Singular Limit for the Compressible Navier–Stokes Equations with the Hard Sphere Pressure Law on Expanding Domains

Abstract

The article is devoted to the asymptotic limit of the compressible Navier–Stokes system with a pressure obeying a hard–sphere equation of state on a domain expanding to the whole physical space $\textbf{R}^3$. Under the assumptions that acoustic waves generated in the case of ill-prepared data do not reach the boundary of the expanding domain in the given time interval and a certain relation between the Reynolds and Mach numbers and the radius of the expanding domain we prove that the target system is the incompressible Euler system on $\textbf{R}^3$. We also provide an estimate of the rate of convergence expressed in terms of characteristic numbers and the radius of domains.

Appendix

The ensuing lemma deals with renormalized solutions of the continuity equation. It collects versions of assertions [34, Lemmas 6.9 and 6.11] adopted for a function b considered in this paper.

Lemma 4.1

Let $T>0$ and $\Omega \subset \textbf{R}^d$, $d\ge 2$, be a bounded Lipschitz domain. Let $\rho \in L^\infty (Q_T)$ be such that $0\le \rho <{\bar{\rho }}$ a.e. in $Q_T$ and $\rho$ together with $u\in L^2(W^{1,2}_0(\Omega )^d)$ satisfy the continuity equation

$$\begin{aligned} \partial _t\rho +{\textrm{div}}(\rho u)=0\text { in }{\mathcal {D}}'(Q_T) \end{aligned}$$

(107)

Let $b\in C^1([0,{\bar{\rho }}))$ be a nonnegative function such that

$$\begin{aligned} b(\rho )\in L^{2}(Q_T), b'(\rho )\in L^2(Q_T) \end{aligned}$$

(108)

and

$$\begin{aligned} b, b'\text { are nondecreasing on }[{\bar{\rho }}-\alpha _0,{\bar{\rho }})\text { for some }\alpha _0\in (0,{\bar{\rho }}). \end{aligned}$$

(109)

Let $\rho$ and u be extended by zero in $(0,T)\times \left( \textbf{R}^d\setminus \Omega \right)$.

• 1. Then the continuity equation (107) holds in the sense of renormalized solutions

  $$\begin{aligned} \partial _tb(\rho )+{\textrm{div}}(b(\rho )u)+(b'(\rho )\rho -b(\rho )){\textrm{div}}u=0\text { in }{\mathcal {D}}'((0,T)\times \textbf{R}^d), \end{aligned}$$

  (110)

• 2. Moreover, for $b_\alpha$ with $\alpha \in (0,\alpha _0)$ defined as

  $$\begin{aligned} b_\alpha (s)={\left\{ \begin{array}{ll} b(s)&{}s\le {\bar{\rho }}-\alpha ,\\ b({\bar{\rho }}-\alpha )&{}s>{\bar{\rho }}-\alpha \end{array}\right. } \end{aligned}$$

  (111)

  the renormalized continuity equation holds in the form

  $$\begin{aligned} \partial _tb_\alpha (\rho )+{\textrm{div}}(b_\alpha (\rho )u)+(b'_\alpha (\rho )\rho -b_\alpha (\rho )){\textrm{div}}u=0\text { in }{\mathcal {D}}'((0,T)\times \textbf{R}^d), \end{aligned}$$

  (112)

  where we set $b'_\alpha (\rho )=0$ in $\{\rho ={\bar{\rho }}-\alpha \}$.

Proof

Applying the extension procedure from [34, Lemma 6.8] we get

$$\begin{aligned} \partial _t\rho +{\textrm{div}}(\rho u)=0\text { in }{\mathcal {D}}'((0,T)\times \textbf{R}^d) \end{aligned}$$

for the extensions $\rho \in L^\infty ((0,T)\times \textbf{R}^d)$ and $u\in L^2(0,T;W^{1,2}_{loc}(\textbf{R}^d))$ of $\rho$ and u from assumptions of the lemma by zero in $(0,T)\times \left( \textbf{R}^d\setminus \Omega \right)$. Regularizing the latter equation over the spatial variables by the usual mollifier $S_\varepsilon$ with $\varepsilon >0$ yields

$$\begin{aligned} \partial _tS_\varepsilon (\rho )+{\textrm{div}}(S_\varepsilon (\rho ) u)=r_\varepsilon (\rho , u)\text { a.e. in }(0,T)\times \textbf{R}^d, \end{aligned}$$

(113)

where

$$\begin{aligned} r_\varepsilon (\rho , u)={\textrm{div}}(S_\varepsilon (\rho ) u-S_\varepsilon (\rho u))\rightarrow 0\text { in }L^r(0,T;L^r_{loc}(\textbf{R}^d))\text { for any }r<2, \end{aligned}$$

(114)

cf. [34, Lemma 6.7]. We observe that

$$\begin{aligned} S_\varepsilon (\rho )<{\bar{\rho }}. \end{aligned}$$

(115)

We fix $\delta \in (0,\alpha _0)$, extend $b(s)=b(-s)$ on $[-\alpha _0,0]$ and define $b_{(\delta )}(\cdot )=b(\cdot -\delta )$. Next we multiply (113) on $b_{(\delta )}'(S_\varepsilon (\rho ))$ and obtain

$$\begin{aligned} \begin{aligned}&\partial _tb_{(\delta )}(S_\varepsilon (\rho ))+{\textrm{div}}(b_{(\delta )}(S_\varepsilon (\rho )) u)+\left( b'_{(\delta )}(S_\varepsilon (\rho ))S_\varepsilon (\rho )-b_{(\delta )}(S_\varepsilon (\rho ))\right) {\textrm{div}}u\\ {}&=b_{(\delta )}'(S_\varepsilon (\rho ))r_\varepsilon (\rho , u)\text { a.e. in }(0,T)\times \textbf{R}^d. \end{aligned} \end{aligned}$$

(116)

The approximating property

$$\begin{aligned} S_\varepsilon (\rho )\rightarrow \rho \text { in }L^q(0,T;L^q_{loc}(\textbf{R}^d))\text { for any }q\in [1,\infty ) \end{aligned}$$

(117)

implies the existence of a nonrelabeled subsequence $\{S_\varepsilon (\rho )\}$ such that $S_\varepsilon (\rho )\rightarrow \rho$ a.e. in $(0,T)\times \textbf{R}^d$. Taking into account the continuity of $b^{(j)}$, where $j\in \{0,1\}$ denotes the order of the derivative, we have $b^{(j)}_{(\delta )}(S_\varepsilon (\rho ))\rightarrow b^{(j)}_{(\delta )}(\rho )$ a.e. in $(0,T)\times \textbf{R}^d$. Taking into consideration also (118120) and (109) it follows that

$$\begin{aligned} \begin{aligned} b_{(\delta )}(S_\varepsilon (\rho ))\le&\max \{\max _{[0,{\bar{\rho }}-\alpha _0+\delta ]} b_{(\delta )},b_{(\delta )}({\bar{\rho }})\},\\ |b'_{(\delta )}(S_\varepsilon (\rho ))|\le&\max \{\max _{[0,{\bar{\rho }}-\alpha _0+\delta ]} |b'_{(\delta )}|,b'_{(\delta )}({\bar{\rho }})\}. \end{aligned} \end{aligned}$$

(118)

We note that (109) implies that $b'\ge 0$ on $({\bar{\rho }}-\alpha _0,{\bar{\rho }})$ so that $|b'_{(\delta )}(S_\varepsilon (\rho ))|=b'_{(\delta )}(S_\varepsilon (\rho ))\le b'_{(\delta )}({\bar{\rho }})$ for $S_\varepsilon (\rho )-\delta >{\bar{\rho }}-\alpha _0$. Hence we can apply the Lebesgue dominated convergence theorem to infer

$$\begin{aligned} \begin{aligned} b_{(\delta )}(S_\varepsilon (\rho ))&\rightarrow b_{(\delta )}(\rho ){} & {} \text { in }L^p(0,T;L^p_{loc}(\textbf{R}^d)),\\ b'_{(\delta )}(S_\varepsilon (\rho ))&\rightarrow b'_{(\delta )}(\rho ){} & {} \text { in }L^p(0,T;L^p_{loc}(\textbf{R}^d)) \end{aligned} \end{aligned}$$

(119)

for any $p\in [1,\infty )$. The latter convergences and (117) imply

$$\begin{aligned} b'_{(\delta )}(S_\varepsilon (\rho ))S_\varepsilon (\rho )-b_{(\delta )}(S_\varepsilon (\rho ))\rightarrow b'_{(\delta )}(\rho )\rho -b_{(\delta )}(\rho )\text { in }L^p(0,T;L^p_{loc}(\textbf{R}^d)). \end{aligned}$$

Combining (116), (114), (119) and the latter convergence one arrives at (110) for $b=b_{(\delta )}$. Now we pass to the limit $\delta \rightarrow 0_+$. We have $b^{(j)}_{(\delta )}(\rho )\rightarrow b^{(j)}(\rho )$ a.e. in $(0,T)\times \textbf{R}^d$, $j\in \{0,1\}$. Moreover, (109) implies

$$\begin{aligned} \begin{aligned} b_{(\delta )}(\rho )\le&\max \{\max _{[0,{\bar{\rho }}-\alpha _0]} b,b(\rho )\},\\ |b'_{(\delta )}(\rho )|\le&\max \{\max _{[0,{\bar{\rho }}-\alpha _0]} |b'|,b'(\rho )\}. \end{aligned} \end{aligned}$$

(120)

Hence we infer by the Lebesgue dominated convergence theorem and (108)

$$\begin{aligned} \begin{aligned} b_{(\delta )}(\rho )&\rightarrow b(\rho ){} & {} \text { in }L^2(0,T;L^2_{loc}(\textbf{R}^d)),\\ b'_{(\delta )}(\rho )&\rightarrow b'(\rho ){} & {} \text { in }L^2(0,T;L^2_{loc}(\textbf{R}^d)). \end{aligned} \end{aligned}$$

Having the latter convergences at hand we pass to the limit $\delta \rightarrow 0_+$ in (110) with $b=b_{(\delta )}$ and the first assertion of the lemma is proved.

In order to prove the second assertion, we begin with the proof of the following auxiliary identity

$$\begin{aligned} ({\bar{\rho }}-\alpha ){\textrm{div}}u=0\text { a.e. in }\{\rho ={\bar{\rho }}-\alpha \}. \end{aligned}$$

(121)

To this end we consider $b\in C^1_c((0,\infty ))$ such that $b(s)=s$ in $[\tfrac{3}{4}({\bar{\rho }}-\alpha ),{\bar{\rho }}-\tfrac{3}{4}\alpha ]$ and b, $b'$ are nondecreasing in $[{\bar{\rho }}-\tfrac{\alpha }{2},{\bar{\rho }}]$. We define $b^+_{\alpha ,\varepsilon }=S_\frac{\varepsilon }{2}(b_{\alpha +\varepsilon })$, $b^-_{\alpha ,\varepsilon }=S_\frac{\varepsilon }{2}(b_{\alpha -\varepsilon })$. Then we have as $\varepsilon \rightarrow 0_+$

(122)

By the first assertion of the lemma we have

$$\begin{aligned} \begin{aligned} \partial _tb^+_{\alpha ,\varepsilon }(\rho )+{\textrm{div}}(b^+_{\alpha ,\varepsilon }(\rho )u)+(\rho (b^+_{\alpha ,\varepsilon })'(\rho )-b^+_{\alpha ,\varepsilon }(\rho )){\textrm{div}}u=0\text { in }{\mathcal {D}}'((0,T)\times \textbf{R}^d),\\ \partial _tb^-_{\alpha ,\varepsilon }(\rho )+{\textrm{div}}(b^-_{\alpha ,\varepsilon }(\rho )u)+(\rho (b^-_{\alpha ,\varepsilon })'(\rho )-b^-_{\alpha ,\varepsilon }(\rho )){\textrm{div}}u=0\text { in }{\mathcal {D}}'((0,T)\times \textbf{R}^d). \end{aligned} \end{aligned}$$

Letting $\varepsilon \rightarrow 0_+$ and employing the convergences from (122) we deduce by the Lebesgue dominated convergence theorem from the latter identities

$$\begin{aligned} \begin{aligned} \partial _tb_\alpha (\rho )+{\textrm{div}}(b_{\alpha }(\rho )u)+(\rho (b_{\alpha })'(\rho )\chi _{\{\rho \ne {\bar{\rho }}-\alpha \}}-b_{\alpha }(\rho )){\textrm{div}}u=0\text { in }{\mathcal {D}}'((0,T)\times \textbf{R}^d),\\ \partial _tb_{\alpha }(\rho )+{\textrm{div}}(b_{\alpha }(\rho )u)+(\rho (b_\alpha )'(\rho )\chi _{\{\rho \ne {\bar{\rho }}-\alpha \}}+({\bar{\rho }}-\alpha )\chi _{\{\rho ={\bar{\rho }}-\alpha \}}-b_\alpha (\rho )){\textrm{div}}u=0\text { in }{\mathcal {D}}'((0,T)\times \textbf{R}^d). \end{aligned} \end{aligned}$$

Subtracting the latter equations we conclude (121).

Next, we consider for fixed $\varepsilon <\alpha$ $S_\varepsilon (b_\alpha )$, the mollification of $b_\alpha$ extended by $b({\bar{\rho }}-\alpha )$ in $({\bar{\rho }},{\bar{\rho }}+1]$ and by 0 outside of $[0,{\bar{\rho }}+1]$. As $S_\varepsilon (b_\alpha )$ is constant in a vicinity of ${\overline{\rho }}$, it fulfills (109) and $S_\varepsilon (b_\alpha )(\rho )$ satisfies (19), it follows that

$$\begin{aligned} \partial _tS_\varepsilon (b_\alpha (\rho ))+{\textrm{div}}(S_\varepsilon (b_\alpha (\rho ))u)+\left( \left( S_\varepsilon (b_\alpha )\right) '(\rho )\rho -S_\varepsilon (b_\alpha (\rho ))\right) {\textrm{div}}u=0\text { in }{\mathcal {D}}'((0,T)\times \textbf{R}^d). \end{aligned}$$

(123)

Furthermore, we have as $\varepsilon \rightarrow 0_+$

Hence we infer that as $\varepsilon \rightarrow 0_+$

(124)

Employing (121) we deduce that $\rho (S_\varepsilon (b_\alpha ))'(\rho ){\textrm{div}}u=0$ a.e. in $\{\rho ={\bar{\rho }}-\alpha \}$. Using the convergences from (124), the uniform bounds with respect to $\varepsilon$ on $S_\varepsilon (b_\alpha ), S_\varepsilon (b_\alpha )'$ and the Lebesgue dominated convergence theorem we pass to the limit $\varepsilon \rightarrow 0_+$ in (123) to conclude (112). $\square$

Lemma 4.2

Let $\Omega \subset \textbf{R}^d$ be a starshaped domain with respect to a ball B possessing the radius R. There exists a linear operator ${\mathcal {B}}: C_c^\infty (\Omega )\rightarrow C_c^\infty (\Omega )^d$ such that ${\textrm{div}}{\mathcal {B}}(f)=f$ provided that $\int _\Omega f=0$. Moreover, ${\mathcal {B}}$ can be extended in a unique way as a bounded linear operator

• 1. ${\mathcal {B}}: L^p(\Omega )\rightarrow W^{1,p}(\Omega )^d$ such that $\Vert B(f)\Vert _{W^{1,p}(\Omega )}\le c\Vert f\Vert _{L^p(\Omega )}$

• 2. ${\mathcal {B}}: \{f\in (W^{1,p'}(\Omega ))':\langle f,1\rangle =0\}\rightarrow L^p(\Omega )^d$ such that $\Vert B(f)\Vert _{L^p(\Omega )}\le c\Vert f\Vert _{(W^{1,p}(\Omega ))'}$

for any $p\in (1,\infty )$ where the constants c take the form

$$\begin{aligned} c=c_0(p,d)\left( \frac{\textrm{diam}(\Omega )}{R}\right) ^d\left( 1+\frac{\textrm{diam}(\Omega )}{R}\right) . \end{aligned}$$

Assertions in the following lemma are based on the results from [23]

Lemma 4.3

Let $v_0\in W^{m,2}(\textbf{R}^3)$ with $m>4$ be such that ${\textrm{div}}v_0=0$ in $\textbf{R}^3$. Then there is $T_{max}>0$ and a classical solution v, unique in the class

$$\begin{aligned} v\in C([0,T_{max}),W^{m,2}(\textbf{R}^3)^3),\ \partial _tv\in C([0,T_{max});W^{m-1,2}(\textbf{R}^3)^3) \end{aligned}$$

to the initial value problem

$$\begin{aligned} \partial _tv+v\cdot \nabla v+\nabla \Pi =0&\text { in }(0,T_{max})\times \textbf{R}^3,\\ v(0,\cdot )=v_0, {\textrm{div}}v_0=0&\text { in }\textbf{R}^3. \end{aligned}$$

Furthermore, the associate pressure $\Pi$ can be expressed as

$$\begin{aligned} \Pi =(-\Delta )^{-1}{\textrm{div}}\, {div}(v\otimes v). \end{aligned}$$

implying particularly that $\Pi \in C^1([0,T];C^1(\textbf{R}^3)\cap W^{1,2}(\textbf{R}^3))$, $T\in (0,T_{max})$.