θ,∇u, ~∆u)‖2H3 + ‖(~∇ρ, ~2∆ρ)‖2H3 ) dt + ε ∫ T∗ 0 ‖(ρ,∇ρ, ~∆ρ, ~2∇∆ρ)‖2H3dt ≤ Cτ ∫ T∗ 0 ‖(fR, ~∇fR)‖2H2dt + Cτd2 0 (∫ T∗ 0 ‖(ρ, θ, u,∇u,∇θ, ~∇ρ, ~∆u, ~2∆ρ)‖2H3dt )1/2 + Cτd0 ∫ T∗ 0 ‖(ρ, θ, u,∇u,∇θ, ~∇ρ, ~∆u, ~2∆ρ)‖2H2dt ≤ Cτ ∫ T∗ 0 ‖(fR, ~∇fR)‖2H2dt+ Cτd3 0, (3.39) thanks to the choice of some suitably small positive constants m, δ, ~ such that those terms without τ ahead disappear. It follows from the above inequality (3.39) that there exists some time t0 ∈ (0, T ∗) such that T ∗‖(ρ, u, θ)(t0)‖2H3 + T ∗‖(~∇ρ, ~∇u, ~2∆ρ)(t0)‖2H3 ≤ Cτ ∫ T∗ 0 ‖(fR, ~∇fR)‖2H2dt+ Cτd3 0. (3.40) 30 M. LI EJDE-2020/103 Combining the results (3.6), (3.12), (3.20) with (3.34), we derive d dt ( ‖(ρ, θ, u)‖2H3 + ‖(~∇ρ, ~∇u, ~2∆ρ)‖2H3 ) ≤ Cτ‖fR‖2H2 + Cτ~2‖∇fR‖2H2 + Cτd0 ( ‖(ρ, u, θ)‖2H3 + ‖(∇u,∇θ, ~∇ρ, ~∆u, ~2∆ρ)‖2H3 ) + Cτd2 0 ( ‖(ρ, θ, u)‖H3 + ‖(∇θ,∇u, ~∇ρ, ~∆u, ~2∆ρ)‖H3 ) . (3.41) Integrating (3.41) from t0 to t (t0 < t ≤ T ∗), then (3.40) guarantees the inequality ‖(ρ, θ, u)(t)‖2H3 + ‖(~∇ρ, ~∇u, ~2∆ρ)(t)‖2H3 ≤ ‖(ρ, θ, u)(t0)‖2H3 + ‖(~∇ρ, ~∇u, ~2∆ρ)(t0)‖2H3 + Cτ ∫ T∗ 0 ‖(fR, ~∇fR)‖2H2dt+ Cτd3 0 ≤ Cτ ∫ T∗ 0 ‖(fR, ~∇fR)‖2H2dt+ Cτd3 0. (3.42) For the reason that (ρ, u, θ) is periodic in T ∗, we have ‖(ρ, θ, u)(0)‖2H3 + ‖(~∇ρ, ~∇u, ~2∆ρ)(0)‖2H3 = ‖(ρ, θ, u)(T ∗)‖2H3 + ‖(~∇ρ, ~∇u, ~2∆ρ)(T ∗)‖2H3 ≤ Cτ ∫ T∗ 0 ‖(fR, ~∇fR)‖2H2dt+ Cτd3 0. (3.43) Therefore, sup t∈[0,T∗] ( ‖(ρ, θ, u)(t)‖2H3 + ‖(~∇ρ, ~∇u, ~2∆ρ)(t)‖2H3 ) ≤ Cτ ∫ T∗ 0 ‖(fR, ~∇fR)‖2H2dt+ Cτd3 0. (3.44) Combining (3.39) with (3.44), and recalling the definition (2.5), we obtain |||(ρ, u, θ)|||2 + ε ∫ T∗ 0 ‖(ρ,∇ρ, ~∆ρ, ~2∇∆ρ)‖2H3dt ≤ Cτ ∫ T∗ 0 ‖(fR, ~∇fR)‖2H2dt+ Cτd3 0 ≤ d2 0 2 , (3.45) in which we have used the assumption that d0 and ∫ T∗ 0 ‖(fR, ~∇fR)‖2H2dt are sufficiently small such that Cλ2 + Cd3 0 ≤ d2 0/2. This prove (3.2). Recalling χ((ρ̃, ũ, θ̃), 0) = 0 in the proof of Lemma 2.3, we have deg(I − χ(·, 1), B̂d0(0), 0) = deg(I − χ(·, 0), B̂d0(0), 0) = deg(I, B̂d0(0), 0) = 1, where we have used the normality and invariance of compact homotopy properties of Leray-Schauder degree deg(I − χ(·, τ), B̂d0(0), 0). Hence, system (2.1) admits a time periodic solution (ρ, u, θ) ∈ XR d0 . � EJDE-2020/103 TIME PERIODIC SOLUTIONS 31 4. Proof of the main results 4.1. Proof of Theorem 1.1. In the following, we will prove the existence of time periodic solution in R3 by a limiting process. Before starting the technical part of the proof, we list an available Lemma 4.1 which has been proved in [11]. Lemma 4.1. Let {fl}∞l=1 be a sequence of functions defined on a set Y . If there exists an exhausting sequence of subsets of Y satisfying ∪∞i=1Yi = Y , and for i > 1, {f il }∞l=1 is a uniformly convergent subsequence of {f i−1 l }∞l=1 in Yi in which we assume f0 l = fl(l = 1, 2, 3, · · · , ), then we can obtain a subsequence {fl′} of {fl} such that {fl′} converges uniformly in Y . In the following, we let Y = R3 and Yi = ΩR. Lemma 4.2. Assume g ∈ L∞(0, T ∗, H2), ∂tg ∈ L2(0, T ∗, L2), we can derive g in C 1 8 , 1 2 ( (0, T ∗) × ΩR ) , where g ∈ C 1 8 , 1 2 ( (0, T ∗) × ΩR ) means for any function g ∈ L∞ ( (0, T ∗)× ΩR ) , it holds sup (x,t)6=(y,s) |g(x, t)− g(y, s)| |x− y|1/2 + |t− s|1/8 ≤ C. Proof. By the condition g(t) ∈ H2 and Sobolev embedding, we have g(t) ∈ C1/2, for any t ∈ (0, T ∗). In the following, we assume 0 < t1 ≤ t2 < T ∗. Without loss of generality, we denote Br by a ball centered at x ∈ ΩR with radius r = |t2 − t1|a. Using the fact ∂tg ∈ L2(0, T ∗, L2), we obtain∫ Br |g(x, t1)− g(x, t2)|dx ≤ ∫ Br ∣∣ ∫ t2 t1 ∂tgdt ∣∣dx ≤ (∫ t2 t1 ∫ Br |∂tg|2dt )1/2(∫ t2 t1 ∫ Br 12dt )1/2 dx ≤ C|t1 − t2| 1 2 + an 2 , (4.1) which implies there exists a point x̃ ∈ Br, such that |g(x̃, t1)− g(x̃, t2)| ≤ C|t1 − t2| 1 2− an 2 . Hence, we deduce that for any x ∈ ΩR, |g(x, t1)− g(x, t2)| ≤|g(x, t1)− g(x̃, t1)|+ |g(x̃, t1)− g(x̃, t2)|+ |g(x̃, t2)− g(x, t2)| ≤2C|t1 − t2| a 2 + C|t1 − t2| 1 2− an 2 . Then we can choose a = 1 4 such that 1 2 − an 2 = a 2 . That is to say, |g(x, t1)− g(x, t2)| ≤ C|t1 − t2| 1 8 . Therefore, we prove that, for any x, y ∈ ΩR and t, s ∈ (0, T ∗), |g(x, t)− g(y, s)| ≤ C|t− s| 18 + C|x− y|1/2. This completes the proof of Lemma 4.2. � Proof of Theorem 1.1. To avoid any confusion, for any fixed R, we denote the solu- tions to the approximated system (2.1) by (ρR,ε, uR,ε, θR,ε). Using Lemma 4.2 and the result (3.45), we deduce ‖(ρR,ε, uR,ε, θR,ε)‖ C 1 8 , 1 2 ( (0,T∗)×ΩR ) ≤ Cd0. 32 M. LI EJDE-2020/103 Then there exists a subsequence {(ρR,εn , uR,εn , θR,εn )}∞n=1 such that (ρR,εn , uR,εn , θR,εn )→ (ρR, uR, θR) uniformly in ΩR as n→∞, (ρR,εn , uR,εn , θR,εn )→ (ρR, uR, θR) strongly in L2(0, T ∗;L2(ΩR), where (ρR, uR, θR) ∈ XR d0 . Letting ε→ 0 (n→∞), we can conclude that the limit function (ρR, uR, θR) ∈ XR d0 . This completes the proof of the time periodic solutions to system (1.2) in the periodic domain ΩR. In what follows, we choose a subsequence Rk such that Rk →∞ (k →∞). Let (ρkn, u k n, θ k n) be the convergent function sequence in ΩRk , and (ρk+1 n , uk+1 n , θk+1 n ) be convergent function sequence in ΩRk+1 , which is also the subsequence of (ρkn, u k n, θ k n). Repeating the process and using the diagonal argument, one can show that there exists a Cantor diagonal subsequence (ρnn, u n n, θ n n) such that (ρnn, u n n, θ n n)→ (ρ, u, θ) as n→∞ in the whole space R3 based on Lemma 4.1 and the bound (3.45) (uniform in R). Hence, we extend the time periodic classical solution (ρ, u, θ) to system (1.2) to the whole space R3. Then the proof is complete. � 4.2. Proof of Theorem 1.2. Assume that (ρ1, u1, θ1) and (ρ2, u2, θ2) are the pe- riodic solutions from Theorem 1.1. Let q = ρ1 − ρ2, v = u1 − u2, ϑ = θ1 − θ2, φ = ψ1 − ψ2. Then system (1.2) can be rewritten as ∂tq + div v = −q div u1 − ρ2 div v − v · ∇ρ1 − u2 · ∇q, (4.2a) ∂tv + v 1 + ρ2 +∇ϑ+ 1 + θ2 1 + ρ2 ∇q − ~2 12 ∇∆q ρ2 + 1 − µ 3 3∆v +∇ div v ρ2 + 1 −∇φ = −~2 12 q∇∆ρ1 (ρ1 + 1)(ρ2 + 1) − v · ∇u1 − u2 · ∇v − (1 + θ1 1 + ρ1 − 1 + θ2 1 + ρ2 ) ∇ρ1 − ~2 12 ( ∇ρ1 (1 + ρ1)2 − ∇ρ2 (1 + ρ2)2 ) (∆ρ1 +∇2ρ1) + ~2 12 ∇ρ2 (1 + ρ2)2 ∆q − µ 3 q(3∆v +∇ div v) (ρ1 + 1)(ρ2 + 1) − ~2 12 |∇ρ2|2 (1 + ρ2)3 ∇q + ~2 12 ( |∇ρ1|2 (1 + ρ1)3 − |∇ρ2|2 (1 + ρ2)3 ) ∇ρ1 + (ρ2 − ρ1)u1 (1 + ρ1)(1 + ρ2) , ∂tϑ+ ϑ+ 2 3 div v − 2κ 3 ∆ϑ ρ2 + 1 − ~2 18 div ∆v − ~2 36 ∆q 1 + ρ2 = −v · ∇θ1 − u2 · ∇ϑ− 2 3 ϑ div u1 − 2 3 θ2 div v + ~2 18 ∇ρ2 ρ2 + 1 ∇ div v + ~2 18 ( ∇ρ1 ρ1 + 1 − ∇ρ2 ρ2 + 1 ) ∇div u1 − 2κ 3 q∆θ1 (ρ1 + 1)(ρ2 + 1) + 2 3 (R̃(u1, ρ1)− R̃(u2, ρ2)) + ~2 36 (ρ2 − ρ1)∆ρ1 (1 + ρ1)(1 + ρ2) , ∆φ = q, (4.2b) EJDE-2020/103 TIME PERIODIC SOLUTIONS 33 where R̃(u, ρ) = µ 2 |∇u+∇u>|2 − 2µ 3 (div u)2 ρ+ 1 + ~2 24 |∇ρ|2 (1 + ρ)2 + 1 2 |u|2. 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Estimates for 3.5. Proof of Theorem ?? 4. Proof of the main results 4.1. Proof of Theorem ?? 4.2. Proof of Theorem ?? Acknowledgments References