Effects of Soft Interaction and Non-isothermal Boundary Upon Long-Time Dynamics of Rarefied Gas

Abstract

In the paper, assuming that the motion of rarefied gases in a bounded domain is governed by the angular cutoff Boltzmann equation with diffuse reflection boundary, we study the effects of both soft intermolecular interaction and non-isothermal wall temperature upon the long-time dynamics of solutions to the corresponding initial boundary value problem. Specifically, we are devoted to proving the existence and dynamical stability of stationary solutions whenever the boundary temperature has suitably small variations around a positive constant. For the proof of existence, we introduce a new mild formulation of solutions to the steady boundary-value problem along the speeded backward bicharacteristic, and develop the uniform estimates on approximate solutions in both $L^2$ and $L^\infty$. Such mild formulation proves to be useful for treating the steady problem with soft potentials even over unbounded domains. In showing the dynamical stability, a new point is that we can obtain the sub-exponential time-decay rate in $L^\infty$ without losing any velocity weight, which is actually quite different from the classical results, such as those in Caflisch (Commun Math Phys 74:97–109, 1980) and Strain and Guo (Arch Ration Mech Anal 187:287–339, 2008), for the torus domain and essentially due to the diffuse reflection boundary and the boundedness of the domain.

Appendix

1.1 An Iteration Lemma

Lemma 6.1

Consider a sequence $\{a_i\}_{i=0}^\infty$ with each $a_i\geqq 0$. For any fixed $k\in {\mathbb {N}}_+$, we denote

$$\begin{aligned} A_i^k=\max \{a_i, a_{i+1},\ldots , a_{i+k}\}. \end{aligned}$$

(1) Assume $D\geqq 0$. If $a_{i+1+k}\leqq \frac{1}{8} A_i^{k}+D$ for $i=0,1,\ldots$, then it holds that

$$\begin{aligned} A_i^k\leqq \left( \frac{1}{8}\right) ^{\left[ \frac{i}{k+1}\right] }\cdot \max \{A_0^k, \ A_1^k, \cdots , \ A_k^k \}+\frac{8+k}{7} D,\quad \text{ for }\quad i\geqq k+1. \end{aligned}$$

(6.1)

(2) Let $0\leqq \eta <1$ with $\eta ^{k+1}\geqq \frac{1}{4}$. If $a_{i+1+k}\leqq \frac{1}{8} A_i^{k}+C_k \cdot \eta ^{i+k+1}$ for $i=0,1,\ldots$, then it holds that

$$\begin{aligned} A_i^k\leqq \left( \frac{1}{8}\right) ^{\left[ \frac{i}{k+1}\right] }\cdot \max \{A_0^k, \ A_1^k, \ldots , \ A_k^k \}+2C_k\frac{8+k}{7} \eta ^{i+k},\quad \text{ for }\quad i\geqq k+1. \end{aligned}$$

(6.2)

Proof

We first show (6.1). By iteration in $i\geqq 0$, we obtain that

$$\begin{aligned} a_{i+1+2k}&\leqq \frac{1}{8} A_{i+k}^k+D=\frac{1}{8} \max \{a_{i+2k}, a_{i+2k-1},\ldots ,a_{i+k}\}+D\nonumber \\&\leqq \frac{1}{8}\max \left\{ a_{i+2k}, \ A_{i+k-1}^k\right\} +D\leqq \frac{1}{8}\max \left\{ \frac{1}{8} A_{i+k-1}^k+D, \ A_{i+k-1}^k\right\} +D\nonumber \\&\leqq \frac{1}{8} A_{i+k-1}^k+\left( 1+\frac{1}{8}\right) D \cdots \leqq \frac{1}{8} A_{i}^k+\left( 1+\frac{k}{8}\right) D. \end{aligned}$$

(6.3)

Similarly, for all $j=0,1,\ldots ,k-1$, we also have

$$\begin{aligned} a_{i+1+j+k}\leqq \frac{1}{8} A_{i}^k+\frac{8+k}{8}D. \end{aligned}$$

(6.4)

Therefore, for $1\leqq \frac{i}{k+1}\in {\mathbb {N}}_+$, it follows from (6.3) and (6.4) that

$$\begin{aligned} A_{i+1+k}^k&=\max \big \{a_{i+1+2k}, \ a_{i+2k},\ \cdots ,\ a_{i+1+k}\big \}\nonumber \\&\leqq \frac{1}{8} A_{i}^k+\frac{8+k}{8}D\leqq \left( \frac{1}{8}\right) ^2A_{i-2(k+1)}^k+\frac{1}{8} \frac{8+k}{8}D+\frac{8+k}{8}D\nonumber \\&=\left( \frac{1}{8}\right) ^2A_{i-2(k+1)}^k+\frac{8+k}{8}\left( 1+\frac{1}{8}\right) D=\cdots \nonumber \\&\leqq \left( \frac{1}{8}\right) ^{\frac{i}{k+1}}A_0^k+\frac{8+k}{8}\left( 1+\frac{1}{8}+\big (\frac{1}{8}\big )^2+\cdots \right) D\nonumber \\&\leqq \left( \frac{1}{8}\right) ^{\frac{i}{k+1}}A_0^k+\frac{8+k}{7}D. \end{aligned}$$

(6.5)

If $\frac{i}{k+1}\notin {\mathbb {N}}_+$ and $i=(k+1) \left[ \frac{i}{k+1}\right] +j$ for some $1\leqq j\leqq k$, then by similar arguments we have

$$\begin{aligned} A_{i+1+k}^k&\leqq \left( \frac{1}{8}\right) ^{\left[ \frac{i}{k+1}\right] +1} A_j^k+\frac{8+k}{7}D. \end{aligned}$$

(6.6)

Hence, from (6.5) and (6.6), we complete the proof of (6.1).

It remains to show (6.2). Noting $\eta <1$ and by similar arguments as in (6.3) and (6.4), we can get

$$\begin{aligned} a_{i+j+k+1}\leqq \frac{1}{8} A_{i}^k +C_k \left( 1+\frac{k}{8}\right) \eta ^{i+k+1},\ \text{ for } \ 0\leqq j\leqq k. \end{aligned}$$

(6.7)

Hence, for $1\leqq \frac{i}{k+1}\in {\mathbb {N}}_+$, noting $\frac{1}{8}\cdot \eta ^{-k-1}\leqq \frac{1}{2}$ and using (6.7), then we have

$$\begin{aligned}&A^k_{i+k+1}=\max \{a_{i+2k+1}, \cdots , a_{i+k+1} \}\leqq \frac{1}{8} A_{i}^k +C_k \left( 1+\frac{k}{8}\right) \eta ^{i+k+1}\nonumber \\&\quad \leqq \cdots \leqq \left( \frac{1}{8}\right) ^{\frac{i}{k+1}} A_0^k+ C_k \left( 1+\frac{k}{8}\right) \eta ^{i+k+1}\cdot \left\{ 1+\frac{1}{8} \eta ^{-k-1}+\left( \frac{1}{8} \eta ^{-k-1}\right) ^2+\cdots \right\} \nonumber \\&\quad \leqq \left( \frac{1}{8}\right) ^{\frac{i}{k+1}} A_0^k+2C_k \left( 1+\frac{k}{8}\right) \eta ^{i+k+1}. \end{aligned}$$

(6.8)

If $\frac{i}{k+1}\notin {\mathbb {N}}_+$ and $i=(k+1) \left[ \frac{i}{k+1}\right] +j$ for some $1\leqq j\leqq k$, then by similar arguments as above, we have

$$\begin{aligned} A^k_{i+k+1} \leqq \left( \frac{1}{8}\right) ^{\left[ \frac{i}{k+1}\right] +1} A_j^k+2C_k \left( 1+\frac{k}{8}\right) \eta ^{i+k+1}. \end{aligned}$$

(6.9)

Thus we prove (6.2) from (6.8) and (6.9). Therefore the proof of lemma 6.1 is complete. $\quad \square$

1.2 Local-in-Time Existence

Proposition 6.2

Let w(v) be the weight function defined in (1.9). Assume

$$\begin{aligned} |\theta -\theta _0|_{L^{\infty }(\partial \Omega )}=\delta \ll 1,\quad F_0(x,v)=F_*(x,v)+\mu ^{\frac{1}{2}}(v)f_0(x,v)\geqq 0 \end{aligned}$$

and $\Vert wf_0\Vert _{L^\infty }:=M_{0}<\infty .$ Then there exists a positive time

$$\begin{aligned} {\hat{t}}:=\bigg [{\hat{C}}\bigg (1+M_{0}\bigg )\bigg ]^{-1} \end{aligned}$$

such that the IBVP (1.1), (1.5) and (1.13) has a unique nonnegative solution

$$\begin{aligned} F(t,x,v)=F_*(x,v)+\mu ^{\frac{1}{2}}(v)f(t,x,v)\geqq 0 \end{aligned}$$

in $[0,{\hat{t}}]$ satisfying

$$\begin{aligned} \sup _{0\leqq t\leqq {\hat{t}}}\bigg \{\Vert wf(t)\Vert _{L^\infty }+|wf(t)|_{L^{\infty }(\gamma )}\bigg \}\leqq 2{\tilde{C}}(M_0+1). \end{aligned}$$

Here ${\tilde{C}}>0$ and ${\hat{C}}>1$ are generic constants independent of $M_0$. Moreover, if the domain $\Omega$ is strictly convex, $\theta (x)$ is continuous over $\partial \Omega$, the initial data $F_0(x,v)$ is continuous except on $\gamma _0$ and satisfies (1.19) then the solution F(t, x, v) is continuous in $[0,{\hat{t}}]\times \{\Omega \times {\mathbb {R}}^3\setminus \gamma _0\}$.

Proof

We consider the following iteration scheme:

$$\begin{aligned} \left\{ \begin{aligned}&\partial _tF^{n+1}+v\cdot \nabla _x F^{n+1}+F^{n+1}\cdot R(F^{n})=Q^+(F^n,F^n),\\&F^{n+1}(t,x,v)\Big |_{t=0}=F_0(x,v)\geqq 0,\\&F^{n+1}(t,x,v)|_{\gamma _-}=\mu _{\theta }(x,v)\int _{n(x)\cdot u>0} F^{n+1}(t,x,u)\{u \cdot n(x)\}\,\mathrm{d}u,\\&F^0(t,x,v)=\mu (v), \end{aligned} \right. \end{aligned}$$

(6.10)

where

$$\begin{aligned} {R}(F^n)(t,x,v)=\int _{{\mathbb {R}}^{3}\times {\mathbb {S}}^{2}}{B}(v-u,\omega )F^n(t,x,u)\,\mathrm{d}u\mathrm{d}\omega . \end{aligned}$$

Let

$$\begin{aligned} f^{n+1}(t,x,v)=\frac{F^{n+1}(t,x,v)-F_{*}(x,v)}{\mu ^{\frac{1}{2}}(v)},\quad h^{n+1}(t,x,v)=wf^{n+1}(t,x,v). \end{aligned}$$

Then the equation of $h^{n+1}$ reads as

$$\begin{aligned} \left\{ \begin{aligned}&\partial _th^{n+1}+v\cdot \nabla _x h^{n+1}+h^{n+1}\cdot {R}(F^{n})=wK_{*}f^n+w\Gamma ^+(f^n,f^n),\\&h^{n+1}(t,x,v)\Big |_{t=0}=h_0(x,v),\\&h^{n+1}\Big |_{\gamma _-}=\frac{1}{{\tilde{w}}(v)}\int _{\{n(x)\cdot u>0\}} h^{n+1} {\tilde{w}}(u)\,\mathrm{d}\sigma (x)+w(v)\frac{\mu _{\theta }-\mu }{\sqrt{\mu }}\int _{n(x)\cdot u>0}h^{n+1}{\tilde{w}}(u)\,\mathrm{d}\sigma (x),\\&h^{0}(t,x,v)=-wf_*(x,v), \end{aligned} \right. \end{aligned}$$

(6.11)

where we have denoted

$$\begin{aligned} K_{*}f^n:=-\sqrt{\mu }^{-1}\left\{ R(\sqrt{\mu }f^n)F_*+\Gamma ^{+}(\sqrt{\mu }f^{n},F_*)+\Gamma ^+(F_*,\sqrt{\mu }f^n)\right\} . \end{aligned}$$

Now we shall use the induction on $n=0,1,\ldots$ to show that there exists a positive time ${\hat{t}}_1>0$, independent of n, such that (6.10) or equivalently (6.11) admits a unique mild solution on the time interval $[0,{\hat{t}}_1]$, and the following uniform bound and positivity hold true:

$$\begin{aligned} \Vert h^n(t)\Vert _{L^\infty }+|h^n(t)|_{L^{\infty }(\gamma )}\leqq 2{\tilde{C}}[\Vert h_0\Vert _{L^\infty }+1], \end{aligned}$$

(6.12)

and

$$\begin{aligned} F^n(t,x,v)\geqq 0, \end{aligned}$$

(6.13)

for $0\leqq t\leqq {\hat{t}}=\left( {\hat{C}}\{1+\Vert h_0\Vert _{L^\infty }\}\right) ^{-1}$ and suitably chosen constants ${\tilde{C}}>0$ and ${\hat{C}}$ independent of t. Thanks to the fact that

$$\begin{aligned} \Vert h^0(t)\Vert _{L^\infty }+|h^0(t)|_{L^\infty (\gamma )}\leqq \Vert wf_*\Vert _{L^\infty }+|wf_*|_{L^\infty (\gamma )}\leqq C\delta , \end{aligned}$$

we see that (6.12) is obviously true for $n=0$. To proceed, we assume that (6.12) holds true up to $n\geqq 0$. Since $F^n\geqq 0$, it holds that ${R}(F^n)\geqq 0$. Then by using a similar argument as in [19, Lemma 3.4], one can construct the solution operator $G^n(t)$ to the following linear problem:

$$\begin{aligned} \left\{ \begin{aligned}&\partial _th+v\cdot \nabla _x h+{R}(F^n)h=0, \quad t>0,\quad x\in \Omega , \quad v\in {\mathbb {R}}^3,\\&h(t,x,v)|_{t=0}=h_0(x,v),\\&h(t,x,v)\Big |_{\gamma _-}=\frac{1}{{\tilde{w}}(v)}\int _{n(x)\cdot u>0} h(t,x,u) {\tilde{w}}(u)\,\mathrm{d}\sigma (x)+w(v)\frac{\mu _{\theta }-\mu }{\sqrt{\mu }}\int _{n(x)\cdot u>0}h(t,x,u){\tilde{w}}(u)\,\mathrm{d}\sigma (x) \end{aligned} \right. \end{aligned}$$

over $(0,\rho )$ for some universal constant $\rho >0$ independent of n, provided that $|\theta -1|_{L^\infty (\partial \Omega )}$ is sufficiently small. Moreover, $G^n(t)$ satisfies the estimate

$$\begin{aligned} \Vert G^{n}(t)h_0\Vert _{L^{\infty }}+|G^n(t)h_0|_{L^\infty (\gamma )}\leqq C_{\rho }\Vert h_0\Vert _{L^{\infty }}. \end{aligned}$$

(6.14)

Here the constant $C_\rho >0$ is independent of n. Then applying Duhamel’s formula to (6.11), we have, for $0<t<\rho ,$ that

$$\begin{aligned} h^{n+1}(t)= {G}^{n}(t,0)h_0+\int _0^t {G}^{n}(t,s)[w K_*f^n(s)+w\Gamma ^+(f^n,f^n)(s)]\mathrm{d}s. \end{aligned}$$

(6.15)

Taking $L^\infty$-norm on the both sides of (6.15) and using (2.16) and (6.14), we have

$$\begin{aligned}&\Vert h^{n+1}(t)\Vert _{L^\infty }+|h^{n+1}(t)|_{L^\infty {(\gamma )}}\nonumber \\&\quad \leqq C_\rho \Vert h_0\Vert _{L^\infty }+C_\rho \int _0^t\Vert wK_{*}f^n(s)\Vert _{L^\infty }+\Vert w\Gamma ^+(f^n,f^n)(s)\Vert _{L^\infty }\mathrm{d}s\nonumber \\&\quad \leqq {C} \Vert h_0\Vert _{L^\infty }+C\int _0^t\Vert h^n(s)\Vert _{L^\infty }+\Vert h^n(s)\Vert ^2_{L^\infty }\mathrm{d}s\nonumber \\&\quad \leqq C_{{3}}\Vert h_0\Vert _{L^{\infty }}+C_{{3}}t\cdot \{\sup _{0\leqq s\leqq t}\Vert h^n(s)\Vert _{L^{\infty }}+\sup _{0\leqq s\leqq t}\Vert h^n(s)\Vert _{L^\infty }^2\} \end{aligned}$$

(6.16)

for some constants $C_3>1$. Now we take ${\tilde{C}}=C_3$ and ${\hat{C}}=8C_3^2$. Then by the induction hypothesis (6.12), for any $0<t<{\hat{t}}$, it follows from (6.16) that

$$\begin{aligned} \begin{aligned} \Vert h^{n+1}(t)\Vert _{L^\infty }+|h^{n+1}(t)|_{L^\infty (\gamma )}&\leqq C_{3}\{\Vert h_0\Vert _{L^{\infty }}+1\}\cdot \{1+2C_3t[1+2C_3]\\&\quad \cdot [1+\Vert h_0\Vert _{L^\infty }]\}\\&\leqq 2C_3\{\Vert h_0\Vert _{L^\infty }+1\}. \end{aligned} \end{aligned}$$

This then proves (6.12) for $n+1$. Next we show the non-negativity (6.13) for $n+1$. We denote that

$$\begin{aligned} I^{n}(t,s):=\exp \left\{ -\int _s^t[R(F^n)](\tau ,X_{cl}(\tau ),V_{cl}(\tau ))\,\mathrm{d}\tau \right\} \end{aligned}$$

and

$$\begin{aligned} \mathrm{d}\Sigma _{l}^n(\tau ):= & {} \left\{ \prod _{j=l+1}^{k-1}\mathrm{d}\sigma _{j}\right\} \cdot I^n(t_l,\tau )[v_l\cdot n(v_l)]\mathrm{d}v_l\\&\quad \cdot \prod _{j=1}^{l-1}\left\{ I^n(t_{j},t_{j+1})\mu _{\theta }(x_{j+1},v_{j})[v_j\cdot n(x_j)]\mathrm{d}v_{j}\right\} . \end{aligned}$$

Then we have the following mild formulation for $F^{n+1}$:

(6.17)

for $t>0$, $x\in {\bar{\Omega }}\times {\mathbb {R}}^3\setminus \gamma _0\cup \gamma _-$ and integer $k\geqq 1$. From (6.12), it holds that

$$\begin{aligned} |F^{n+1}(t,x,v)|=\left| F_{*}(x,v)+\mu ^{\frac{1}{2}}(v)\frac{h^{n+1}(t,x,v)}{w(v)}\right| \leqq C(1+\Vert h_0\Vert _{L^{\infty }})\mu ^{\frac{1}{2}}(v) \end{aligned}$$

(6.18)

for some constant $C>0$. Furthermore, a direct computation shows that

$$\begin{aligned} 0<\mu _{\theta }(x_{j+1},v_{j})\leqq \frac{1}{(1-\delta )^2}\exp \left\{ \frac{\delta |v_j|^2}{2(1-\delta )}\right\} \mu (v_j). \end{aligned}$$

Then similarly as for (4.12), we have, for sufficiently large $T_0>0$ and for $k={\hat{C}}_5T_0^{5/4}$, that

$$\begin{aligned} \int _{\prod _{j=1}^{k-2}{\mathcal {V}}_j}{\mathbf {1}}_{\{t_{k-1}>0\}}\prod _{j=1}^{k-2}\mu _{\theta }(x_{j+1},v_j)\{n(x_j)\cdot v_j\}\mathrm{d}v_j\leqq \left( \frac{1}{2}\right) ^{{\hat{C}}_6T_0^{5/4}} \end{aligned}$$

for some generic constant ${\hat{C}}_5>0$ and ${\hat{C}}_6>0$. Then, by (6.17) and (6.18), we have

$$\begin{aligned} \begin{aligned} F^{n+1}&\geqq -C\mu _\theta (x_{1},v)\{1+\Vert h_0\Vert _{L^\infty }\} \int _{\prod _{j=1}^{k-2}{\mathcal {V}}_j}{\mathbf {1}}_{\{t_{k-1}>0\}}\\&\quad \prod _{j=1}^{k-2}\mu _{\theta }(x_{j+1},v_j)\{n(x_j)\cdot v_j\}\mathrm{d}v_j\\&\geqq -C\mu _\theta (x_{1},v)\{1+\Vert h_0\Vert _{L^\infty }\}\cdot \left( \frac{1}{2}\right) ^{{\hat{C}}_6T_0^{5/4}}. \end{aligned} \end{aligned}$$

(6.19)

Since $T_0>0$ can be taken arbitrarily large, we have $F^{n+1}\geqq 0.$ This then proves (6.12) and (6.13). Finally, with the uniform estimates (6.12) in hand, we can use a similar argument as one in [19, Theorem 4.1] to show that $h^n,$$n=0,1,2\cdots ,$ is a Cauchy sequence in $L^\infty$. We omit here for brevity. The solution is obtained by taking the limit $n\rightarrow \infty$. If $\Omega$ is convex and the compatibility condition (1.19) holds, the continuity is a direct consequence of the $L^\infty$-convergence. The uniqueness is standard. The proof of Proposition 6.2 is complete. $\quad \square$