On Hyperbolicity of 13-Moment System

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.

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. Definition 2.5. Let p(z) and q(z) be two polynomials defined as p(z) = p 0 + p 1 z + p 2 z 2 + · · · + p m z m , q(z) = q 0 + q 1 z + q 2 z 2 + · · · + q n z n . (2) The (m + n) × (m + n) matrix p m · · · · · · p 0 . . . . . .
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: res(p, q) := det (Syl(p, q)) .
The following lemma is to be used to tell if two polynomials have common roots. 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 13moment 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 The right hand side Q(f, f ) describes the interaction between particles: 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.
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 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: Following the conventional style, we denote 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 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 : where f M is the Maxwellian, defined as 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 where φ = (1, C 1 , C 2 , C 3 , By simplifications of the expression obtained after the integrations, the above system can be explicitly given by Here, 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.
In order to check the diagonalizability ofM(ŵ), we calculate its characteristic polynomial as (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 arê Therefore, in this case,M(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 [9].
3.4. 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 Now w is a vector with 13 entries: 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 The characteristic polynomial of All roots of the above polynomial are Thus the eigenvalues of M 1 are all real. In order to check its diagonalizability, let 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: When f is the following Gaussian distribution: 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 Obviously q(λ) and det(λI − M 1 ) share the same roots. Direct calculation of q(M 1 ) gives us that 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.
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.

4.1.
Derivation of the modified system. The modified 13-moment system is based on the following assumption of the phase density: where s = (s 1 , s 2 , s 3 ) T , and f G is a Gaussian distribution: 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 For the postulate (23), the relation between s and the heat flux q is Similar as the derivation of Grad's 13-moment system, the new moment system can be written as
Since the linear space spanned by all the components ofφ is rotationally invariant, the moment equations (25) are also rotationally invariant. Therefore, below we focus on the first coefficient matrixM 1 (w).
Therefore s T Θ −1 = (k, 0, 0), and then This completes the proof of the lemma. Now we claim that the modified 13-moment system (25) is locally hyperbolic around the equilibrium. Precisely, we have the following major theorem of this section: Proof. Let According to Lemma 4.1, we have η 1 η 2 < δ. By direct calculation, the characteristic polynomial ofM 1 (w) is where .
This can be verified by direct calculation. Second case: s 1 = 0 and s 2 2 + s 2 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 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: Obviously, when δ is small enough,p 1 andp 2 are polynomials with all their roots positive. Ifp 1 andp 2 have no common roots, then p 1 and p 2 have no common roots. 2. The polynomialp 1 (z) is a linear function, and its only root is (175 − 8η 2 )/125.
Final conclusion. For all the three cases listed above, it has been proven that when δ is small, all the eigenvalues ofM 1 are real, and the matrixM 1 is diagonalizable. Thus the proof of Theorem 4.2 is completed. Proof. The hyperbolicity of the moment system (25) is equivalent to the diagonalizability of the matrix n kMk (w) for all unit vectors n = (n 1 , n 2 , n 3 ) T ∈ R 3 .
Since δ max ≈ 0.095, we have that C hyp ≈ 0.065. Thus the temperature is allow to change around 6.5% 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 [8] show that such a criterion is satisfied even for very fast plate velocities.

5.
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.