Statistical solution and Liouville type theorem for the Klein-Gordon-Schrödinger equations

Statistical solution and Liouville type theorem for the Klein-Gordon-Schr¨odinger equations∗ Caidi Zhaoa† ,Tom´as Caraballob‡ ,Grzegorz Lukaszewiczc§ aDepartment of Mathematics, Wenzhou University, Wenzhou, Zhejiang Province, 325035, P. R. China bDepartmento de Ecuaciones Diferenciales y An´alisis Num´erico Facultad de Matem´aticas, Universidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla, Spain cInstitute of Applied Mathematics and Mechanics, University of Warsaw Banacha 2, 02-097 Warsaw, Poland November 23, 2020 Abstract In this article, the authors investigate the system of Schr¨odinger and KleinGordon equations with Yukawa coupling. They first prove the existence of pullback attractor and construct a family of invariant Borel probability measures. Then they establish that this family of probability measures satisfies a Liouville type theorem and is indeed a statistical solution for the coupling equations. Further, they reveal that the invariant property of the statistical solution is a particular situation of the Liouville type theorem. Keywords: Klein-Gordon-Sch¨odinger equations; Statistical solution; Pullback attractor; Invariant measure; Liouville type theorem MSC2010: 35B41, 34D35, 76F20 ∗Supported by NSF of China with No.11971356, 11271290 and by NSF of Zhejiang Province with No.LY17A010011, the Spanish Ministerio de Ciencia, Innovaci´on y Universidades (MCIU), Agencia Estatal de Investigaci´on (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-096540-B-I00, and National Science Center (NCN) of Poland under project No. DEC2017/25/B/ST1/00302. †Corresponding author E-mail: zhao[email protected] or zhao[email protected] ‡E-mail: [email protected] §E-mail: [email protected]. 1 1 Introduction In this article, we investigate the following non-autonomous weakly dissipative Klein-Gordon-Sch¨odinger (KGS for short) equations utt +νut−∆u+µu −β|z|2=g(x, t), x ∈Ω, t > τ, (1.1) izt+ ∆z+iαz +zu =f(x, t), x ∈Ω, t > τ, (1.2) with initial and boundary conditions (u(x, t), ut(x, t), z(x, t))t=τ= (uτ, u0τ, zτ), x ∈Ω,(1.3) u(x, t)∂Ω=z(x, t)∂Ω= 0,(1.4) where Ω is a bounded domain in R3with smooth boundary ∂Ω. Equations (1.1)-(1.2) describe the interaction of scalar nucleons with neutral mesons through Yukawa coupling (see [3]), where u=u(x, t) and z=z(x, t) denote a real meson field and a complex scalar nucleon field, respectively, the parameters α > 0, ν > 0 denote the dissipative mechanism of the system, µ > 0, β > 0 are constants representing the damping coefficients, and the real-valued function g(x, t) and complexvalued function f(x, t) are the time-dependent external forces. The autonomous KGS equations (1.1)-(1.2) and its related versions were extensively studied, one can see [1, 3, 17, 21, 23] and the references therein. For example, when Ω⊂R3is a bounded smooth domain, Biler in [3] established the existence of global attractor in the weak topologies of E=H1 0(Ω) ×L2(Ω) ×H1 0(Ω) (1.5) and E1= (H2(Ω) ∩H1 0(Ω)) ×H1 0(Ω) ×(H2(Ω) ∩H1 0(Ω)).(1.6) Later, these results were improved by [29]. Also, Lange and Wang in [19] proved the regularity of the global attractor when Ω ⊂Ris a bounded interval. The Cauchy problem associated to equations (1.1)-(1.2) were investigated in [21, 23]. For instance, Li and Guo in [21] used a Strichartz type inequality and some suitable decomposition to prove the asymptotic smoothing effect for the solutions. However, to the best of our knowledge, there are only some references concerning the asymptotic behavior of solutions for the non-autonomous KGS equations (1.1)-(1.2). The motivation of the current article is to investigate the statistical solutions for the non-autonomous KGS equations (1.1)-(1.2). We are interested in the probability distribution of solutions within the phase space E. In turbulent flow regimes, the physical properties are universally recognized as randomly varying and characterized by some suitable probability distribution functions. In the theory of fluid mechanics, 2 the invariant measures and statistical solutions have proven to be very useful in the understanding of turbulence (see Foias et al. [11]). The main reason is that the measurements of several important aspects (such as mass and velocity) of turbulent flows are actually measurements of time-average quantities. Statistical solutions have been introduced as a rigorous mathematical notion to formalize the object of ensemble average in the conventional statistical theory of turbulence. Nowadays, invariant measures and statistical solutions are widely used to describe certain characteristics of the fluids in the real world. There are two prevalent notions of statistical solutions. The one is the so-called Foias-Prodi statistical solutions introduced by Foias and Prodi in [10] and the other is the so-called Vishik-Fursikov statistical solutions given by Vishik and Fursikov in [28]. The Foias-Prodi statistical solutions are a family of Borel measures parametrized by the time variable and defined on the phase space of the Navier-Stokes equations, representing the probability distribution of the velocity field of the flow at each time. The Vishik-Fursikov statistical solutions are a single Borel measure on the space of trajectories, representing the probability distribution of the space-time velocity field. The invariant measures for well-posed dissipative systems were studied in a series of references (see [9,22,24–27,30]). For instance, Lukaszewicz, Real and Robinson [25] used the notion of Generalized Banach limit to construct the invariant measures for general continuous dynamical systems on metric spaces. Later, Chekroun and Glatt-Holtz [9] improved the results of [25] to construct invariant measures for a broad class of dissipative autonomous dynamical systems. Recently, Lukaszewicz and Robinson [26] extended the result of [9] to construct invariant measures for dissipative non-autonomous dynamical systems. The result of [26] was used to investigate the invariant measure for the three-dimensional (3D for short) globally modified Navier-Stokes equations and regularized MHD equations in [31,37]. There also are some references investigating the statistical solutions and trajectory statistical solutions for some model evolution equations that possess global weak solutions but without a known result of global uniqueness. For instance, Foias, Rosa and Temam studied systematically the statistical solutions for the 3D Navier-Stokes equations in [12–16]. Bronzi, Mondaini and Rosa in [4, 6] proved an abstract framework for the theory of statistical solutions and trajectory statistical solutions for general evolution equations, including those with properties similar to the 3D Navier-Stokes equations. Bronzi and Rosa studied the convergence of statistical solutions of the 3D Navier-Stokes-αmodel as αvanishes in [5]. Caraballo, Kloeden and Real investigated the invariant measure and statistical solution for the 3D globally modified Navier-Stokes equations in [7]. Kloeden, Rubio and Real studied the equivalence of invariant measure and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations in [18]. Zhao and Caraballo in [32] constructed the trajectory statistical solutions for the 3D globally modified Navier-Stokes equations. Zhao, Song and Caraballo 3 in [34] constructed the strong trajectory statistical solutions for the 2D dissipative Euler equations. Zhao, Li and Caraballo in [33] proved sufficient conditions ensuring the existence of trajectory statistical solutions for autonomous evolution equations. In addition, Zhao, Li and Song in [35] constructed the trajectory statistical solutions for the 3D Navier-Stokes equations via the trajectory attractor approach. Also, Zhao, Jiang and Caraballo constructed in [36] the trajectory statistical solutions for the nonlinear wave equations with polynomial growth The main result of the current article is to prove the existence of the statistical solution for the non-autonomous KGS equations (1.1)-(1.2). This statistical solution describes the probability distribution of the meson field and nucleon field in the phase space. We will first use the abstract theory for dissipative non-autonomous system in [26, Theorem 3.1] to obtain the existence of a family of invariant Borel probability measures {mt}t∈R. Then we establish that {mt}t∈Rsatisfies a Liouville type theorem and is indeed a statistical solution for equations (1.1)-(1.2). Finally, we reveal that the invariant property of the statistical solution is a particular situation of the Liouville type theorem. To apply the abstract theory of [26, Theorem 3.1] to obtain the existence of a family of invariant Borel probability measures, we shall prove that the solution operators associated to problem (1.1)-(1.4) generate a continuous process {U(t, τ)}t⩾τin the phase space Eand (1) the process {U(t, τ)}t⩾τis pullback strongly bounded in E; (2) the process {U(t, τ)}t⩾τis pullback asymptotically compact in E; (3) for each given t∈Rand given ψ∗∈E, the E-valued function τ7→ U(t, τ)ψ∗is continuous and bounded on (−∞, t]. By definition, a continuous process {U(t, τ)}t⩾τin the phase space Emeans {U(t, τ)}t⩾τ is a two-parameter family of mappings in Esatisfying: (a) U(t, s)U(s, τ) = U(t, τ), ∀t⩾s⩾τ, τ ∈R; (b) U(τ, τ) = Id (identity operator), τ∈R; (c) For given tand τwith t⩾τ, the mapping U(t, τ) is continuous from Eto E. It is not a standard fact to prove above assertions (1)-(3) for the process {U(t, τ)}t⩾τ. Firstly, on the one hand, it is not a direct generalization of the dynamics from autonomous system to non-autonomous system. On the other hand, in [3], the authors established the existence of global attractor with the conditions f, g ∈ Cb(R+;L2(Ω)) or f, g ∈L2(Ω), which implies that fand gare uniformly bounded in L2(Ω) with respect to time t. In this article, the conditions imposed on fand gare weaker than those in [3]. In fact, we allow that the external forces are unbounded and actually 4 even exponentially growing time-dependent functions. At the same time, the nonlinear terms |z|2and zu produce some difficulties when we estimate the solutions and prove the pullback strongly boundedness of {U(t, τ)}t⩾τin E. Secondly, it is not easy to prove directly the pullback asymptotically compactness of {U(t, τ)}t⩾τin Ebecause of the special coupling of a hyperbolic equation with a parabolic one. Here we will employ some delicate decomposition of the process {U(t, τ)}t⩾τ. Precisely, we decompose the addressed system into two equations, and simultaneously split the nonlinear term |z|2into Re(zz1) and Re(zz2) (here z=z1+z2). By this way we can prove that the solutions of the first decomposed equations pullback decay exponentially, whereas the solutions of the second decomposed equations are pullback bounded in E1. Note that the embedding E ,→E1is compact. We then obtain the pullback asymptotically compactness of {U(t, τ)}t⩾τin Eby the abstract theory of [8, Theorem 3.2]. Thirdly, to prove assertion (3) the key step is to establish the continuous dependence of the solutions on the initial data in E. This continuous dependence has been proved by Wang and Lange in [29, Theorem 3.4] via the method of energy equation. Here we will present a simple and direct proof to this continuous dependence. The main technique we used is to construct a suitable space Eµwhich is equivalent to the usual phase space E. Then we establish that the abstract operator corresponding to the linear part of equations (1.1)-(1.2) is coercive on Eµ. This coerciveness allows us to prove directly the continuous dependence of the solutions on the initial data in the norm of Eµ, which is equivalent to the continuous dependence of the solutions on the initial data in the norm of E. To establish that {mt}t∈Ris a statistical solution of the KGS equations, the important step is to prove that {mt}t∈Rsatisfies a Liouville type theorem similar to that from Statistical Mechanics. Fortunately, the form of the construction of the Borel probability measures {mt}t∈Rplays essential role in our proof. We also want to point out an interesting relation between the invariant property and the Liouville type theorem for the statistical solution. We all know that Liouville theorem from Statistical Mechanics indicates that the distribution of a set in the phase space could change with the evolution of time, but its Liouville measures is conserved. We will discover in this article that the invariant property of the statistical solution describes exactly that the shape of the pullback attractor ADδ(τ) could change with the evolution of time from τto t, but the measure of ADδ(τ) and ADδ(t) coincides with each other. The rest of the article is arranged as follows. In the next section, we estimate the solutions and then show the global well-posedness of problem (1.1)-(1.4). In Section 3, we establish the existence of the pullback attractor for the process {U(t, τ)}t⩾τassociated to problem (1.1)-(1.4). In Section 4, we first construct a family of invariant Borel probability measures for the process {U(t, τ)}t⩾τ. Then we establish that this family of probability measures satisfies a Liouville type theorem and is indeed a statistical 5 solution for the KGS equations. Further, we reveal that the invariant property of the statistical solution is a particular situation of the Liouville type theorem. 2 Estimates and global well-posedness of solutions We first introduce some notations. Let Lp(Ω), H1 0(Ω) and Wm,p(Ω) denote the usual Lebesgue and Sobolev spaces with norms k ·kLp(Ω),k · kH1 0(Ω) and k· kWm,p(Ω), respectively. Especially, Hm(Ω) = Wm,2(Ω) and k · kL2(Ω) =k·k. Throughout this article, we will use the function spaces E(see (1.5)) and E1(see (1.6)) and the norms are defined respectively as kψkE= (k∇uk2+kvk2+k∇zk2)1/2,for ψ= (u, v, z)T∈E, (2.1) kψkE1= (k∆uk2+k∇vk2+k∆zk2)1/2,for ψ= (u, v, z)T∈E1.(2.2) In addition, we will employ the notation a.b(also a&b) to mean that a⩽cb (also a⩾cb) for a universal constant c > 0 that only depends on the parameters coming from the problem. Put v=v(t) = ut+δu, (2.3) where δ > 0 is some constant that will be specified later. Then problem (1.1)-(1.4) is equivalent to the following problem ut+δu −v= 0, t > τ, (2.4) vt−∆u−δ(ν−δ)u+µu + (ν−δ)v=β|z|2+g(x, t), t > τ, (2.5) zt−i∆z+αz =izu −if(x, t), t > τ, (2.6) (u(x, t), v(x, t), z(x, t))t=τ= (uτ, vτ, zτ), x ∈Ω,(2.7) (u(x, t), v(x, t), z(x, t))∂Ω= (0,0,0),(2.8) hereinafter vτ=u0τ+δuτand δis the constant from (2.3). Denote ψ=ψ(x, t)=(u(x, t), v(x, t), z(x, t))T and Θ =   δI −I0 −∆−δ(ν−δ)I+µI (ν−δ)I0 0 0 −i∆ + αI  ,(2.9) F(ψ, t) = (0, β|z|2+g(x, t), izu −if(x, t))T,(2.10) where Iin the matrix Θ is the identity operator. Then problem (2.4)-(2.8) can be written as dψ dt+ Θψ=F(ψ, t), t > τ, (2.11) ψ(τ) = ψτ= (uτ, vτ, zτ)T.(2.12) 6 We next estimate the solutions of problem (2.11)-(2.12). Lemma 2.1. Let f(x, t),ft(x, t),g(x, t)belong to L2 loc(R;L2(Ω)). Then for any ψτ= (uτ, vτ, zτ)T∈E, every solution ψ(x, t)=(u(x, t), v(x, t), z(x, t))Tof problem (2.11)- (2.12) corresponding to ψτsatisfies kz(t)k2.kzτk2e−α(t−τ)+e−αt Zt τ eαskf(s)k2ds, ∀t⩾τ, (2.13) kψ(t)k2 E.Υ(τ)e−δ(t−τ)+e−δt Zt τ eδskG(s)k2ds +e−δt Zt τ eδskz(s)k6ds+kz(t)k6+kf(t)k2,∀t⩾τ, (2.14) where Υ(τ) = k∇zτk2+1 2k∇uτk2+1 2kvτk2+ 2Re ZΩ zτf(τ)dx−ZΩ|zτ|2uτdx+µ 2kuτk2, (2.15) kG(s)k2=kf(s)k2+k∂f(x, s) ∂t k2+kg(s)k2.(2.16) Proof. Let the assumption of this lemma hold. Then the existence and uniqueness of the solution ψ(x, t) corresponding to the initial data ψτ= (uτ, vτ, zτ)T∈Ecan be proved in a standard way, using the Galerkin approximations as in [3]. The estimate (2.13) is easily obtained by taking the scalar product of (2.6) with z(t) and the real part of the resulting equality. These details are omitted and here we prove (2.14). In fact, in [3, (2.10)], it is proved that dΥ(t) dt+δΥ(t) + (2α−3δ)k∇z(t)k2+ (ν−3δ 2)kv(t)k2+δ 2k∇u(t)k2+δµ 2ku(t)k2 =2αZΩ|z(t)|2u(t)dx−2(α−δ)Re ZΩ z(t)f(x, t)dx+δ(ν−δ)ZΩ u(t)v(t)dx +ZΩ v(t)g(x, t)dx+ 2Re ZΩ z(t)∂f(x, t) ∂t dx, ∀t⩾τ, (2.17) where Υ(t) = k∇z(t)k2+1 2k∇u(t)k2+1 2kv(t)k2+2Re ZΩ z(t)f(x, t)dx−ZΩ|z(t)|2u(t)dx+µ 2ku(t)k2. We next estimate the terms on the right-hand side of (2.17). By H¨older’s inequality, Gagliardo-Nirenberg’s inequality and the embedding H1 0(Ω) ,→L6(Ω), we obtain ZΩ|z(t)|2u(t)dx.kz(t)k2 L12/5(Ω)ku(t)kL6(Ω) .kz(t)k3/2k∇z(t)k1/2k∇u(t)k, and thus 2αZΩ|z(t)|2u(t)dx.α 2k∇z(t)k2+δ 4k∇u(t)k2+kz(t)k6.(2.18) 7 The other terms on the right-hand side of (2.17) are simpler to deal with:                        2(α−δ)Re ZΩ z(t)f(x, t)dx⩽(α−δ)k∇z(t)k2+4(α−δ) λ2 1kf(t)k2, 2Re ZΩ z(t)∂f(x, t) ∂t dx⩽α 2k∇z(t)k2+2 αλ2 1k∂f(x, t) ∂t k2, ZΩ v(t)g(x, t)dx⩽ν 4kv(t)k2+1 νkg(t)k2, δ(ν−δ)ZΩ u(t)v(t)dx⩽ν−δ 2kv(t)k2+δ2(ν−δ) 2ku(t)k2, (2.19) where we have used the following Poincar´e inequality kuk2⩽λ−1 1k∇uk2,∀u∈H1 0(Ω). Inserting (2.18)-(2.19) into (2.17) yields dΥ(t) dt+δΥ(t)+(α 2−2δ)k∇z(t)k2+ (ν 4−δ)kv(t)k2+δ 4k∇u(t)k2+δ 2(µ−δ(ν−δ))ku(t)k2 .kz(t)k6+kG(t)k2,∀t⩾τ, (2.20) where G(·) is defined by (2.16). We now choose δsuch that 0< δ < min{α 4,ν 4,µ ν}. Then (2.20) implies dΥ(t) dt+δΥ(t).kz(t)k6+kG(t)k2,∀t⩾τ, and applying Gronwall’s inequality we deduce Υ(t).Υ(τ)e−δ(t−τ)+e−δt Zt τ eδskG(s)k2ds+e−δt Zt τ eδskz(s)k6ds, ∀t⩾τ. (2.21) Combining (2.21), the estimates similar to (2.18) and the first inequality in (2.19), we obtain (2.14). The proof of Lemma 2.1 is completed. According to the estimates in Lemma 2.1, we next analyze under what assumptions on the data there exists a pullback absorbing set for the process associated to problem (2.11)-(2.12). From (2.15) we see that Υ(τ).kψτk2 E+kzτk6+kf(τ)k2.(2.22) From (2.14) and (2.22) it follows that the following assumptions lim τ→−∞ kψτk2 Eeδτ 3= 0,(2.23) lim τ→−∞ kf(τ)k2eδτ = 0,(2.24) Zt −∞ eδskG(s)k2ds < +∞,for each t∈R,(2.25) 8 are needed. We next analyze the third term on the right-hand side of (2.14). In fact, it follows from (2.13) that e−δt Zt τ eδskz(s)k6ds.ρ1(t, τ) + ρ2(t, τ) + ρ3(t, τ) + ρ4(t, τ), where                        ρ1(t, τ) = e−δt Zt τ eδskzτk6e−3α(s−τ)ds, ρ2(t, τ) = e−δt Zt τ eδskzτk4e−2α(s−τ)e−αs Zs τ eαθkf(θ)k2dθds, ρ3(t, τ) = e−δt Zt τ eδskzτk2e−α(s−τ)e−αs Zs τ eαθkf(θ)k2dθ2ds, ρ4(t, τ) = e−δt Zt τ eδse−αs Zs τ eαθkf(θ)k2dθ3ds. By (2.23), lim τ→−∞ kzτk6eδτ .lim τ→−∞(kψτk2eδτ 3)3= 0, and thus when τ→ −∞, ρ1(t, τ)=(kzτk2eδτ 3)3e−δt Zt τ eδse−3α(s−τ)e−δτ ds.(kzτk2eδτ 3)3e−δt −→ 0.(2.26) We write ρ2(t, τ) in the form (kzτk2eδτ 3)2e−2δτ 3e−δt Zt τ eδse−2α(s−τ)e−αs Zs τ eαθkf(θ)k2dθds. (2.27) Thus, ρ2(t, τ)−→ 0 as τ→ −∞ if the integral in (2.27) stays bounded as τ→ −∞. For this purpose, we assume e(2δ 3−2α)sZs −∞ eαθkf(θ)k2dθ⩽K(s),(2.28) where K(s) is a continuous function on the real line which is bounded on every interval of the form (−∞, t). If (2.28) holds, then ρ2(t, τ).(kzτk2eδτ 3)2e(2α−2δ 3)τe−δt Zt τ e−(3α−δ)se(2α−2δ 3)sK(s)ds .(kzτk2eδτ 3)2e(α−δ 3)τe K(t)−→ 0, τ → −∞,(2.29) where e K(t) is a bounded quantity depending only on tand the function K(·). Also, if (2.28) holds, then for K1(s) = K(s)e(α−δ 3)swe have e(2δ 3−2α)sZs −∞ eαθkf(θ)k2dθ2.K2 1(s), and thus e(δ 3−α)sZs −∞ eαθkf(θ)k2dθ.K1(s). 9 Lemma 3.4. Let assumption (H) hold. Then for any given b D={D(s)|s∈R} ∈ Dδand ψτ= (uτ, vτ, zτ)T∈D(τ), there exists a time τ2=τ2(t, b D)such that the solution ψ2(x, t) = u2(x, t), v2(x, t), z2(x, t)Tof problem (3.7)-(3.11) corresponding to ψτsatisfies k∆u2(x, t)k2+k∇v2(x, t)k2+k∆z2(x, t)k2 .1 + kf(t)k2+kg(t)k2+Zt −∞ e−δ(t−s)kf(s)k2+k∂f(s) ∂t k2+ +k∂g(s) ∂t k2ds +Zt −∞ e−δ(t−s)Zs −∞ e−α(s−θ)kf(θ)k2+k∂f(θ) ∂t k2dθds, ∀τ⩽τ2.(3.29) Proof. Let b D={D(s)|s∈R}∈Dδand ψτ= (uτ, vτ, zτ)T∈D(τ) be given. We first estimate k∆z2(t)k2. Lemma 3.3 shows that the solution ψ2(x, t)=(u2(x, t), v2(x, t), z2(x, t))T of problem (3.7)-(3.11) corresponding to ψτsatisfies kψ2(x, t)k2 E=k∇u2(x, t)k2+kv2(x, t)k2+k∇z2(x, t)k2.1,∀τ⩽τ0.(3.30) Note that z2(x, τ) = 0 for x∈Ω due to (3.10). Thus ∆z2(x, τ) = 0. We now differentiate equation (3.9) with respect to time tand find that z2tis a solution of the following problem iz2tt + ∆z2t+iαz2t+z2tu+utz2=∂f(x, t) ∂t , t > τ, (3.31) z2t(x, τ) = −if(x, τ).(3.32) Multiplying (3.31) by 2z2t, integrating over Ω and then taking the imaginary part of the resulting equality, we obtain d dtkz2tk2+ 2αkz2tk2=−2Im ZΩ utz2z2tdx+ 2Im ZΩ ∂f(x, t) ∂t z2tdx .kutkkz2kL∞(Ω)kz2tk+α 3kz2tk2+k∂f(x, t) ∂t k2.(3.33) From (3.9) we see k∆z2k.kz2tk+kz2k+kukL4(Ω)kz2kL4(Ω) +kf(t)k.(3.34) Using Gagliardo-Nirenberg’s inequality, (3.30) and (3.34), we have kz2kL∞(Ω) .kz2k1 4k∆z2k3 4.kz2tk+1+kf(t)k3 4 .kz2tk3 4+ (1 + kf(t)k)3 4,∀τ⩽τ0.(3.35) 16 It then follows from (3.33) and (3.35) that d dtkz2tk2+ 2αkz2tk2.kz2tk7 4+ (1 + kf(t)k)3 4kz2tk+α 3kz2tk2+k∂f(x, t) ∂t k2 .αkz2tk2+ (1 + kf(t)k)3 2+k∂f(x, t) ∂t k2. Thus d dtkz2t(t)k2+αkz2tk2.1 + kf(t)k2+k∂f(x, t) ∂t k2,∀τ⩽τ0.(3.36) Using again Gronwall’s inequality to (3.36) and then using (3.32) yield kz2t(t)k2.kz2t(τ)k2e−α(t−τ)+Zt τ e−α(t−s)1 + kf(s)k2+k∂f(x, s) ∂t k2ds .1 + kf(τ)k2e−α(t−τ)+Zt τ e−α(t−s)kf(s)k2+k∂f(x, s) ∂t k2ds, ∀τ⩽τ0. Combining above estimate and (3.34) we see that there is a τ1=τ1(t, b D) such that k∆z2(t)k2.1 + kf(t)k2+Zt τ e−α(t−s)kf(s)k2+k∂f(x, s) ∂t k2ds, ∀τ⩽τ1,(3.37) since, by (2.33), lim τ→−∞ kf(τ)k2eδτ = 0. Secondly, we estimate k∆u2(t)k2+k∇v2(t)k2. Multiplying (3.8) with −∆v2and then integrating the resulting equality over Ω yield 1 2 d dtk∇v2k2+ (ν−δ)k∇v2k2+δ(ν−δ)(u2,∆v2)−µ(u2,∆v2) + (∆u2,∆v2) = ZΩ−∆v2Re(zz2)dx−(g, ∆v2).(3.38) Direct computations give            (u2,∆v2) = −1 2 d dtk∇u2k2−δk∇u2k2, (∆u2,∆v2) = 1 2 d dtk∆u2k2+δk∆u2k2, (g, −∆v2) = −δZΩ ∆u2gdx−d dtZΩ ∆u2gdx+ZΩ ∂g(x, t) ∂t ∆u2dx. (3.39) Inserting (3.39) into (3.38) yields 1 2 d dtk∇v2k2+k∆u2k2+ (µ−δ(ν−δ)))k∇u2k2+ 2 ZΩ ∆u2gdx + (ν−δ)k∇v2k2+δk∆u2k2+δ(µ−δ(ν−δ))k∇u2k2+δZΩ ∆u2gdx =ZΩ∇v2∇Re(zz2)dx+ZΩ ∂g(x, t) ∂t ∆u2dx. (3.40) 17 Set H3(t) =k∇v2k2+ (µ−δ(ν−δ))k∇u2k2+k∆u2k2+ZΩ ∆u2gdx, J3(t) =2 ZΩ∇v2∇Re(zz2)dx+ 2 ZΩ ∂g(x, t) ∂t ∆u2dx −(2ν−3δ)k∇v2k2−δk∆u2k2−δ(µ−δ(ν−δ))k∇u2k2. Then from (3.40) it follows that dH3(t) dt+δH3(t) = J3(t).(3.41) By H¨older’s inequality, Sobolev’s embedding theorem and (3.28), 2ZΩ∇v2∇(Re(zz2))dx.k∇v2kk∇zkkz2kL∞(Ω) +kzkL6(Ω)k∇z2kL3(Ω) .k∇v2kk∆z2k.(2ν−3δ)k∇v2k2+k∆z2k2,∀τ⩽τ0. Thus by (3.30), (3.37) and Cauchy’s inequality, J3(t).(2ν−3δ)k∇v2k2+k∆z2k2+ 2 ZΩ ∂g(x, t) ∂t ∆u2dx −(2ν−3δ)k∇v2k2−δk∆u2k2−µ−δ(ν−δ)k∇u2k2 .1 + k∂g(x, t) ∂t k2+k∆z2k2 .1 + kf(t)k2+k∂g(x, t) ∂t k2 +Zt τ e−α(t−s)kf(s)k2+k∂f(x, s) ∂t k2ds, ∀τ⩽τ2,(3.42) where τ2= min{τ0(t, b D), τ1(t, b D)}. Note that for x∈Ω we have (u2(x, τ), v2(x, τ)) ≡ (0,0) due to (3.10). Thus ∆u2(x, τ) = 0 and k∇u2(x, τ)k=k∆u2(x, τ)k=k∇v2(x, τ)k= 0. These gives the fact that H3(τ) =k∇v2(x, τ)k2+ (µ−δ(ν−δ))k∇u2(x, τ)k2+k∆u2(x, τ)k2+ZΩ ∆u2(x, τ)g(x, τ)dx =0. Applying Gronwall’s inequality to (3.41), then using (3.42) and the fact that H3(τ) = 0, 18 we obtain H3(t).H3(τ)e−δ(t−τ)+Zt τ e−δ(t−s)J3(s)ds .1 + Zt τ e−δ(t−s)kf(s)k2+k∂g(s) ∂t k2ds +Zt τ e−δ(t−s)Zs τ e−α(s−θ)(kf(θ)k2+k∂f(θ) ∂t k2)dθds .1 + Zt −∞ e−δ(t−s)kf(s)k2+k∂g(s) ∂t k2ds +Zt −∞ e−δ(t−s)Zs −∞ e−α(s−θ)(kf(θ)k2+k∂f(θ) ∂t k2)dθds, ∀τ⩽τ2.(3.43) From assumption (H) we see that the right-hand side of inequality (3.43) is a bounded quantity which is independent of τ. Hence k∆u2k2+k∇v2k2.H3(t) + k∇u2(t)k2+ZΩ ∆u2gdx.H3(t) + 1 + 1 2k∆u2k2+kgk2, that is k∆u2k2+k∇v2k2.H3(t) + 1 + kg(t)k2,∀τ⩽τ2.(3.44) We obtain (3.29) from (3.37) and (3.43)-(3.44). The proof of Lemma 3.4 is completed. From Lemma 3.2 and Lemma 3.4 we see that the solutions operators of problem (3.2)-(3.6) and problem (3.7)-(3.11) generate in the space Econtinuous processes, {S(t, τ)}t⩾τand {T(t, τ)}t⩾τ, respectively, that is the process {U(t, τ)}t⩾τcan be decomposed as U(t, τ) = S(t, τ) + T(t, τ). Moreover, Lemma 3.2 shows that {S(t, τ)}t⩾τpullback decays exponentially in the manner kS(t, τ)ψτk2 E.kψτk2 E(1 + kψτk 3 4 E)e−δ(t−τ),∀τ⩽τ0. Lemma 3.4 indicates that {T(t, τ)}t⩾τis pullback strongly bounded in E1. Notice that the embedding E1,→Eis compact. Thus {T(t, τ)}t⩾τis pullback-Dδcompact in Ein the sense that for each given t∈R, each b D={D(s)|s∈R} ∈ Dδand any sequence {τn}n⩾1in (−∞, t] such that τn→ −∞ as n→ ∞, the sequence {T(t, τn)ψ(τn)}with any ψ(τn)∈D(τn) possesses a convergent subsequence in E. Combining these analyses and [8, Theorem 3.2], we obtain the following result. Lemma 3.5. Let assumption (H) hold. Then the process {U(t, τ)}t⩾τis pullback-Dδ asymptotically compact in E. 19 At this stage, we use the results of Lemma 2.2, Lemma 3.1, Lemma 3.5 and [8, Theorem 3.1] to obtain the main results of this section. Theorem 3.1. Let assumption (H) hold. Then the process {U(t, τ)}t⩾τpossesses a pullback-Dδattractor b ADδ={ADδ(t)|t∈R}in Esatisfying (a) Compactness: for every t∈R,ADδ(t)is a nonempty compact subset of E; (b) Invariance: U(t, τ)ADδ(τ) = ADδ(t),∀τ⩽t; (c) Pullback attraction: ADδ(t)is pullback-Dδattracting in the following sense, lim τ→−∞ DistEU(t, τ)D(τ),ADδ(t)= 0,∀b D={D(s)|s∈R}∈Dδ, t ∈R. 4 Invariant measure, Liouville type theorem and statistical solution The goal of this section is to construct the statistical solution for the KGS equations. To this end, we will first prove the existence of the family of invariant Borel probability measures for the process {U(t, τ)}t⩾τ. Then we establish that this family of probability measures satisfies a Liouville type theorem and is indeed a statistical solution for the KGS equations. We first prove that the solutions of problem (2.11)-(2.12) depend continuously on the initial data. For any u, v ∈H1 0(Ω), define (u, v)µ=µ(u, v)+(∇u, ∇v), where µis the positive constant from equation (1.1) and (·,·) is the inner product of L2(Ω). Obviously, we have k∇uk2⩽kuk2 µ.(1 + µ)k∇uk2,∀u∈H1 0(Ω), which means that (·,·)µis an inner product in H1 0(Ω) and Hµ= (H1 0(Ω),(·,·)µ) is a Hilbert space equivalent to H1 0(Ω) with the usual inner product. Set Eµ=Hµ×L2(Ω) ×H1 0(Ω) and equip it with the inner product and norm as (φ, ϕ)Eµ= (φ1, ϕ1)µ+ (φ2, ϕ2)+(∇φ3,∇ϕ3), φ = (φ1, φ2, φ3)T, ϕ = (ϕ1, ϕ2, ϕ3)T∈Eµ, kφk2 Eµ= (φ, φ)Eµ, φ = (φ1, φ2, φ3)T∈Eµ. Obviously, (Eµ,(·,·)Eµ) is a Hilbert space equivalent to Ewith the usual inner product. 20 Lemma 4.1. For any ψ= (u, v, z)T∈Eµ, there holds Re(Θψ, ψ)Eµ⩾ϑ(kuk2 µ+kvk2) + ν 2kvk2+αk∇zk2,(4.1) where Θis the operator defined by (2.9) and 0< ϑ =µν pν2+ 4µ(pν2+ 4µ+ν)∈(0,ν 4).(4.2) Proof. Direct computations imply Re(Θψ, ψ)Eµ=µδkuk2+δk∇uk2+δ(δ−ν)(u, v) + αk∇zk2+ (ν−δ)kvk2. We now choose ϑas in (4.2) and 0< δ =µν ν2+ 4µ<ν 4.(4.3) Then δ > ϑ and Re(Θψ, ψ)Eµ−ϑ(kuk2 µ+kvk2)−ν 2kvk2−αk∇zk2 =(δ−ϑ)kuk2 µ+ (ν 2−δ−ϑ)kvk2+δ(δ−ν)(u, v) ⩾(δ−ϑ)kuk2 µ+ (ν 2−δ−ϑ)kvk2−δν √µkukµkvk⩾0, since 4(δ−ϑ)(ν 2−δ−ϑ) = δ2ν2 µ. This ends the proof. With the above coercivity of the operator Θ in Eµ, we can prove directly the continuous dependence of the solutions of problem (2.11)-(2.12) on the initial data. Lemma 4.2. Let ψ(1)(t) = ψ(1)(x, t)and ψ(2)(t) = ψ(2)(x, t)be two solutions of problem (2.11)-(2.12) corresponding to the initial data ψ(1) τand ψ(2) τ, respectively. Then kψ(1)(t)−ψ(2)(t)k2 Eµ.kψ(1) τ−ψ(2) τk2 Eµexp nZt τkψ(1)(s)kE+kψ(2)(s)kEdso. (4.4) Proof. Let ψ(k)(t) = ψ(k)(x, t;τ, ψ(k) τ) = u(k)(x, t), v(k)(x, t), z(k)(x, t)T, k = 1,2, be two solutions of problem (2.11)-(2.12) corresponding to the initial data ψ(1) τand ψ(2) τ, respectively. Put        eu(t) = eu(x, t) = u(1)(x, t)−u(2)(x, t), ev(t) = ev(x, t) = v(1)(x, t)−v(2)(x, t), ez(t) = ez(x, t) = z(1)(x, t)−z(2)(x, t), e ψ(t) = ψ(1)(t)−ψ(2)(t). 21 Then e ψ(t) satisfies d dte ψ(t)+Θe ψ(t) = F(ψ(1)(t), t)−F(ψ(2)(t), t), t > τ, (4.5) e ψ(τ) = e ψτ=ψ(1) τ−ψ(2) τ.(4.6) By Lemma 4.1, Re(Θ e ψ, e ψ)Eµ⩾ϑ(keuk2 µ+kevk2) + ν 2kevk2+αk∇ezk2,∀t⩾τ. (4.7) At the same time, direct computations imply kF(ψ(1)(t), t)−F(ψ(2)(t), t)k2 Eµ =β2 |z(1)|2−|z(2)|2 2+k|∇(z(1)u(1) −z(2)u(2))k2 .kz(1) −z(2)k2 |z(1)|+|z(2)| 2+k∇z(1)k2ku(1) −u(2)k2+k∇u(2)k2kz(1) −z(2)k2 +kz(1)k2k∇u(1) −∇u(2)k2+ku(2)k2k∇z(1) −∇z(2)k2 .kψ(1) −ψ(2)k2 Eµ(kψ(1)(t)k2 E+kψ(2)(t)k2 E),∀t⩾τ. (4.8) Taking the real part of the inner product (·,·)Eµof equation (4.5) with e ψ(t) first and then using (4.7)-(4.8), we obtain d dske ψ(s)k2 Eµ+σke ψ(s)k2 Eµ.ke ψ(s)k2 Eµ(kψ(1)(s)kE+kψ(2)(s)kE),∀s > τ, (4.9) where σ= min{2ϑ, 2α}>0. Integrating (4.9) over [τ, t] yields ke ψ(t)k2 Eµ.ke ψ(τ)k2 Eµ+Zt τke ψ(s)k2 Eµkψ(1)(s)kE+kψ(2)(s)kEds, ∀t > τ. (4.10) Applying Gronwall’s inequality to (4.10) gives (4.4). The proof of Lemma 4.2 is completed. Lemma 4.3. Let assumption (H) hold. Then for every ψ∗∈Eand every t∈R, the E-valued function τ7−→ U(t, τ)ψ∗is continuous and bounded on (−∞, t]. Proof. Let ψ∗= (u∗, v∗, z∗)T∈Eand t∈Rbe given. For any s∗∈(−∞, t] we next prove that U(t, τ)ψ∗is continuous at τ=s∗. To this end, we shall establish that for any  > 0 there exists some η=η()>0, such that if r < t with |r−s∗|< η then kU(t, r)ψ∗−U(t, s∗)ψ∗kE< . (4.11) We assume r < s∗without loss of generality. Notice that the norm k·kEµis equivalent to k·kE. Employing (4.4) and the continuity property of the process, we have kU(t, r)ψ∗−U(t, s∗)ψ∗k2 E =kU(t, s∗)U(s∗, r)ψ∗−U(t, s∗)U(r, r)ψ∗k2 E .kU(s∗, r)ψ∗−U(r, r)ψ∗k2 Eexp nZt s∗kU(θ, r)ψ∗kE+kU(θ, s∗)ψ∗kEdθo.(4.12) 22 Remember that (2.36) shows that U(·, r)ψ∗and U(·, s∗)ψ∗belong to C([s∗, t], E). Hence Zt s∗kU(θ, r)ψ∗kE+kU(θ, s∗)ψ∗kEdθ < +∞.(4.13) So from (2.36) and (4.13) we conclude that the right hand side of (4.12) is as small as needed if |r−s∗|is small enough, that is (4.11) holds true. Therefore, the E-valued function τ7−→ U(t, τ)ψ∗is continuous on (−∞, t]. We next prove that the Eµ-valued function τ7−→ U(t, τ)ψ∗is bounded on (−∞, t]. In fact, for above ψ∗∈Eand t∈R, we see from Lemma 2.1 that lim τ→−∞ kU(t, τ)ψ∗k2 E .e−δt Zt −∞ eδskG(s)k2ds+e−δt Zt −∞ eδse−αs Zs −∞ eαθkf(θ)k2dθ3ds +e−αt Zt −∞ eαskf(s)k2ds3+kf(t)k2, t ∈R.(4.14) The right-hand side of (4.14) is a bounded quantity which is independent of τ. From this fact and the continuity of τ7−→ U(t, τ)ψ∗on (−∞, t] we obtain the desired result. The proof of Lemma 4.3 is completed. We next recall the definition of generalized Banach limit and a useful property. Definition 4.1. ( [11,26]) A generalized Banach limit is any linear functional, which we denote by LIMt→+∞, defined on the space of all bounded real-valued functions on [0,+∞)that satisfies (1) LIMt→+∞h(t)⩾0for nonnegative functions h(·)on [0,+∞); (2) LIMt→+∞h(t) = lim t→+∞h(t)if the usual limit lim t→+∞h(t)exists. Let B+be the collection of all bounded real-valued functions on [0,+∞). For any generalized Banach limit LIMt→+∞, the following useful property |LIMt→+∞h(t)|⩽lim sup t→+∞|h(t)|,∀h(·)∈B+,(4.15) is presented in [11, (1.38)] and in [9, (2.3)]. Remark 4.1. Notice that we consider the “pullback” asymptotic behavior and we require generalized limits as τ→ −∞. For a given real-valued function ϕdefined on (−∞,0] and a given Banach limit LIMT→+∞, we define LIMt→−∞ϕ(t) = LIMt→+∞ϕ(−t).(4.16) Combining Lemma 2.2, Theorem 3.1, Lemma 4.3 and [26, Theorem 3.1], we obtain the following result. 23 Theorem 4.1. Let assumption (H) hold. Let {U(t, τ)}t⩾τbe the process associated to problem (2.11)-(2.12) and b ADδ={ADδ(t)|t∈R}the pullback-Dδattractor obtained in Theorem 3.1. Then for a given generalized Banach limit LIMt→+∞and a continuous map ξ:R7−→ Ewith ξ(·)∈ Dδ, there exists a unique family of Borel probability measures {mt}t∈Rin Esuch that the support of the measure mtis contained in ADδ(t) and LIMτ→−∞ 1 t−τZt τ Ψ(U(t, s)ξ(s))ds =ZADδ(t) Ψ(ψ)dmt(ψ) = ZE Ψ(ψ)dmt(ψ) (4.17) =LIMτ→−∞ 1 t−τZt τZE Ψ(U(t, s)ξ(s))dms(ψ)ds, (4.18) for any real-valued continuous functional Ψon E. Moreover, mtis invariant in the sense that ZADδ(t) Ψ(ψ)dmt(ψ) = ZADδ(τ) Ψ(U(t, τ)ψ)dmτ(ψ), t ⩾τ. (4.19) We next introduce the class Tof test function associated to the definition of statistical solutions for equation (2.11). We write (2.11) as dψ dt=F(ψ, t),(4.20) where F(ψ, t) = −Θψ+F(ψ, t). Then F(ψ, t) : E×R7−→ E∗, here E∗is the dual space of E. We expect that the function Φ ∈ T satisfies d dtΦ(ψ(t)) = hΦ0(ψ(t)),F(ψ(t), t)i,(4.21) for every global solution ψ(t) of equation (2.11), where h·,·i is the dual pairing between Eand E∗. Definition 4.2. (cf. [11, page 178, Definition 1.2]) We define the class Tof test functions to be the set of real-valued functionals Φ = Φ(ψ)on Ethat are bounded on bounded subset of Eand satisfy (a) for any ψ∈E, the Frech´et derivative Φ0(ψ)exists: for each ψ∈Ethere exists an element Φ0(ψ)such that |Φ(ψ+ϕ)−Φ(ψ)−hΦ0(ψ), ϕi| kϕkE−→ 0as kϕkE→0, ϕ ∈E; (b) Φ0(ψ)∈Efor all ψ∈E, and the mapping ψ7−→ Φ0(ψ)is continuous and bounded as a function from Eto E; 24 (c) for every global solution ψ(t)of equation (2.11),(4.21) holds true. For example, we can consider the cylindrical test function defined on E. Let ϕ1,ϕ2 and ϕ3belong to Eand γbe a continuously differentiable real-valued function on R3 with compact support. For each ψ∈E, define Φ(ψ) via Φ(ψ) = γ(hψ, ϕ1i,hψ, ϕ2i,hψ, ϕ3i), where hψ, ϕjiis the dual pairing between ψ∈E⊂E∗and ϕj∈E. Then the function Φ(·) is obviously continuous from Eto Rand in fact is differentiable in E, with differential Φ0(·) at ψ∈Egiven by Φ0(ψ) = 3 X j=1 ∂jγ(hψ, ϕ1i,hψ, ϕ2i,hψ, ϕ3i)ϕj.(4.22) where ∂jγdenotes the derivative of γwith respect to its j-th coordinate. (4.22) shows that Φ0(·)∈E. Above analyses show that the cylindrical test functions of above form satisfy Definition 4.2. We now introduce the definition of statistical solution for equation (4.20) and prove its existence. Definition 4.3. A family {ρt}t∈Rof Borel probability measures in Eis said to be a statistical solution in the phase space E(or simply a statistical solution) of equation (4.20) if the following conditions are satisfied: (a) the function t7→ ZE Γ(ψ)dρt(ψ)is continuous for every Γ∈ C(E)(the collection of continuous and bounded functions on E); (b) for almost t∈R, the function ψ7→ hF(ψ(t), t), φiis ρt-integrable for every φ∈E. Moreover, the map t7→ ZEhF(ψ(t), t), φidρt(ψ) belongs to L1 loc(R)for every φ∈E; (c) for any cylindrical test function Φ∈ T , it follows that ZE Φ(ψ)dρt(ψ)−ZE Φ(ψ)dρτ(ψ) = Zt τZEhF(ψ(s), s),Φ0(ψ)idρs(ψ)ds, for all t, τ ∈R. Theorem 4.2. Let assumption (H) hold. Then the family of Borel probability measures {mt}t∈Robtained in Theorem 4.1 is a statistical solution of equation (4.20). 25