GLOBAL EXISTENCE AND LONG TIME BEHAVIOR OF THE ELLIPSOIDAL-STATISTICAL-FOKKER-PLANCK MODEL FOR DIATOMIC GASES

. We are concerned with the global existence and long time behavior of the solutions to the ES-FP model for diatomic gases proposed in [22]. The global existence of the solutions for this model near the global Maxwellian is established by nonlinear energy method based on the macro-micro decomposi- tion. An algebraic convergence rate in time of the solutions to the equilibrium state is obtained by constructing the compensating function. Since the den- sity distribution function F ( t,x,v,I ) also contains energy variable I , we derive more general Poincar´e inequality including variables v,I on R 3 × R + to establish the coercivity estimate of the linearized operator.


1.
Introduction. Recently, the Ellipsoidal-Statistical-Fokker-Planck (ES-FP) model of the Boltzmann equation for polyatomic gases is proposed in [22] to obtain the correct Prandtl number in the Compressible Navier-Stokes asymptotics for polyatomic gases, which can be thought of as an extension of the previous model in [21] from monoatomic gases to polyatomic gases. The ES-FP model takes the form where F (t, x, v, I) ≥ 0 is the number density function of the particles at time t > 0 with the position x ∈ R 3 , the velocity v ∈ R 3 and a non-translational internal energy parameter I ∈ R + . C(F ) is the collision operator defined by where the constant τ > 0 is the relaxation time, R > 0 is the gas constant, and δ > 0 is linked to the number of degrees of freedom for the polyatomic gases (δ = 2 for diatomic gases). The bulk velocity U = U (t, x) is defined by with the macroscopic density defined by ρ(t, x) = R 3 ×R + F (t, x, v, I)dvdI. The tensor Π = Π(t, x) is the combination among the stress tensor Θ, the translational temperature tensor RT tr I and the temperature tensor RT I: The relaxation temperature T rel is introduced by the non-translational internal energy temperature T int and the temperature T as follows where the coefficients ν, 0 < θ < 1 are some free parameters, I is identity matrix, and the macroscopic quantities Θ = Θ(t, x), T tr = T tr (t, x), T rel = T rel (t, x) and T = T (t, x) are defined by x, v, I)dvdI, Notice that it satisfies from (4) that It is proved in [22] that for ν ∈ (− 1 2 , 1), Π is symmetric positive definite and the collision operator C(F ) satisfies the following conservation laws of the mass, momentum, and energy: and the dissipation of the entropy The inequality above also leads to the equilibrium property . In the sequel, for simplicity we assume that both τ and R equal to 1. In this paper, we consider the Cauchy problem for Eq. (1) with initial data and the purpose is to study the global existence and long time behavior of the solutions to this Cauchy problem for δ = 2 (corresponding to the diatomic gases case), when the initial data (6)  Before stating the main results, we reformulate the Cauchy problem (1) and (6) around the equilibrium state as follows. Let By (8), the Cauchy problem (1) and (6) is changed into where L ν,θ denotes the linearized collision operator and N (f ) denotes the nonlinear part with a = By straightforward calculations, we have The two equalities above will be used in the estimates of Lemma 2.1 and Proposition 3.
We now state the main results as follows.
There exists a positive constant δ 1 small enough such that if f 0 2 L 2 v,I (H 3 x ) ≤ δ 1 , then the Cauchy problem (9)-(10) admits a unique global solution with C > 0 a generic constant. Moreover, there exists a positive constant δ 2 small enough, such that if f 0 with C > 0 a generic constant. Remark 1. Theorem 1.1 implies the global existence, uniqueness and an algebraic convergence rate in time of the solution to the Cauchy problem (1) and (6).

Remark 2.
Physically, for diatomic molecular gases, two important parameters Prandtl number P r 0.7 and bulk viscosity is in 0.6-0.8. Correspondingly, the value of θ should be around 0.5. The condition of Theorem 1.1 conforms this requirement.

Remark 3.
In the periodic box case [5,36], the Poincaré inequality was used to obtain the estimate of Pf in terms of its derivative, then the convergence rate is exponential. In the current whole space case, only the derivative of Pf can be directly estimated from the macroscopic equations (41)-(46), and the Poincaré inequality does not hold to estimate Pf itself, thus the time-decay rate is algebraic here.
Remark 4. The Prandtl number P r satisfies 6 5 < P r < 3 for ν and θ in Theorem 1.1, where P r is defined by (53) in [22]. Actually, P r can be chosen arbitrarily smaller than 6 5 for any ν ∈ (−∞, − 1 2 ], θ ∈ (0.5, 1). By Proposition 2.4 in [22], we can take ν ∈ (− RTtr+ θ 1−θ RT λmax−RTtr , 1) for the positive definiteness of Π, where T tr and T are defined by (4), and λ max > 0 is the maximum eigenvalue of the tensor Θ which is defined by (4). By definition λ max > RT tr , and both RT tr and λ max get close to RT when f is close to the local Maxwellian M p (v, I), then the value of − λmax−RTtr can be as small as possible. Consequently, this shows that ν can take any value between −∞ and 6 5 when f is close to the local Maxwellian equilibrium. In particular, for any ν ∈ (−∞, − 1 2 ], θ ∈ (0.5, 1), the coercivity estimate of the linearized operator L ν,θ (22) is still satisfied. We can prove the supremum norm of f is small by the energy estimates method when the initial data is a small perturbation of the Maxwellian equilibrium [12,13,14], we omit the details here.
The proof of Theorem 1.1 consists of two parts. On the one part, we construct the global solution motivated by the L 2 energy method based on the macro-micro decomposition developed by Guo in the analysis of the well-posedness of the Boltzmann equation [12,13,14]. A crucial step in this process is to derive an energy estimate so that the a-priori estimate can be closed. On the other part, to establish the time decay rate of the global solutions, we employ the method of compensating function together with the energy estimates introduced by Kawashima in [17]. This method is first used for the Boltzmann equation [10,17], and then is extended to other kinetic equations [9,29,30,31,32,33].
However, different from the usual situation, because the introduction of internal energy parameter I (cf. [22]) and more complicated structure which the equation have, we face some new difficulties and deal with them as follows. Firstly, the structure of the linearized collision operator L ν,θ given by (11) is more complicated than the linearized Fokker-Planck operator. The coercivity estimates of the linearized Fokker-Planck operator cannot be used here, so we have to re-establish the coercivity estimate of L ν,θ which is crucial to the theory of well-posedness. To achieve this purpose, the most important step is to derive more general Poincaré inequality (25) including variables v, I on R 3 × R + . For the diatomic case (δ = 2), inspired by [19,28], using the method of Sturm-Liouville theory, and choosing Laguerre function as an orthonormal basis, we develop the inequality (25) which we need. However, so far as I know, for more general case (δ ≥ 3), the approach above establishing the inequality (25) will no longer work. How to obtain the coercivity estimates of the linearized operator for all cases of polyatomic gases and build the existence of solutions, will be our further work to study. Secondly, because of appearance of I, we get fourteen moments rather than thirteen moments as usual, so we build new macro-micro system (cf. [12]) to finish energy estimates, and reconstruct the corresponding compensating function to obtain the time convergence rate of the solutions to the equilibrium. Thirdly, we should mention here that the energy estimates obtained for the global existence of the solutions are not sufficient enough for the time decay rate because of the additional difficulties caused by the second order derivative terms with respect to variable v, I in N (f ). We need to establish some new estimates containing higher order derivatives with respect to variable v, I and x, which is the main reason why we impose more spatial regularity assumptions on the initial data in Theorem 1.1.
At the end of the introduction, we also review some works related to the Ellipsoidal Statistical model. In [2,6,7,15], the BGK model was extended to the ES-BGK model which was proved to retain the elementary properties of the Boltzmann operator (collisional invariants, Maxwellian equilibrium) and H-theorem. In the past ten years, some works about ES-BGK were studied. Yun et al. proved the existence of global in time of the smooth solutions or the weak solutions to the Cauchy problem and the stationary weak solutions to the boundary value problem of the ES-BGK model for monoatomic gases [3,24,34,35], and the global existence of the classical solutions or mild solutions to the Cauchy problem of the ES-BGK model for polyatomic gases [25,36]. For the ES-FP model, in a previous paper [26], we have established the global existence of the unique solution of this model for monoatomic gases, and obtained its algebraic time convergence rate to the equilibrium state.
The rest of the paper is organized as follows. The uniform a-priori estimates are established in Section 2. The global existence of the unique solution is constructed in Section 3. The time decay estimate is obtained in Section 4.
Notation. Throughout this paper, the constant C > 0 denotes a generic constant. We introduce P as the macroscopic projection operator, defined by Then for any fixed point (t, x), any function f (t, x, v, I) ∈ L 2 v,I (H 3 x ) can be uniquely decomposed into where Pf and (I−P)f are the macroscopic and microscopic components of the function f , respectively. Corresponding to the linearized operator L ν,θ , the dissipation rate is given by the following norm: The inner product ·, · on L 2 (R 3 × R + ) is defined by and the space L 2 x,v,I = L 2 v,I (L 2 x ). Notice that, by the definitions of (11) ,(13) and straightforward calculations, we can obtain that the linearized operator L ν,θ is selfadjoint: L ν,θ f, g = L ν,θ g, f , and L ν,θ (Pf ) = 0.
The Fourier transformf = Ff of an integrable function f is defined bŷ 2. The uniform a-priori estimates. Our goal of this section is to establish the uniform a-priori estimates for the smooth solution f = f (t, x, v, I) to the Cauchy problem (9)- (10) for any given constant T > 0 and the constant N 0 > 0 small enough. In this section, the main result is as follows.
where a, b and c are defined by (17) and C > 0, σ 0 > 0 are generic positive constants.
The proof of Proposition 1 is mainly based on the properties of the linearized collision operator L ν,θ and the nonlinear energy method. We state the details in the following two subsections.
Proof. By straightforward calculations, we are able to obtain Applying the classical Poincaré inequality with respect to the measure µ v (v) (cf. [11,28]) and the inequality with respect to the measure µ I (I) (the proof is given in Appendix) and motivated the approach (cf. [4]) which generalized Poincaré inequality from R to R 3 , we are able to derive the following generalized inequality on for any suitable function h(v, I) and µ(v, , we get from (26) that
with C > 0 a generic constant.
To estimate the terms of N (f ), L ν,θ (f ) using Lemma 2.2, by straightforward calculations, we can also get Our first result is about the basic energy estimate on f .
x, v, I) be a smooth solution to the Cauchy problem (9)-(10) satisfying (20) Then, it holds that where a, b and c are defined by (17), C > 0 and σ 2 are generic constants, and N 0 is small enough.
Proof. The proof is divided into two steps.
Step 1. Estimate on f (t) 2 x ) . Multiplying (9) by f and taking the integrations over R 3 By (22), the second term of (33) is estimated by By Corollary 1 and Lemma 2.2, and using Hölder's and Young's inequalities, one can estimate the last term of (33) as follows where the constant ε > 0 is to be chosen small enough. Substituting (34)- (35) into (33), we can obtain with C > 0 a generic constant, where the inequality holds due to the fact that N 0 is small enough.
Step 2. Estimate on . Taking the spatial derivatives ∂ α to (9) with 1 ≤ |α| ≤ 3, multiplying by ∂ α f and taking the integrations over R 3 By (22), the second term of (37) is estimated by By Corollary 1 and Lemma 2.2, and using Hölder's and Young's inequalities, one can estimate the last term of (37) as follows where the constant ε > 0 is to be chosen small enough. Substituting (38)-(39) into (37) and taking the summation over 1 ≤ |α| ≤ 3, we can obtain the estimate d dt 1≤|α|≤3 with C > 0 a generic constant. The combination of (36) and (40) gives rise to the estimate (32) due to N 0 small enough. The proof of Lemma 2.3 is completed.
We next estimate the macroscopic quantities a, b and c which are defined by (17). Inspired by [12], using L ν,θ (Pf ) = 0 and pluging f = Pf + (I − P)f into (9) to express the macroscopic part Pf through the microscopic part (I − P)f , we get By re-writing the left hand side and right hand side of (9) with respect to the following basis we obain the coupled micro-macro system Our second result is about the energy estimates of a, b and c.
Step 1. Estimate on ∇ x c(t) 2 H 2 x . Taking the spatial derivative ∂ α to (46) with |α| ≤ 2, multiplying by ∂ α ∂ xj c, and taking integration over R 3 x , we have The first term on the right hand side of (48) can be estimated as follows. It holds that Multiplying (43) by 1 √ 5 , multiplying (44) by 2 √ 10 , and combine the two equalities together, we get where the constant ε is to be chosen small enough. For the rest terms on the right hand side of (48), applying Lemma 2.2, and using Hölder's and Young's inequalities, we have Substituting (49), (50) and (51) into (48), and taking the summation over α, j with |α| ≤ 2 and 1 ≤ j ≤ 3, we can obtain the estimate Notice that Step 2. Estimate on Taking the spatial derivatives ∂ α to (45) with |α| ≤ 2, multiplying by ∂ α (∂ xi b j + ∂ xj b i ), and taking integrations over R 3 x , we have (53)

LEI JING AND JIAWEI SUN
The first term on the right hand side of (53) can be estimated as follows. It holds that By (42), and using Hölder's and Young's inequalities, one can estimate the last term of (54) as follows where the constant ε is to be chosen small enough. By Lemma 2.2, and using Hölder's and Young's inequalities, the rest terms on the right hand side of (53) can be estimated as follows Substituting (54), (55) and (56) into (53) and taking the summation over α, i, j with |α| ≤ 2 and 1 ≤ i < j ≤ 3, we can obtain the estimate |α|≤2 1≤i<j≤3 Step 3. Estimate on ∇ x · b 2 H 2 x . Taking spatial derivatives ∂ α to (43) with |α| ≤ 2, multiplying by ∂ α ∂ xj b j , and taking integration over R 3 x , we have √ 2 (58) The first two terms on the right hand side of (58) can be estimated as follows. It holds that Applying (42), using Hölder's and Young's inequalities, one can estimate the last term of (59) as follows where the constant ε is small enough to be determined later. By Lemma 2.2, and using Hölder's and Young's inequalities, the rest terms on the right hand side of (58) can be estimated as follows

LEI JING AND JIAWEI SUN
Substituting (59), (60) and (61) into (58) and taking the summation over α, j with |α| ≤ 2 and 1 ≤ j ≤ 3, we can obtain the estimate (62) Step 4. Estimate on ∇ x a 2 H 2 x . Taking the spatial derivatives ∂ α to (42) with |α| ≤ 2, multiplying by ∂ α ∂ xj a, and taking integration over R 3 x , we have (64) Applying (41), one can estimate the first term on the right hand side of (64) as follows By Lemma 2.2, and using Hölder's and Young's inequalities, the remaining terms on the right hand side of (64) can be estimated as follows where the constant ε is to be chosen small enough.
Proof of Proposition 1. We combine Lemma 2.3 and Lemma 2.4 to prove Proposition 1. Actually, define Obviously , x ) for enough small constant ε. Taking (32) + ε(47) with ε > sufficiently small, and choosing N 0 small enough for each fixed ε, we get Integrating with respect to the time over [0, t], we conclude the following estimates . It gives rise to the estimate (21) by taking supremum on [0, T ]. The proof of Proposition 1 is completed.
3. Global existence. We first construct the local solution to the Cauchy problem (9)-(10) with the initial data f 0 ∈ L 2 v,I (H 3 x ). Considering the following problem where L ν,θ f ε and N (f ε ) are defined by (11) and (12), and with η the standard mollifier. We can prove the following local existence results to the Cauchy problem (69)-(70) (cf. [13], [14]). There exists a constant T 0 = T 0 (ε) > 0 small enough, such that there is a unique solution f ε = f ε (t, x, v, I) to the Cauchy problem(69)-(70) with C > 0 a generic constant independent of ε. Based on the uniformly bounded estimate (71), we can pass into the limit ε → 0 and obtain that there ). In addition, f = f (t, x, v, I) is the unique solution to the Cauchy problem (9)-(10) for t ∈ (0, T 1 ]. By repeating above approach with the help of the uniform a-priori estimates proved in Proposition 1 and the standard continuity argument, we can extend the local solution globally in time so as to obtain the global solution to the Cauchy problem (9)-(10). 4. The time decay rate. In this section, we will combine the compensating function with the energy estimates to obtain the time decay rate of the global solutions to the Cauchy problem (9)- (10). Let us recall the definition of compensating function introduced by Kawashima [17]. 2. iS(ω) is self-adjoint on L 2 v,I for all ω ∈ S 2 . 3. There is a constant c 1 > 0 such that for all f ∈ L 2 v,I , ω ∈ S 2 . The main idea to construct S(ω) owns to [17]. Indeed, let X be the closed subspace spanned by fourteen moments consisting of the kernel of L ν,θ and the images of linear operator v j (j = 1, 2, 3) from Ker(L ν,θ ) → L 2 v,I denoted by X = span{e j |j = 1, 2, . . . , 14}, where the orthonormal basis {e j } for X is given by , by a straightforward calculation, one has with the constant α > 0 to be determined later, ω = (ω 1 , ω 2 , ω 3 ) = ξ |ξ| , and with β > 0 some constant to be determined later. It is easy to verify (cf. [17]) that there exist a constant α > 0 and a constant β > 0 small enough, such that S(ω) defined by (73) is a compensating function to (9).
Taking the Fourier transform in x for equation (9), we have Taking the inner product of (74) with ((1 + |ξ| 2 ) − iκS(ω))f and using the property (72) of the compensating function, we obtain where with κ > 0 a constant small enough. The left of (75) is bounded below by because of the coercivity estimate (22) of L ν,θ and inequality (72).
On the other hand, by using Corollary 1 and integration by parts, the absolute value of the first term on the right of (75) satisfies By the definition (73) and integration by part, the absolute value of the third term on the right of (75) equals Notice that and where we used the self-adjoint property of L ν,θ and the exponential decay of e l (v, I) in v, I. For the absolute value of the second term on the right of (75), we have Therefore for any ε > 0, the right of (75) is not more than Now take ε, κ small enough such that ε = min{ c1 2 , σ1 6 } , and then 0 < κ ≤ κ 2 , where Finally, we get With the help of (80), we are able to establish the following estimates.
) is a solution of (9), we have Proceeding with the estimates similar to Lemma 2.3-Lemma 2.4, we can obtain the following the fourth-order energy estimate similar to (68). Proposition 3. Let − 1 2 < ν < 1, 0.5 < θ < 1, T > 0, and f (t, x, v, I) be a smooth solution to the Cauchy problem (9)-(10) satisfying sup (82) where a, b and c are defined by (17), k > 0, σ 0 > 0 are generic constants and x ) . In this section, the main result is as follows.
Similarly, taking ∂ α x to (9) with 1 ≤ |α| ≤ 3, multiplying by |v| 2 ∂ α x f , taking integrations over R 3 x × R 3 v × R + I , and taking the summation over α, we obtain Taking ∂ α x ∂ I to (9) with 1 ≤ |α| ≤ 3, multiplying by I∂ α x ∂ I f , taking integrations over R 3 x × R 3 v × R + I , and taking the summation over α, we obtain Taking ∂ α x to (9) with 1 ≤ |α| ≤ 3, multiplying by I∂ α x f , taking integrations over R 3 x × R 3 v × R + I , and taking the summation over α, we obtain When |α| = 0, we are able to get inequalities similar completely to (85)-(88) except that there are some subtle differences produced in the process of enlarging the inequalities, which are as the case of the proof of Lemma 2.3.
By Proposition 3, multiplying (85)-(88) and the case of |α| = 0 by a small constant κ > 0, adding them to (82), and using the assumption that N 1 is small enough, one can remove the terms that are not needed, and get We have proved in Proposition 2 that By straightforward calculations from the definition of (12) and (76), we have furthermore, by (90),(91),(93), we get ). Finally, we obtain from (92) that (1 + t) Thus, if is small enough, we are able to obtain (83) from (94). The proof of the Proposition 4 is completed.