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The Boltzmann Equation for Uniform Shear Flow

Abstract

The uniform shear flow for rarefied gas is governed by the time-dependent spatially homogeneous Boltzmann equation with a linear shear force. The main feature of such flow is that the temperature may increase in time due to the shearing motion that induces viscous heat, and the system strays far from equilibrium. For Maxwell molecules, we establish the unique existence, regularity, shear-rate-dependent structure and non-negativity of self-similar profiles for any small shear rate. The non-negativity is justified through the large time asymptotic stability even in spatially inhomogeneous perturbation framework, and the exponential rates of convergence are also obtained with the size proportional to the second order shear rate. This analysis supports the numerical result that the self-similar profile admits an algebraic high-velocity tail that is the key difficulty to overcome in the proof.

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Acknowledgements

RJD was partially supported by the General Research Fund (Project No. 14302817) from RGC of Hong Kong and the Direct Grant (4053397) from CUHK. SQL was supported by grants from the National Natural Science Foundation of China (contracts: 11971201 and 11731008). SQL would like to thank Department of Mathematics, CUHK for their hospitality during his visit in January–March in 2020. RJD would thank Florian Theil for introdcuing to him the problem in 2015, and also thank Alexander Bobylev for stimulating discussions on [11] during the conference “Advances in Kinetic Theory” hosted by Chongqing University in October 2019. The authors would like to thank the anonymous referees for all valuable and helpful comments on the manuscript.

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Communicated by J. Bedrossian.

Appendix

Appendix

In this section, we collect some known basic estimates which have been used in the previous sections. The following lemma can be found in [24, Lemmas 3.2, 3.3, pp.638-639], where the more general hard sphere case is proved:

Lemma 6.1

In the Maxwell molecular case, there is a constant \(\delta _0>0\) such that

$$\begin{aligned} \langle Lf,f\rangle =\langle L{\mathbf {P}}_1f,{\mathbf {P}}_1f\rangle \geqq \delta _0\Vert {\mathbf {P}}_1f\Vert ^2. \end{aligned}$$

Moreover, for \(\gamma >0\) and \(l\geqq 0\),

$$\begin{aligned} \langle w^2_l\partial _v^\gamma L f,\partial _v^\gamma f\rangle \geqq \delta _0\Vert w_l\partial _v^\gamma f\Vert ^2-C\Vert f\Vert ^2. \end{aligned}$$

The following lemma is concerned with the integral operator K given by (2.2), and its proof in case of the hard sphere model has been given by [25, Lemma 3, pp.727].

Lemma 6.2

Let K be defined as (2.2), then it holds that

$$\begin{aligned} Kf(v)=\int _{{\mathbb {R}}^3}{\mathbf {k}}(v,v_*)f(v_*)\,\mathrm{d}v_*\end{aligned}$$

with

$$\begin{aligned} |{\mathbf {k}}(v,v_*)|\leqq C\{1+|v-v_*|^{-2}\}e^{- \frac{1}{8}|v-v_*|^{2}-\frac{1}{8}\frac{\left| |v|^{2} -|v_*|^{2}\right| ^{2}}{|v-v_*|^{2}}}. \end{aligned}$$

Moreover, let \( {\mathbf {k}}_w(v,v_*)=w_{l}(v){\mathbf {k}}(v,v_*)w_{-l}(v_*) \) with \(l\geqq 0\), then it also holds that

$$\begin{aligned} \int _{{\mathbb {R}}^3} {\mathbf {k}}_w(v,v_*)e^{\frac{\varepsilon |v-v_*|^2}{8}}\mathrm{d}v_*\leqq \frac{C}{1+|v|}, \end{aligned}$$

for \(\varepsilon \geqq 0\) small enough.

For the velocity weighted derivative estimates on the nonlinear operator \(\Gamma \), one has

Lemma 6.3

In the Maxwell molecular case, it holds that

$$\begin{aligned} \Vert w_l\partial _v^\gamma \Gamma (f,g)\Vert _{L_v^2}\leqq C\sum \limits _{\gamma _1\leqq \gamma }\Vert w_l\partial _v^{\gamma _1}f\Vert _{L_v^2}\Vert w_l\partial _v^{\gamma -\gamma _1}g\Vert _{L_v^2}, \end{aligned}$$
(6.1)

and

$$\begin{aligned} \Vert w_l\partial _v^\gamma \Gamma (f,g)\Vert _{L^\infty }\leqq C\sum \limits _{\gamma _1\leqq \gamma }\Vert w_l\partial _v^{\gamma _1}f\Vert _{L^\infty }\Vert w_l \partial _v^{\gamma -\gamma _1}g\Vert _{L^\infty }, \end{aligned}$$
(6.2)

for any multiple index \(\gamma \) and any \(l\geqq 0\).

Proof

The proof of (6.1) and (6.2) is similar as that of [26, Lemma 2.3, pp.1111] and [25, Lemma 5, pp.730], respectively. Thus we omit the details for brevity. \(\quad \square \)

The following Lemma on the velocity weighted derivative estimates for the original Boltzmann equation Q can be verified by using the parallel argument as obtaining [1, Proposition 3.1, pp.397] where the hard potential case and the case \(|\gamma |=0\) were proved.

Lemma 6.4

In the Maxwell molecular case, for \(l>\frac{3}{2}\) and \(|\gamma |\geqq 0\), it holds that

$$\begin{aligned} \Vert w_{l} \partial _v^\gamma Q(F_1,F_2)\Vert _{L^\infty }\leqq C\sum \limits _{\gamma _1\leqq \gamma }\Vert w_{l} \partial _v^{\gamma -\gamma _1} F_1\Vert _{L^\infty }\Vert w_{l} \partial _v^{\gamma _1}F_2\Vert _{L^\infty }. \end{aligned}$$

We now give the following two useful results concerning the second momentum invariant property of the linearized operator L in the case of Maxwell molecules. The first one is due to [28, Proposition 4.10, pp.804].

Lemma 6.5

Let \(W_{ij}(v)\) be quadratic functions in the form of \(W_{ij}(v)=v_iv_j\) \((1\leqq i,j\leqq 3)\) and define

$$\begin{aligned} T_{ij}=\frac{1}{2}\int _{{\mathbb {S}}^2}\mathrm{d}\omega \,B_0(\cos \theta )\left[ W_{i,j}(v')+W_{i,j}(v_*')-W_{i,j}(v)-W_{i,j}(v_*)\right] , \end{aligned}$$
(6.3)

where \((v,v_*)\) and \((v',v_*')\) satisfies (1.3). Then it holds that

$$\begin{aligned} T_{ij}=-b_0\left[ (v-v_*)_i(v-v_*)_j-\frac{\delta _{ij}}{3}|v-v_*|^2\right] , \end{aligned}$$
(6.4)

with \(b_0\) given in (1.15).

Based on the above nice lemma, we can obtain

Lemma 6.6

Let L be defined as (2.1), then it holds that for all \(1\leqq i,j\leqq 3\),

$$\begin{aligned} L(v_iv_j\mu ^{1/2})=2b_0\left( v_iv_j-\frac{\delta _{ij}}{3}|v|^2\right) \mu ^{1/2}. \end{aligned}$$
(6.5)

Proof

For \(f=\mu ^{1/2}W\) with a general function \(W=W(v)\), one has

$$\begin{aligned} Lf=-\mu ^{1/2}\int \mu _*\,\mathrm{d}v_*\int \mathrm{d}\omega \,B_0(\cos \theta ) [W'+W_*'-W-W_*]. \end{aligned}$$

In particular, letting \(W=W_{ij}(v)=v_iv_j\) and applying Lemma 6.5, we have

$$\begin{aligned} L(\mu ^{1/2}W_{ij})=- 2\mu ^{1/2} \int \mu _*T_{ij} \,\mathrm{d}v_*, \end{aligned}$$
(6.6)

where \(T_{ij}\) is given by (6.3). Plugging (6.4) into (6.6), one sees that (6.5) is valid. This completes the proof of Lemma 6.6. \(\quad \square \)

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Duan, R., Liu, S. The Boltzmann Equation for Uniform Shear Flow. Arch Rational Mech Anal 242, 1947–2002 (2021). https://doi.org/10.1007/s00205-021-01717-5

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