ON HYPERBOLICITY OF 13-MOMENT SYSTEM. Zhenning Cai. Yuwei Fan. Ruo Li. (Communicated by Yan Guo)

Transcription

1 Manuscript submitted to AIMS Journals Volume X, Number 0X, XX 20XX doi: /xx.xx.xx.xx pp. X XX ON HYPERBOLICITY OF 13-MOMENT SYSTEM Zhenning Cai CAPT & School of Mathematical Sciences Peking University Yiheyuan Road, Beijing, China Yuwei Fan School of Mathematical Sciences Peking University Yiheyuan Road, Beijing, China Ruo Li HEDPS, CAPT & School of Mathematical Sciences Peking University Yiheyuan Road, Beijing, China (Communicated by Yan Guo) Abstract. We point out that the thermodynamic equilibrium is not an interior point of the hyperbolicity region of Grad s 13-moment system. With a compact expansion of the phase density, which is compacter than Grad s expansion, we derived a modified 13-moment system. The new 13-moment system admits the thermodynamic equilibrium as an interior point of its hyperbolicity region. We deduce a concise criterion to ensure the hyperbolicity, thus the hyperbolicity region can be quantitatively depicted. 1. Introduction. Grad s 13-moment system [4] has been studied for over 0 years. This system was derived by utilizing the isotropic Hermite expansion [3] of the phase density in the Boltzmann equation, and such an idea opened a brand new direction in the gas kinetic theory. In the subsequent years, a number of defects of this model were discovered, one of which was that Grad s 13-moment system is not globally hyperbolic, and for 1D flows, the hyperbolicity can only be obtained near the equilibrium [8]. The loss of hyperbolicity directly breaks the well-posedness of the partial differential equations, and thus the capability of this model is strictly limited. Extensions of this model include the regularized Burnett equations [], regularized 13-moment equations [11, 12], and the Pearson-13-moment equations [14]. These methods may alleviate the problem of hyperbolicity to some extent [, 13, 14]. However, all the analysis on the hyperbolicity is restricted to the 1D flows, while there are no comments indicating that the 3D case is the same or similar as the 1D case. Actually, Grad s paper [4] has pointed out that in comparison with the 1D flows, an additional soundspeed appears in the 2D flows. In this paper, we are concerned with the hyperbolicity for the full 3D flows Mathematics Subject Classification. 82C40, 3L0. Key words and phrases. Grad s moment system, hyperbolicity, modified 13-moment system. The research is supported in part by the National Basic Research Program of China (2011CB309704). 1

2 2 ZHENNING CAI AND YUWEI FAN AND RUO LI We first point out that in the 3D Grad s 13-moment equations, for each equilibrium state, none of its neighbourhoods is contained in the hyperbolicity region. This reveals that the manifold formed by all the equilibrium states is on the boundary of the hyperbolicity region, and thus an arbitrary small perturbation of the equilibrium state may lead to the loss of hyperbolicity, which indicates the instability of Grad s 13-moment equations. More precisely, we prove that if an arbitrary small anisotropic perturbation is applied to the phase density from the equilibrium, the hyperbolicity may be broken down. Noticing that the anisotropy plays an essential role in breaking down the hyperbolicity, we then propose a new modified 13-moment model such that the equilibrium state lies in the interior of the hyperbolicity region. This modified system is derived by an anisotropic Hermite expansion instead of the isotropic Hermite expansion in Grad s method, where the anisotropy is specified by the temperature tensor. We find out a dimensionless quantity which can prescribe the departure of the phase density from the equilibrium state. It is proven that if this quantity is controlled above by a threshold, the full 3D system is hyperbolic. The value of this threshold is approximately given, and the size of the hyperbolicity region is depicted using the Chapman-Enskog type expansion. The rest of this paper is as follows: in Section 2, some algebraic lemmas are given as preliminaries; in Section 3, the hyperbolicity of Grad s 13-moment equations is discussed for both 1D and 3D flows; Section 4 gives a modified 13-moment system, and its hyperbolicity is discussed in detail; finally, some concluding remarks are given in. 2. Some preliminary results in linear algebra. In this paper, we will focus on the hyperbolicity of moment systems, i.e. the real diagonalizability of the coefficient matrices in these systems. In order to make our later derivation self-consistent, we present some lemmas about matrices and polynomials as preliminaries and most of the proofs can be found in textbooks of linear algebra. If not specified, we are considering real matrices and polynomials with real coefficients. Lemma 2.1 (Cayley-Hamilton). For any square matrix A and its characteristic polynomial p(λ) = det(λi A), we have p(a) = 0. Lemma 2.2. For a square matrix A, the following three statements are equivalent: 1. λ is a root of the minimal polynomial of A, 2. λ is a root of the characteristic polynomial of A, 3. λ is an eigenvalue of A. Lemma 2.3. A square matrix A is diagonalizable if and only if the minimal polynomial of A is the product of distinct linear functions. The following corollary can be derived from the above results: Corollary 1. For a square matrix A, suppose p(λ) is its characteristic polynomial. If there exists a polynomial q(λ) such that p(λ) and q(λ) share the same roots, and q(a) 0, then A is not diagonalizable. Proof. Let us prove it by contradiction. We suppose A is diagonalizable. According to Lemma 2.2 and Lemma 2.3, the minimal polynomial of A can be written as m(λ) = (λ λ 1 ) (λ λ n ), (1) where λ 1,, λ n are all the distinct eigenvalues of A. Since p(λ) and q(λ) share the same roots, we have m(λ) q(λ), and thus q(a) = 0, which violates the condition q(a) 0.

3 HYPERBOLICITY OF 13-MOMENT SYSTEMS 3 We will use Lemma 2.3 when proving a matrix is diagonalizable, and use Corollary 1 when proving a matrix is not diagonalizable. The following lemma can be used to tell if a polynomial has multiple roots. Lemma 2.4. Let p(z) be a polynomial defined on complex numbers. Then p(z) has no multiple roots if and only if p(z) and d p(z) are coprime. dz Definition 2.. Let p(z) and q(z) be two polynomials defined as p(z) = p 0 + p 1 z + p 2 z p m z m, q(z) = q 0 + q 1 z + q 2 z q n z n. (2) The (m + n) (m + n) matrix p m p 0 Syl(p, q) := p m p 0 q n q q n q 0 n rows m rows is called the Sylvester matrix associated to p(z) and q(z). The resultant of p(z) and q(z) is defined as the determinant of the above matrix: (3) res(p, q) := det (Syl(p, q)). (4) The following lemma is to be used to tell if two polynomials have common roots. Lemma 2.. Two nonzero polynomials are coprime if and only if their resultant is nonzero. 3. Boltzmann equation and Grad s 13-moment system. From this section, we start our discussion on the kinetic models. Section 3.1 and 3.2 give brief introductions to the Boltzmann equation and Grad s 13-moment system respectively, and Section 3.3 reviews the classical knowledge on the hyperbolicity of Grad s 13- moment system for 1D flows. In the last part of this section, we present our new findings that an intrinsic difference exists between the hyperbolicity regions in the 1D and 3D cases for Grad s system Boltzmann equation. The Boltzmann equation is a fundamental physical model in the gas kinetic theory. Suppose f(t, x, ξ) is the function of phase density, where t is the time, x = (x 1, x 2, x 3 ) T is the spatial coordinates, and ξ = (ξ 1, ξ 2, ξ 3 ) T stands for the velocity of gas molecules. Then the Boltzmann equation reads f t + ξ xf = Q(f, f). () The right hand side Q(f, f) describes the interaction between particles: 2π π/2 Q(f, f) = (f f f f) ξ ξ σ sin Θ dθ dε dξ, () R where f = f(t, x, ξ ), f = f(t, x, ξ ), f = f(t, x, ξ ), and the velocities ξ, ξ and ξ, ξ are the pre- and post-collision velocities of a colliding pair of molecules, and σ is the differential cross-section. In this paper, we consider only the Maxwell molecules, for which ξ ξ σ is a function of Θ. We refer the readers to [2, 1] for more details of the collision term and the Maxwell molecules.

4 4 ZHENNING CAI AND YUWEI FAN AND RUO LI The gas kinetic theory describes the fluid states in a microscopic view, while the macroscopic quantities such as density, velocity and temperature can be obtained by integrations. Define h = m h dξ, (7) R 3 where m is the mass of a single gas molecule. Then the relations between the density function f and some common macroscopic quantities are as follows: Density: ρ = f ; Velocity: u = (u 1, u 2, u 3 ) T = ρ 1 ξf ; Temperature: T = (3ρk B /m) 1 ξ u 2 f, where k B is the Boltzmann constant; Temperature tensor: T ij = (ρk B /m) 1 (ξ i u i )(ξ j u j )f, i, j = 1, 2, 3; Heat flux: q = (q 1, q 2, q 3 ) T = ξ u 2 (ξ u)f/2. Following the conventional style, we denote θ = k B m T, θ ij = k B m T ij, Θ = (θ ij ) 3 3. (8) It can be derived from the positivity of the density function f that Θ is symmetric positive definite. For simplicity, below we denote the relative velocity ξ u by C, and the norm of C is denoted by C. For example, we have θ = (3ρ) 1 C 2 f, θ ij = ρ 1 C i C j f. (9) 3.2. Grad s 13-moment system. The high dimensionality of the Boltzmann equation introduces extreme difficulties to its numerical treatment. In order to simplify the model, Grad proposed a 13-moment system [4], in which the velocity variable ξ was eliminated, while only 13 equations are presented. These equations are derived by assuming the following particular form of the phase density f: [ f 13 = 1 + θ ij δ ij θ ( Ci 2θ 2 C j δ ij C 2) + 2 ( q k C 2 ρθ 2 C k 2θ )] f M, (10) 2 where f M is the Maxwellian, defined as ) ρ f M = exp ( C2, (11) (2πθ) 3/2 2θ and in (10), the Einstein summation convention is assumed. Accordingly, when an index appears twice in a single term, it implies summation of that term over all the values of the index. By (10), Grad s 13-moment system can be written as a closed system as φ f 13 + φ(ξ x f 13 ) = φq(f 13, f 13 ), (12) t where φ = (1, C 1, C 2, C 3, C 2 1, C 2 2, C 2 3, C 1 C 2, C 1 C 3, C 2 C 3, C 2 C 1, C 2 C 2, C 2 C 3 ) T.

5 HYPERBOLICITY OF 13-MOMENT SYSTEMS By simplifications of the expression obtained after the integrations, the above system can be explicitly given by dρ dt + ρ u k = 0, x k du i dt + θ ik ρ + θ ik = 0, i = 1, 2, 3, ρ x k x k dθ ij u j) + 2θ k(i + 1 ( 4 q (i + 2 ) dt x k ρ x j) δ q k ij = ρθ x k µ (θ ij δ ij θ), i, j = 1, 2, 3, dq i dt (θ ijθ jk 2θθ ik + θ 2 δ ik ) ρ + 7 x k q u k i + 7 x k q u i k + 2 x k q u k k x ( i θjk ρθ ik 7 ) ( θ jj θik + 2ρθ 1 ) θ jj = 2 ρθ x j x k x k 3 x i 3 µ q i, i = 1, 2, 3. (13) Here, d dt = t + u k x k is the material derivative, and the brackets around indices denote the symmetrization of a tensor. The symbol µ denotes the coefficient of viscosity. For Maxwell molecules, µ is proportional to θ Local hyperbolicity of 1D Grad s moment system. It is well-known that the 1D Grad s moment system is hyperbolic only when the fluid is near the thermodynamic equilibrium state. For 1D flows, the 13-moment system reduces to a smaller system containing only five equations, which are obtained by setting u 2 = u 3 = θ 12 = θ 13 = θ 23 = q 2 = q 3 = 0 and θ 33 = θ 22 in (13). Such operation eliminates eight of the thirteen variables, and results in the 1D system ŵ t + ˆM(ŵ) ŵ x = ˆQ(ŵ), (14) where ŵ = (ρ, u 1, θ 11, θ 22, q 1 ) T, ˆQ(ŵ) = ( 0, 0, ρθ(θ θ 11 )/µ, ρθ(θ θ 22 )/µ, 2 3 ρθq 1/µ ) T, and u 1 ρ θ 11 /ρ u ˆM(ŵ) = 0 2θ 11 u 1 0 /(ρ) u 1 2/(ρ). 4(θ 11 θ 22 ) 2 /9 1q 1 / ρ(11θ θ 22 )/18 ρ(17θ 11 8θ 22 )/9 u 1 (1) The system (14) is hyperbolic if and only if ˆM(ŵ) is real diagonalizable. In order to check the diagonalizability of ˆM(ŵ), we calculate its characteristic polynomial as [ det(λi ˆM) = (λ u 1 ) (λ u 1 ) (101θ θ 22 )(λ u 1 ) 2 9 q 1 2 ρ (λ u 1) + 1 ] (1) 1 (3θ2 11 1θ 11 θ θ22) 2.

6 ZHENNING CAI AND YUWEI FAN AND RUO LI We introduce the dimensionless quantity ˆλ = (λ u 1 )/ θ, and then the equation det(λi ˆM) = 0 becomes [ ˆλ ˆλ θ θ 22 ˆλ 2 9 q 1 4 θ 2 ρθ ˆλ + 1 3θ11 2 1θ 11 θ θ 2 ] 22 3/2 1 θ 2 = 0. (17) Consider the special case θ 11 = θ 22 = θ and q 1 = 0, which implies the fluid is in its local equilibrium, all solutions of the above equation are ˆλ 1, = ±, ˆλ2,4 = ±, ˆλ3 = 0. (18) Therefore, in this case, ˆM(ŵ) has no multiple eigenvalues, thus is real diagonalizable. If (θ 11 θ 22 )/θ and q 1 /(ρθ 3/2 ) are small enough, the roots of (17) are small perturbations of (18), which are still real and separable. This shows that there is a hyperbolicity region for 1D moment system around the thermodynamic equilibrium, and the Maxwell distribution is an interior point of the hyperbolicity region. A precise depiction of the hyperbolicity region can be found in [8] Lack of hyperbolicity of 3D Grad s 13-moment system. To the best of our knowledge, there has not been any published investigation on the hyperbolicity of the full 3D Grad s system. One may take it for granted that the full 3D case is similar as the 1D case and there exists a neighbourhood of the equilibrium such that the system is hyperbolic. Unfortunately, this is not true. In this section, we are going to show that Maxwellian is on the boundary of the hyperbolicity region. The analysis below contains some tedious calculations, which are carried out by the computer algebra system Mathematica [10]. In the 3D case, Grad s 13-moment equations can also be written in the quasilinear form as w t + M k(w) w = Q(w). (19) x k Now w is a vector with 13 entries: w = (ρ, u 1, u 2, u 3, θ 11, θ 22, θ 33, θ 12, θ 13, θ 23, q 1, q 2, q 3 ) T. The expressions of the matrices M k and the operator Q can be obtained from (13). Since Grad s moment system is rotationally invariant, in order to check the hyperbolicity of (19), we only need to check the diagonalizability of M 1. As a reference, the precise form of M 1 (w) is given on page 8. When w represents the equilibrium state, which means θ 12 = θ 13 = θ 23 = q 1 = q 2 = q 3 = 0, θ 11 = θ 22 = θ 33 = θ. (20) The characteristic polynomial of M 1 (w) is det(λi M 1 ) = 1 12 (λ u 1) [(λ u 1 ) 2 7θ] 2 [(λ u 1 ) 4 2θ(λ u 1 ) 2 + 1θ 2 ]. All roots of the above polynomial are u 1, u 1 ± 7 θ, u 1 ± θ, u 1 ± θ. Thus the eigenvalues of M 1 are all real. In order to check its diagonalizability, let q(λ) = 1 2 (λ u 1)[(λ u 1 ) 2 7θ] [(λ u 1 ) 4 2θ(λ u 1 ) 2 + 1θ 2 ].

7 HYPERBOLICITY OF 13-MOMENT SYSTEMS 7 Direct verification shows q(m 1 ) = 0. According to Lemma 2.3, M 1 (w) is real diagonalizable at the equilibrium state. In order to show that the equilibrium is on the boundary of the hyperbolicity region, we consider the following case: θ 13 = θ 23 = q 1 = q 2 = q 3 = 0, θ 11 = θ 22 = θ 33 = θ. (21) When f is the following Gaussian distribution: ( ρ f = exp 1 det(2πθ) 2 CT Θ 1 C ), Θ = θ θ 12 0 θ 12 θ θ, (22) the relation (21) is satisfied. When θ 12 < θ, the matrix Θ is positive definite, and thus the distribution function (22) can be a physical configuration. Substituting (21) into (3.4) and calculating the characteristic polynomial of M 1 (w), one has ( (λ u1 ) 2 where Let det(λi M 1 ) = 1 12 (λ u 1) 3 [(λ u 1 ) 2 7θ] r ( ) ( ) r(x) = 2x 4 1x θ2 12 θ 2 x θ2 12 θ 2 10 x 28 θ2 12 θ 2. ( (λ q(λ) = (λ u 1 )[(λ u 1 ) 2 u1 ) 2 ) 7θ] r. θ Obviously q(λ) and det(λi M 1 ) share the same roots. Direct calculation of q(m 1 ) gives us that q(m 1 ) = θ2 θ12 3 (ρθe 10,4 E 10,13 ), ρ where E i,j = e i e T j, and e j is the unit vector with the j-th entry being 1. According to Corollary 1, if θ 12 0, then M 1 (w) is not diagonalizable. Actually, one may find that r(x) have at least one negative root since r( ) > 0 and r(0) < 0, and therefore M 1 (w) has eigenvalues with nonzero imaginary parts, which also violates the hyperbolic condition. The above analysis shows that when (21) and θ 12 0 holds, the hyperbolicity of (19) breaks down, no matter how small the value of θ 12 is. It turns out that there does not exist a neighbourhood of the equilibrium such that all the states in this neighbourhood lead to the hyperbolicity of Grad s 13-moment system. Without the hyperbolicity in a neighbourhood of the equilibrium, the wellposedness of the Grad s 13-moment system is not guranteed even if the phase density is extremely close to the equilibrium. This severe drawback may be the possible reason why there are hardly any positive evidences for the Grad s 13-moment system in the last decades. θ ),

8 8 ZHENNING CAI AND YUWEI FAN AND RUO LI M1(w) = u1 ρ θ11 ρ u θ12 ρ 0 u θ13 ρ 0 0 u θ u ρ 0 0 2θ u ρ θ u ρ 0 θ12 θ u ρ 0 θ13 0 θ u θ13 θ u ρ (θ θ11) 2 (θ θ 2 13) θ12θ33 θ13θ23 θθ12 θ13θ22 θ12θ23 θθ13 1q1 7q2 2q2 7q3 0 2q3 7q1 0 7q1 ρ(θ11+8θ) ρθ12 ρθ13 ρ(7θ11 4θ) 7ρθ12 7ρθ13 ρ(7θ11 4θ) ρθ12 ρθ13 0 u ρθ12 ρ(2θ θ22) ρθ u1 0 7ρθ13 ρθ23 ρ(2θ θ33) u1

9 HYPERBOLICITY OF 13-MOMENT SYSTEMS 9 4. Modified 13-moment System. The results in Section 3.4 reveal a crucial issue of Grad s original system. In order to establish the local hyperbolicity around the equilibrium state, we derive a modified 13-moment system in this section, which is hyperbolic for any states close enough to the equilibrium. The proofs will be given in detail, and the size of the hyperbolicity region will be discussed Derivation of the modified system. The modified 13-moment system is based on the following assumption of the phase density: f 13 = [ ( 1 ρ st Θ 1 C 2 CT Θ 1 C )] f G, (23) 2 where s = (s 1, s 2, s 3 ) T, and f G is a Gaussian distribution: f G = ( ρ m det(2πθ) exp 1 ) 2 CT Θ 1 C. (24) Comparing with f M, the function f G incorporates the whole temperature tensor into the exponent, and thus it can be expected that such an approximation includes more nonlinearity than (10), and is more suitable for describing anisotropic density functions. In order to meet the requirement of orthogonality, the vector s should be related to the density function by s = 1 2 C2 GCf, where C 2 G = CT Θ 1 C. For the postulate (23), the relation between s and the heat flux q is q j = 3 s (iθ ij). Similar as the derivation of Grad s 13-moment system, the new moment system can be written as φ f 13 + φ(ξ x f 13 ) = φq( f 13, t f 13 ), where φ = (1, C 1, C 2, C 3, C1, 2 C2, 2 C3, 2 C 1 C 2, C 1 C 3, C 2 C 3, CGC 2 1, CGC 2 2, CGC 2 3 ) T.

10 10 ZHENNING CAI AND YUWEI FAN AND RUO LI We reformulate the resulting system in explicit form as dρ dt + ρ u k = 0, (2a) x k du i dt + θ ik ρ + θ ik = 0, i = 1, 2, 3, (2b) ρ x k x k dθ ij u j) + 2θ k(i + ( ) θ jk) s k) s (i + θ (ij = ρθ dt x k ρ x k x k µ (θ ij δ ij θ), i, j = 1, 2, 3, (2c) ds j dt u k) θik θ l(i s j 18 ( ) θ kl) s l) x l 2ρ θik s (i s j + s (i θ jk x l x l + 1 ( ρθ ik θ ik θ jl + 2ρ θ ) jl + 2 ( ) u l) 7s (j + θ ik u k) θ jl s (i = 2 x l x l x l x Q j, j = 1, 2, 3. l (2d) In equation (2d), θ ij stands for the (i, j) entry of matrix Θ 1, and Q j = ρθ ( 71 µ 30 s j 9 10 θθ(ii s j) 1 ) 1 θii θ jk s k. The expressions of Qj are obtained by using the following properties of Maxwell molecules: C i C j Q(f, f) = ρθ (C i C j 13 ) µ C2 δ ij f, C 2 C j Q(f, f) = 2 ρθ 3 µ C2 C j f, (C i C j C k 3 ) C2 C (i δ jk) Q(f, f) = 3 ρθ (C i C j C k 3 ) 2 µ C2 C (i δ jk) f. The system (2) can also be written in a quasi-linear form: and we choose w t + M k ( w) w x k = Q( w), (2) w = (ρ, u 1, u 2, u 3, θ 11, θ 22, θ 33, θ 12, θ 13, θ 23, s 1, s 2, s 3 ) T. Since the linear space spanned by all the components of φ is rotationally invariant, the moment equations (2) are also rotationally invariant. Therefore, below we focus on the first coefficient matrix M 1 ( w) Local hyperbolicity of the modified system. Before establishing the local hypebolicity of (2), we provide a technical lemma first: Lemma 4.1. For a given symmetric positive definite matrix Θ = (θ ij ) 3 3, the inequality θ 1 11 s2 1 s T Θ 1 s (27) holds for any vector s = (s 1, s 2, s 3 ) T R 3. The equality holds if and only if there exists a constant k such that s 1 = kθ 11, s 2 = kθ 12, s 3 = kθ 13. (28) Proof. We first prove θ 12 θ 12 + θ 13 θ Let S = θ 12 θ 12 + θ 13 θ 13. Then we have θ 12 (θ 23 θ 13 θ 12 θ 33 ) + θ 13 (θ 12 θ 23 θ 22 θ 13 ) = Sdet(Θ). (29)

11 HYPERBOLICITY OF 13-MOMENT SYSTEMS 11 This equation can be considered as a quadratic equation of θ 13, and its discriminant is = (2θ 12 θ 23 ) 2 θ 22 [4θ 2 12θ 33 + Sdet(Θ)] = 4θ 2 12(θ 2 23 θ 22 θ 33 ) Sθ 22 det(θ). Since Θ is positive definite, the following inequalities hold: θ 2 23 θ 22 θ 33 < 0, θ 22 > 0, det(θ) > 0. Thus, in order that (29) is not less than zero, S 0 must hold. Moreover, if S = 0, θ 12 must also be zero, and then (29) becomes θ 22 θ 2 13 = 0, which means θ 13 = 0. Obviously when θ 12 = θ 13 = 0, one has S = 0. Therefore we finally conclude that S 0, and the equality holds if and only if θ 12 = θ 13 = 0. Now let D = s T Θ 1 s θ 1 11 s2 1, which can be written as (θ 11 θ 11 1)s θ 11 (θ 12 s 2 + θ 13 s 3 )s 1 + θ 11 (θ 22 s θ 23 s 2 s 3 + θ 33 s 2 3) = θ 11 D. (30) We consider the following two cases: If θ 11 θ 11 1 = 0, since θ 1k θ 1k = 1, one has θ 12 θ 12 + θ 13 θ 13 = 0. In this case, θ 12 = θ 13 = 0, and therefore θ 12 = θ 13θ 23 θ 12 θ 33 det(θ) = 0, θ 13 = θ 12θ 23 θ 13 θ 22 det(θ) = 0. Thus (30) becomes θ 22 s θ 23 s 2 s 3 + θ 33 s 2 3 = D, which is equivalent to s T Θ 1 s = D for s = (0, s 2, s 3 ) T. Since Θ is positive definite, Θ 1 is also positive definite. This shows that D 0, and the equality holds if and only if s 2 = s 3 = 0. In the case of D = s 2 = s 3 = θ 12 = θ 13 = 0, the relation (28) holds with k = s 1 /θ 11. If θ 11 θ , then θ 11 θ 11 1 = (θ 12 θ 12 + θ 13 θ 13 ) > 0. In this case, (30) is a quadratic equation of s 1, whose discriminant is = [2θ 11 (θ 12 s 2 + θ 13 s 3 )] 2 4θ 11 (θ 11 θ 11 1)(θ 22 s θ 23 s 2 s 3 + θ 33 s 2 3 D) = 4θ 11 (θ 12 s 3 θ 13 s 2 ) 2 /det(θ) + 4θ 11 D(θ 11 θ 11 1). In order that s 1 is real, D 0 must hold. And if D = 0, θ 12 s 3 θ 13 s 2 must be zero. Thus when D is zero, there exist a constant k such that s 2 = kθ 12, s 3 = kθ 13. (31) Substitute (31) and D = 0 into (30), it can be solved that s 1 = kθ 11. In both cases, (27) holds, and it has been demonstrated that if θ11 1 s2 1 = s T Θ 1 s, then (28) holds. It only remains to prove that θ11 1 s2 1 = s T Θ 1 s is a conclusion of (28). If (28) holds, then Therefore s T Θ 1 = (k, 0, 0), and then (k, 0, 0)Θ = k(θ 11, θ 12, θ 13 ) = s T. s T Θ 1 s = (k, 0, 0)s = k 2 θ 11 = θ 1 11 s2 1. (32) This completes the proof of the lemma. Now we claim that the modified 13-moment system (2) is locally hyperbolic around the equilibrium. Precisely, we have the following major theorem of this section: Theorem 4.2. There exists a positive constant δ > 0, such that if ρ 2 s T Θ 1 s < δ, M 1 ( w) is real diagonalizable.

12 12 ZHENNING CAI AND YUWEI FAN AND RUO LI Proof. Let η 1 := ρ 2 θ 1 11 s2 1, η 2 := ρ 2 s T Θ 1 s, ζ = λ u 1 θ11. According to Lemma 4.1, we have η 1 η 2 < δ. By direct calculation, the characteristic polynomial of M1 (w) is p(λ) := det(λi M 1 ) = ( θ 11 ) [p 11(ζ) + η 2 p 12 (ζ)][p 21 (ζ) + η 2 p 22 (ζ)], (33) where p 11 (ζ) = 2ζ 2 (ζ 2 7) 130 η 1 ζ 3 + 4η 1 (ζ 2 + 7), p 12 (ζ) = 8ζ 2, p 21 (ζ) = 2ζ 3 (2ζ 1ζ ζ 2 10) 20η 1/2 1 ζ 2 (110ζ 311ζ ζ 2 10) + 100η 1 ζ(111ζ + 447ζ 4 209ζ ) 40η 3/2 1 (18ζ + 97ζ ζ ) + 9η 2 1ζ(1ζ 2 + 3), p 22 (ζ) = 8ζ [ 2ζ 2 (23ζ 4 73ζ 2 + 3) 10 η 1 ζ(27ζ 4 + 7ζ 2 18) + 12η 1 (48ζ 2 7) ]. (34) Below we divide the proof into three cases. First case: s 1 = s 2 = s 3 = 0. In this case, η 1 = η 2 = 0, and p(λ) = ( θ 11 ) 13 ζ (ζ 2 7) 2 (ζ 4 2ζ 2 + 1). 12 Obviously the roots of p(λ) are all real. According to Lemma 2.3, we only need to prove ˆp( M 1 ) = 0 for ˆp(λ) = ( θ 11 ) 13 ζ(ζ 2 7)(ζ 4 2ζ 2 + 1). This can be verified by direct calculation. Second case: s 1 = 0 and s s2 3 > 0. In this case, η 1 = 0, while the SPD property of Θ gives η 2 > 0. The characteristic polynomial p(λ) can be simplified as where p(λ) = p 1 (ζ) = 2(ζ 2 7) + 8η 2, ζ 7812 p 1(ζ)p 2 (ζ), p 2 (ζ) = 2(ζ 2 7)(ζ 4 2ζ 2 + 1) + 8η 2 (23ζ 4 73ζ 2 + 3). When η 2 equals zero, all the roots of p 1 and p 2 are single and nonzero. Thus, when η 2 < δ for δ small enough, the roots of p 1 and p 2 are also single and nonzero. Furthermore, we claim that p 1 (ζ) and p 2 (ζ) have no common roots when δ is small enough. This can be proven following these steps: 1. Let p 1 (z) = p 1 ( z), p 2 (z) = p 2 ( z). Obviously, when δ is small enough, p 1 and p 2 are polynomials with all their roots positive. If p 1 and p 2 have no common roots, then p 1 and p 2 have no common roots. 2. The polynomial p 1 (z) is a linear function, and its only root is (17 8η 2 )/12. The value of p 2 at this point is ( ) 17 8η2 p 2 = 8η 2(112η η 2 437). (3) 12 12

13 HYPERBOLICITY OF 13-MOMENT SYSTEMS Since 0 < η 2 < δ, the value of (3) is negative if δ < ( )/288, which means p 1 and p 2 have no common roots. The above analysis shows when δ is small, p 1 (ζ)p 2 (ζ) has no multiple roots, and ζ = 0 is not a root of p 1 (ζ)p 2 (ζ). Thus, the polynomial q(ζ) := ζp 1 (ζ)p 2 (ζ) (3) has no multiple roots if δ is small. Finally, it is verified by computer algebra system ( ) M1 u 1 I q = 0, if s 1 = 0. (37) θ11 Hence, according the Lemma 2.3, the matrix M 1 is diagonalizable. Third case: s 1 0. In this case, η 1 > 0 and η 2 > 0. We first prove when δ is small, both p 11 (ζ) + η 2 p 12 (ζ) and p 21 (ζ) + η 2 p 22 (ζ) have no multiple or imaginary roots. When η 1 = η 2 = 0, p 11 (ζ) + η 2 p 12 (ζ) = 2ζ 2 (ζ 2 7), p 21 (ζ) + η 2 p 22 (ζ) = 2ζ 3 (ζ 2 7)(ζ 4 2ζ 2 + 1). (38a) (38b) Both polynomials have only real roots, and both of them have only one multiple roots ζ = 0. Thus, for small δ, if η 1 and η 2 are nonzero, then the multiple or imaginary roots must be around ζ = 0 if they exist. For p 11 (ζ) + η 2 p 12 (ζ), when δ is small, one can obtain its values at some particular points around ζ = 0: ζ = η 1 : p 11 ( η 1 ) + η 2 p 12 ( η 1 ) = 279η η 1 + 8η 1 η 2 < 0, ζ = 0: p 11 (0) + η 2 p 12 (0) = 28η 1 > 0, ζ = η 1 : p 11 ( η 1 ) + η 2 p 12 ( η 1 ) = 19η η 1 + 8η 1 η 2 < 0. This tells us that there are two distinct real roots of p 11 (ζ)+η 2 p 12 (ζ) around ζ = 0. Noting that ζ = 0 is a root of multiplicity 2 of (38a), we conclude that in the case of 0 < η 1 < δ and 0 < η 2 < δ, p 11 (ζ) + η 2 p 12 (ζ) has no multiple or imaginary roots. Similarly, for p 11 (ζ) + η 2 p 12 (ζ), when δ is small, one has ζ = η 1 : p 21 ( η 1 ) + η 2 p 22 ( η 1 ) = η 3/2 1 [494η 3 1 ( η 2 )η ( η 2 )η 1 ( η 2 )] > 0, ζ = 0: ζ = 2 η1/ η3/2 1 : ( 2 p 21 η1/ η3/2 1 4η 7/2 1 p 21 (0) + η 2 p 22 (0) = 4200η 3/2 1 < 0, ) ( 2 + η 2 p 22 η1/ η3/2 1 [ = 409η η η ( η 2 )η ( η 2 )η ( η 2 )η ( η 2 )η ( η 2 )η ( η 2 )η ( η 2 )η ( η 2 ) ] > 0, )

14 14 ZHENNING CAI AND YUWEI FAN AND RUO LI ζ = η 1 : p 21 ( η 1 ) + η 2 p 22 ( η 1 ) = η 3/2 1 [149η ( η 2 )η 2 1 ( η 2 )η 1 + ( η 2 )] < 0. This reveals that there are three distinct real roots of p 21 (ζ)+η 2 p 22 (ζ) around ζ = 0. Until now, the statement at the beginning of this paragraph has been proven. The subsequent proof is divided into two parts: 1. If η 1 = η 2, then p 11 (ζ) + η 2 p 12 (ζ) is a factor of p 21 (ζ) + η 2 p 22 (ζ), and we actually have p 11 (ζ) + η 2 p 12 (ζ) = (2ζ 3 1 η 1 ζ 2 3ζ 14 η 1 )(ζ 2 η 1 ), (39) p 21 (ζ) + η 2 p 22 (ζ) = [2ζ(ζ 4 2ζ 2 + 1) + 30 η 1 (3ζ 4 + ζ 2 + ) 192η 1 ζ] Thus we need to verify (2ζ 3 1 η 1 ζ 2 3ζ 14 η 1 )(ζ 2 η 1 ). (40) ( ) ( ) M1 u 1 I M1 u p 21 1 I + η 2 p 22 = 0. (41) θ11 θ11 According to Lemma 4.1, the condition η 1 = η 2 is equivalent to (28). Substitute (28) into the expression of M1, and (41) then can be directly verified. 2. If η 1 η 2, the resultant of p 11 + η 2 p 12 and p 21 + η 2 p 22 is calculated as res(p 11 + η 2 p 12, p 21 + η 2 p 22 ) = η 3 1(η 1 η 2 ) r(η 1, η 2 ), (42) where r(η 1, η 2 ) is r(η 1, η 2 ) = η η 2 η η η 2 2η η 2 η η η 3 2η η 2 2η η 2 η η η 4 2η η 3 2η η 2 2η η 2 η η η η η η η Evidently when δ is small, r(η 1, η 2 ) < 0. Noting that η 1 > 0 and η 1 η 2, we conclude (42) is nonzero. According to Lemma 2., p 11 (ζ) + η 2 p 12 (ζ) and p 21 (ζ) + η 2 p 22 (ζ) have no common roots. Thus, the characteristic polynomial p(λ) has no multiple roots, which gives us the diagonalizability of M1. Final conclusion. For all the three cases listed above, it has been proven that when δ is small, all the eigenvalues of M1 are real, and the matrix M 1 is diagonalizable. Thus the proof of Theorem 4.2 is completed. Theorem 4.3. There exists a positive constant δ > 0, such that if ρ 2 s T Θ 1 s < δ, the moment system (2) is hyperbolic. Proof. The hyperbolicity of the moment system (2) is equivalent to the diagonalizability of the matrix n k Mk ( w) for all unit vectors n = (n 1, n 2, n 3 ) T R 3.

15 HYPERBOLICITY OF 13-MOMENT SYSTEMS 1 The rotational invariance of (2) implies that for any unit vector n, there exists a constant square matrix R such that n k Mk ( w) = R 1 M1 (R w)r. Actually, R can be constructed as follows: 1. Construct an orthogonal matrix G = (g ij ) 3 3 such that the first row of G is (n 1, n 2, n 3 ). 2. Define the rotated moments w as where w = (ρ, u 1, u 2, u 3, θ 11, θ 22, θ 33, θ 12, θ 13, θ 23, s 1, s 2, s 3) T, ρ = ρ, u i = g ij u j, θ ij = g ik g jl θ kl, s i = g ij s j. (43) 3. The matrix R is the unique matrix such that w = R w for all w. According to Theorem 4.2, there exists a constant positive number δ such that the matrix n k Mk ( w) is diagonalizable if ρ 2 s T (Θ ) 1 s < δ, where Θ = (θ ij ) 3 3. Using (43), we have ρ 2 s T (Θ ) 1 s = ρ 2 (Gs) T (GΘG T ) 1 (Gs) = ρ 2 s T Θ 1 s. Thus the theorem is proven Quantification of the hyperbolicity region. The proof of Theorem 4.3 reveals that the maximal value of δ (denoted by δ max below) in Theorem 4.3 equals that in Theorem 4.2. Below we give a rough estimation of δ max. Let p = (p 11 + η 2 p 12 )(p 21 + η 2 p 22 ), where p 11, p 12, p 21, p 22 are defined in (34). We denote the domain on which the polynomial p has no imaginary roots to be Σ, and thus where Σ = {(η 1, η 2 ) I(η 1, η 2 ) = 0}, I(η 1, η 2 ) := max{ Im(z) z is the root of p}, 0 η 1 η 2. Since I is continuous, Σ has to be a closed region. We plot the domain Σ as the green area in Figure 1(a). The horizontal line η 2 = δ is tangent to the red curve. We have δ max δ and δ We denote the domain S to be the domain on which the polynomial p has multiple roots. According to Lemma 2.4 and Lemma 2., we have that and S = {(η 1, η 2 ) R(η 1, η 2 ) = 0}, R(η 1, η 2 ) := res( p, p ), p (ζ) = d dζ p(ζ), 0 η 1 η 2. Due to the continuity of the roots of polynomials with respect to its coefficients, we have Σ S. Figure 1(b) shows part of S. Comparing Figure 1(a) and Figure 1(b), we conclude that if 0 < η 1 < η 2 < δ, which implies that (η 1, η 2 ) is an interior point of Σ below the line η 2 = δ, then M 1 ( w) is real diagonalizable. In order to determine δ max, we have to consider two additional cases: (1) η 1 = 0, (2) η 1 = η 2 > 0. They correspond to the straight red lines in Figure 1. It can be argued as below for these cases:

16 1 ZHENNING CAI AND YUWEI FAN AND RUO LI η 2 η η 2 = δ 0.08 η 2 = δ (a) The region Σ η (b) The zero level set of R η 1 Figure 1. The x-axis stands for η 1, and the y-axis stands for η 2 For the case η 1 = 0, if η 2 = 0, the real diagonalizability of M1 has been proven. If η 2 > 0, since (37) always holds, we only need to consider whether the polynomial q(ζ) defined in (3) has multiple roots. Figure 2 gives the plots of res(q, q ) for η 2 [0, 0.1], where q (ζ) = d q(ζ). It is found that if dζ 0 < η 2 < δ < 0.1, then res(q, q ) > 0, thus q(ζ) has no multiple roots. Then M 1 is real diagonalizable. res(q, q ) η Figure 2. Plots of res(q, q ) in the case of η 1 = 0 For the case η 1 = η 2 > 0, we have to study the multiplicities of the roots of (40). Denote the polynomial (40) by q, and let q (ζ) = d q(ζ). The values of dζ res( q, q ) for η 2 [0, 0.1] are given in Figure 3. It can also be observed that when 0 < η 1 = η 2 < δ < 0.1, res( q, q ) is greater than zero, which results in the real diagonalizability of M1. As a summary, we claim that if 0 η 1 η 2 < δ, the moment system (2) is hyperbolic. Thus δ max = δ In order to give a more precise description of the size of the hyperbolicity region, we apply the Chapman-Enskog method to the modified 13-moment system (2).

17 res( q, q ) HYPERBOLICITY OF 13-MOMENT SYSTEMS η 1 = η 2 Figure 3. Plots of res( q, q ) in the case of η 1 = η 2 Apply the transformation t = εt and x = εx to (2), and then the moment system becomes w t + M k ( w) w x = 1 k ε Q( w). (44) For small ε, we formally expand w as w = w (0) + ε w (1) + ε 2 w (2) +. The Chapman-Enskog expansion fixes the leading order term w (0) to be the equilibrium part of w: ρ = ρ (0), u = u (0), Θ = θi + εθ (1) + ε 2 Θ (2) +, s = εs (1) + ε 2 s (2) +. (4) Substituting (4) into (44) and balancing the zeroth order terms on both sides of (44), one may conclude ( ) θ (1) ij = 2µ v(i 1 v k, s (1) j = 1µ θ. ρ 3 4θ x j) x k These are equivalent to the well-known Navier-Stokes and Fourier laws. According to the expansion (4), we have Θ 1 = θ 1 I ε θ 2 Θ(1) + O(ε 2 ), and thus ρ 2 s T Θ 1 s = ε 2 ρ 2 θ 1 s (1) 2 + O(ε 3 ) = 22 1 ε2 µ 2 ρ 2 θ 3 x θ 2 + O(ε 3 ). (4) For Maxwellian molecules, the viscosity µ can be related to the mean free path l mfp by πθ µ = ρ l mfp 2. Thus, (4) is simplified as ρ 2 s T Θ 1 s = 22π 32 ε2 x i ( ) 2 x θ l mfp + O(ε 3 ). θ Neglecting the high order terms, we get ρ 2 s T Θ 1 s 22π ( ) 2 x θ 32 ε2 l mfp = 22π θ 32 ( ) 2 x θ l mfp. (47) θ

18 18 ZHENNING CAI AND YUWEI FAN AND RUO LI Thus the hyperbolicity condition ρ 2 s T Θ 1 s < δ max is approximately given as 32δmax x θ < C hyp θ/l mfp, C hyp = 22π. Since δ max 0.09, we have that C hyp 0.0. Thus the temperature is allow to change around.% of its value in one mean free path in order to ensure the hyperbolicity. Consider the symmetric plane Couette flow problem. The Navier- Stokes equations together with the first-order slip boundary condition is valid only for l mfp 0.1L, where L is the distance between plates [7]. For Kn = l mfp /L, in order to satisfy the criterion (47), the ratio of the temperature in the middle of the two plates to the temperature on each plate must not exceed Kn 1 C hyp. The numerical results in [9] show that such a criterion is satisfied even for very fast plate velocities.. Conclusion. We find that for Grad s 13-moment system, the equilibrium is always on the boundary of its hyperbolicity region. A modified 13-moment system is proposed so that the local hyperbolicity around the equilibrium states can be achieved. The derivation of this new model is almost the same as the original one, except that the basis functions used in the expansions of the distribution functions are different. Obviously, this new model is far away from perfection; most of the classical criticism on Grad s 13-moment system still applies to this new model. However, due to the similarity of these two systems, the techniques developed for Grad s 13-moment system may also apply to this new model. This modified system enriches the 13-moment family, and some interesting aspects are found for this new member. REFERENCES [1] G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford: Clarendon Press, [2] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases, Third Edition, Cambridge University Press, [3] H. Grad, Note on N-dimensional Hermite polynomials, Comm. Pure Appl. Math., 2 (1949), [4] H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), [] S. Jin, L. Pareschi, and M. Slemrod, A relaxation scheme for solving the Boltzmann equation based on the Chapman-Enskog expansion, Acta Math. Appl. Sin.-E., 18 (2002), [] S. Jin and M. Slemrod, Regularization of the Burnett equations via relaxation, J. Stat. Phys., 103 (2001), [7] G. Karniadakis, A. Beskok, and N. Aluru, Microflows and Nanoflows: Fundamentals and Simulation, Interdisciplinary Applied Mathematics, vol. 29, Springer, New York, U.S.A., 200. [8] I. Müller and T. Ruggeri, Rational Extended Thermodynamics, Second Edition, Springer tracts in natural philosophy, vol. 37, Springer-Verlag, New York, [9] L. Mieussens and H. Struchtrup, Numerical comparison of Bhatnagar-Gross-Krook models with proper Prandtl number, Phys. Fluids, 1 (2004), [10] Wolfram Research, Mathematica 9, [11] H. Struchtrup and M. Torrilhon, Regularization of Grad s 13 moment equations: Derivation and linear analysis, Phys. Fluids, 1 (2003), [12] H. Struchtrup, Derivation of 13 moment equations for rarefied gas flow to second order accuracy for arbitrary interaction potentials, Multiscale Model. Simul., 3 (200),

19 HYPERBOLICITY OF 13-MOMENT SYSTEMS 19 [13] M. Torrilhon, Regularized 13-moment-equations, Rarefied Gas Dynamics: 2th International Symposium (M. S. Ivanov and A. K. Rebrov, eds.), 200, pp [14] M. Torrilhon, Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson-IV-distributions, Commun. Comput. Phys., 7 (2010), Received xxxx 20xx; revised xxxx 20xx. address: caizn@pku.edu.cn address: ywfan@pku.edu.cn address: rli@math.pku.edu.cn