Abstract
In this paper, we establish the existence of weak solutions to the ellipsoidal BGK model (ES-BGK model) of the Boltzmann equation with the correct Prandtl number, which corresponds to the case when the Knudsen parameter is \(-1/2\).
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1 Introduction
This paper studies the global in time existence of weak solutions to the Cauchy problem of the ES-BGK model:
in the critical case \((\nu =-1/2)\). The particle distribution function f(x, v, t) is the number density of the molecules on the position \(x\in \mathbb {R}^{3}\), with the velocity \(v\in \mathbb {R}^{3}\) at time \(t \ge 0\). The Knudsen parameter \(\nu \) is chosen in the range \(-1/2\le \nu <1\), and \(A_{\nu }=1/(1-\nu )\). The non-isotropic Gaussian \(\mathcal {M}_{\nu }(f)\) parametrized by \(\nu \) is defined by
Here the local density \(\rho \), momentum U, temperature T and stress tensor \(\Theta \) are defined through the following relations:
Note that elements of \(\Theta \) are given by \((1\le i,j \le 3)\)
The temperature tensor \(\mathcal {T}_{\nu }\) is given as a linear combination of the temperature and the stress tensor:
where Id is the \(3\times 3\) identity matrix. We note that on (x, t) where \(\rho =0\), \({\mathcal M}_{\nu }(f)\) is defined to be zero. The range of \(\nu \) is restricted to \(1/2\le \nu <1\) since it is the minimum condition that guarantees the non-negative definiteness of the temperature tensor at least at the formal level [2]. We also mention that the horizontal cross-section of the non-isotropic Gaussian \(\mathcal {M}_{\nu }(f)\) is an ellipsoid, whereas the horizontal-cross section of the usual Maxwellian is a sphere. This is why the model is called the ellipsoidal BGK model.
A direct computation shows that the ellipsoidal Gaussian satisfies
which leads to the conservation laws of mass, momentum and energy:
The celebrated H-theorem was verified by Andries et al [2]:
The Boltzmann equation is the fundamental model for the description of gases at the mesoscopic level. In practice, the BGK model [4] is widely used in place of the Boltzmann equation due to its reliable performance in numerical simulations at much lower computational costs. But the compressible Navier-Stokes limit of the original BGK model shows that the Prandtl number—The ratio between the heat conductivity and the viscosity—is not computed correctly. Holway managed this problem by introducing a free parameter \(\nu \in [-1/2,1)\) and generalizing the local Maxwellian into a non-isotropic Gaussian [13]. When \(\nu =0\), (1.1) reduces to the original BGK model [4] and \(\nu =-1/2\) is the choice that yields the correct Prantl number. The ES-BGK model, however, was not employed popularly in the community since the H-theorem was not known. The H-theorem was verified later in [2], and the model got popularized [1, 10,11,12, 15, 18, 23]. To motivate the current work, we briefly review the results that are directly relevant to this work. Brull et al. derived ES-BGK model systematically using an entropy minimization argument [5]. The entropy production estimate for ES-BGK model was obtained in [22] for the non-critical case \(-1/2<\nu <1\) and in [14] for the critical case \(\nu =-1/2\). The weak solutions and the unique mild solution in the non-critical case, were established in [16], and [8, 19, 20] respectively. The existence of classical solutions near-equilibrium was studied in [21] for \(-1/2\le \nu <1\). The results on the stationary solution for the ES-BGK in a bounded interval can be found in [3] for the non-critical case and in [6] for the critical case.
All in all, the existence of the ES-BGK model in the non-critical case has been rather thoroughly studied, while many problems remain open for the critical case. One of the main reasons is that, in the non-critical case \((-1/2<\nu <1)\), the temperature tensor enjoys the following equivalence type estimate [6, 19, 21]:
Therefore, many estimates of the temperature tensor can be reduced to similar estimates of the local temperature. In the critical case \(\nu =-1/2\), however, such estimate breaks down, and the temperature tensor has to be treated with more care. Especially, the existence of weak solutions for (1.1) in the critical case \((\nu =-1/2)\) has not been addressed, which is the main purpose of this work. In this regard, our main result is as follows:
Theorem 1.1
Let \(\nu =-1/2\). Suppose that \(f_0(x,v)\ge 0\) satisfies
Then, for any final time \(T^f\)there exists a non-negative weak solution \(f(x,v,t)\in L^1([0,T^f],\mathbb {R}^3\times \mathbb {R}^3)\) to (1.1):
for every \(\phi \in C^1_c(\mathbb {R}^3\times \mathbb {R}^3\times \mathbb {R}^+)\) with \(\phi (x,v,T^f)=0\). Moreover, f satisfies
the conservation laws:
and the entropy dissipation \((t_2\ge t_1\ge 0)\):
2 Proof of Theorem 1.1
2.1 Approximate problem
For \(n=1,2,\cdots \), we set up our approximate problem of (1.1) by
where \(f^{n}_0\) is the regularized initial data:
with m(x, v) is defined by \((q>5)\)
and \(\mathcal {M}_{-1/2+1/n}(f_n)\) corresponds to the non-isotropic Gaussian defined in (1.2) with \(\nu =-1/2+1/n\):
where \(\rho _n\), \(U_n\), \(T_n\) and \(\Theta _{n}\) are the macroscopic fields constructed from the particle distribution function \(f_n\) through the relation (1.3), and \(\mathcal {T}_{-1/2+1/n,n}\) is the temperature tensor constructed from \(f_n\) in the case \(\nu =-1/2+1/n\):
We note that the approximate equation (2.1) corresponds to the ES-BGK model with non-critical Prandtl parameter \((-1/2<\nu <1)\), whose existence theory is considered in [16]:
Proposition 2.1
Let \(T^f\) be any final time. For each \(n=1,2,3,\cdots \), there exists a global weak solution \(f_n(x,v,t)\ge 0\) to (2.1):
for every \(\phi \in C^1_c(\mathbb {R}^3\times \mathbb {R}^3\times \mathbb {R}^+)\) with \(\phi (x,v,T^f)=0\). Moreover
-
1.
\(f_n\) satisfies
$$\begin{aligned} \int ^{T^f}_0\int _{\mathbb {R}^{6}}(1+|v|^2+|x|^2+|\ln f_n|)f_n dxdvdt < C, \end{aligned}$$for some \(C>0\) independent of n.
-
2.
The conservation laws hold:
$$\begin{aligned} \int _{\mathbb {R}^{6}}f_n(t)\left( \begin{array}{c}1\\ v\\ |v|^2\end{array}\right) dxdv =\int _{\mathbb {R}^{6}}f_{0,n}\left( \begin{array}{c}1\\ v\\ |v|^2\end{array}\right) dxdv. \end{aligned}$$ -
3.
\(f_n\) satisfies the entropy dissipation:
$$\begin{aligned} \int _{\mathbb {R}^{6}}f_n(t_2)\ln f_n(t_2) dxdv \le \int _{\mathbb {R}^{6}}f_n(t_1)\ln f_n(t_1) dxdv.\quad (t_2\ge t_1) \end{aligned}$$ -
4.
For any compact set \(K_x\subseteq \mathbb {R}^3_x\), \(f_n\) satisfies the following moment estimate:
$$\begin{aligned} \int _{0}^{T^f} \int _{K_{x}}\int _{\mathbb {R}^3}|v|^{3}f_{n}(x,v,t)dvdxdt \le C_{K_{x}}. \end{aligned}$$ -
5.
\(\mathcal {T}_{-1/2+1/n,n}\) is strictly positive definite:
$$\begin{aligned} \kappa ^{\top } \mathcal {T}_{-1/2+1/n,n}(x,t)\kappa \ge C_{T^f,f_{0,n},n}{(1+|x|^2)^{-q/2}}>0, \text{ for } \text{ any } \kappa \in \mathbb {S}^2. \end{aligned}$$
Remark 2.1
(1) The 3rd moment is established by Perthame in [17]. (2) The strictly positive definiteness in (5) holds due to the fact that the regularized initial data \(f_{0,n}\) has a strict lower bound. See Theorem 2.1. in [16].
The following estimate is also crucially used for the weak \(L^1\) compactness of \(\mathcal {M}_{-1/2+1/n}\).
2.2 Weak compactness of \(f_n\) and \(\mathcal {M}_{-1/2+1/n}(f_n)\)
We deduce from Proposition 2.1 and Dunford-Pettis theorem [7, 9] that there exists \(f\in L^1\) such that \(f_n\), \(f_n v\) converge to f, fv weakly \(L^1(\mathbb {R}^3 \times \mathbb {R}^3 \times [0,T^f] )\). This, combined with the velocity averaging lemma gives
Similarly, but this time combined with Proposition 2.1 (4), it can be shown that
in \(L^1([0,T^f],K_x \times \mathbb {R}^3)\), so that
in \(L^1([0,T^f],K_x \times \mathbb {R}^3)\). Therefore, we have almost everywhere convergence of macroscopic fields on a set where \(\rho \) does not vanish:
where \(\mathbb {E}\) is defined by
On the other hand, the weak compactness of \(\mathcal {M}_{-1/2+1/n}(f_n)\) in \(L^1((0,T^f)\times \mathbb {R}^3\times \mathbb {R}^3)\) follows from the following inequality established in Lemma 2.3 of [16] with a \(C>0\) independent of n:
Therefore, we can find \(M\in L^1([0,T^f],\mathbb {R}^3\times \mathbb {R}^3)\) such that \(\mathcal {M}_{-1/2+1/n}\) converges weakly in \(L_1\) to M as \(n \rightarrow \infty \).
2.3 Conclusion of the proof
It remains to check that
For this, we define
and consider (Recall that \(\mathbb {E}\) is defined in (2.3).)
Below, we consider each integrals separately to show that M coincides with \(\mathcal {M}_{-1/2}\) on each subset of \(\mathbb {R}^3 \times \mathbb {R}^3\).
\(\bullet \) \(I_1\): Since \(\rho \ne 0\), we find from (2.2) that \(\mathcal {M}_{-1/2+1/n}(\rho _n,U_n,{\mathcal {T}}_{-1/2+1/n,n})\) converges almost everywhere to \(\mathcal {M}_{-1/2}(\rho ,U,\mathcal {T}_{-1/2})\). Therefore, using Fatou’s Lemma, we get
But we have from the definition of M that
This yields
To show the reverse inequality, we choose \(\phi =1\) and observe from the definition of M that
Since \(\mathcal {T}_{-1/2+1/n}\) is strictly positive definite by Proposition 2.1 (5), we can take the change of variable:
to compute
The last line comes from (2.2). Now, since \(\mathcal {T}_{-1/2}(x,t)\) is also strictly positive definite on \(\mathbb {A}\). We can take the change of variable:
to get
In summary, we have on \(\mathbb {A}\cap \mathbb {E}\)
From (2.4) and (2.5), we conclude that
almost everywhere on \(\mathbb {A}\cap \mathbb {E}\).
\(\bullet \) \(I_2\): Using the same argument of case \(I_1\), we find
Therefore, \(M=0=\mathcal {M}_{-1/2}(\rho ,U,\mathcal {T}_{-1/2})\).
\(\bullet \) \(I_3\) : Since \(\rho =0\), we have \(\mathcal {M}_{-1/2}(f)=0 \) by definition. Therefore, by the Fatou’s lemma and the fact that \(\mathcal {M}_{\nu }(f_n)\) converges in weak \(L^1\) to M, we have
On the other hand, fixing \(\phi \) to 1 and proceeding as in the previous case, we get
\(\bullet \) \(I_4\): \((x,t)\in \mathbb {A}^c\) means that there exists a non-zero vector \(k(x,t)\in \mathbb {R}^3\) such that
We can find through an explicit computation using
Since \((x,t)\in \mathbb {A}\), the statement \(k^{\top } \mathcal {T}_{-1/2}k=0\), is equivalent to \( k^{\top }\left\{ \rho \mathcal {T}_{-1/2}\right\} k=0\). But
Recalling
One finds that
on \((t,x,v)\in \mathbb {A}^c\cap \mathbb {E}\times \mathbb {R}^3_v\). If f is identically zero on the set, we are done. If not, there exists a measurable set B of strictly positive measure such that
Therefore,
on B, which is possible only when \(v-U(x,t)\) and k(x, t) are parallel on B. Combining the conclusion This is contradiction since k does not depend on v. From this, we conclude that
Therefore, we have desired result from the same argument as in the case of \(I_2\).
Combining the arguments above, we conclude that \(M=\mathcal {M}_{-1/2} \text{ on } \mathbb {R}^3\times \mathbb {R}^3\), and the proof of main theorem is completed.
Data Availability Statement
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Andries, P., Bourgat, J.-F., Le Tallec, P., Perthame, B.: Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases. Comput. Methods Appl. Mech. Engrg. 191(31), 3369–3390 (2002)
Andries, P., Le Tallec, P., Perlat, J.-P., Perthame, B.: The Gaussian-BGK model of Boltzmann equation with small Pranl number. Eur. J. Mech. B. Fluids 19(6), 813–830 (2000)
Bang, J., Yun, S.-B.: Stationary solution for the ellipsoidal BGK model in slab. J. Differ. Equ. 261(10), 5803–5828 (2016)
Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. Small amplitude process in charged and neutral one-component systems. Phys. Rev. 94, 511–525 (1954)
Brull, S., Schneider, J.: A new approach for the ellipsoidal statistical model. Contin. Mech. Thermodyn. 20(2), 63–74 (2008)
Brull, S. and Yun, S.-B.: Stationary flows of the ES-BGK model with the correct Prandtl number. Submitted
Cercignani, C., Illner, R., and Pulvirenti, M.: The mathematical theory of dilute gases. Applied Mathematical Sciences, 106. Springer-Verlag, New York, 1994. viii+347 pp
Chen, Z.: Smooth solutions to the BGK equation and the ES-BGK equation with infinite energy. J. Differ. Equ. 265(1), 389–416 (2018)
Dunford, N., Schwartz, J.T.: Linear operators. Part I General theory. A Wiley-Interscience Publication. John Wiley & Sons Inc, New York (1988)
Filbet, F., Jin, S.: An asymptotic preserving scheme for the ES-BGK model of the Boltzmann equation. J. Sci. Comput. 46(2), 204–224 (2011)
Filbet, F., Russo, G.: Semilagrangian schemes applied to moving boundary problems for the BGK model of rarefied gas dynamics. Kinet. Relat. Models 2(1), 231–250 (2009)
Galli, M.A., Torczynski, R.: Investigation of the ellipsoidal-statistical Bhatnagar-Gross-Krook kinetic model applied to gas-phase transport of heat and tangential momentum between parallel walls. Phys. Fluids 23, 030601 (2011)
Holway, L.H.: Kinetic theory of shock structure using and ellipsoidal distribution function. Rarefied Gas Dynamics, Vol. I (Proc. Fourth Internat. Sympos., Univ. Toronto, : Academic Press. N. Y. 1966, 193–215 (1964)
Kim, D., Lee, M.-S., and Yun, S.-B.: Entropy production estimate for the ES-BGK model with the correct Prandtl number. Submitted. Available at arXiv:2104.14328
Meng, J., Wu, L., Reese, J.M., Zhang, Y.: Assessment of the ellipsoidal-statistical Bhatnagar-Gross-Krook model for force-driven Poiseuille flows. J. Comput. Phys. 251, 383–395 (2013)
Park, S.J., Yun, S.-B.: Cauchy problem for the ellipsoidal-BGK model of the Boltzmann equation. J. Math. Phys. 57(8), 081512 (2016)
Perthame, B.: Global existence to the BGK model of Boltzmann equation. J. Differ. Equ. 82(1), 191–205 (1989)
Russo, G., Yun, S.-B.: Convergence of a semi-Lagrangian scheme for the ellipsoidal BGK model of the Boltzmann equation. SIAM J. Numer. Anal. 56(6), 3580–3610 (2018)
Yun, S.-B.: Classical solutions for the ellipsoidal BGK model with fixed collision frequency. J. Differ. Equ. 259(11), 6009–6037 (2015)
Yun, S.-B.: Ellipsoidal BGK model for polyatomic molecules near Maxwellians: a dichotomy in the dissipation estimate. J. Differ. Equ. 266(9), 5566–5614 (2019)
Yun, S.-B.: Ellipsoidal BGK model near a global Maxwellian. SIAM J. Math. Anal. 47(3), 2324–2354 (2015)
Yun, S.-B.: Entropy production for ellipsoidal BGK model of the Boltzmann equation. Kinet. Relat. Models. 9(3), 605–619 (2016)
Zheng, Y., Struchtrup, H.: Ellipsoidal statistical Bhatnagar-Gross-Krook model with velocity dependent collision frequency. Phys. Fluids 17, 127103 (2005)
Acknowledgements
Seok-Bae Yun is supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1801-02.
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This article is part of the topical collection “T.C.: Kinetic Theory” edited by Seung-Yeal Ha, Marie-Therese Wolfram, Jose Carrillo and Jingwei Hu.
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Son, Sj., Yun, SB. Cauchy problem for the ES-BGK model with the correct Prandtl number. Partial Differ. Equ. Appl. 3, 41 (2022). https://doi.org/10.1007/s42985-022-00175-2
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DOI: https://doi.org/10.1007/s42985-022-00175-2
Keywords
- BGK model
- Ellipsoidal BGK model
- Boltzmann equation
- Kinetic theory of gases
- Cauchy problem
- Correct Prandtl number