Finite time blow up of compressible Navier-Stokes equations on half space or outside a fixed ball

TL;DR

This paper proves that classical solutions to the compressible Navier-Stokes equations with certain initial conditions on a half space or outside a ball must blow up in finite time, extending previous results to cases with physical boundaries.

Contribution

It extends the finite-time blow-up results for compressible Navier-Stokes equations from the Cauchy problem to bounded domains with physical boundaries.

Findings

Classical solutions blow up in finite time under specified conditions.

Extension of Xin's results to bounded domains with boundary.

Conditions include positive initial mass and bounded initial entropy.

Abstract

In this paper, we consider the initial-boundary value problem to the compressible Navier-Stokes equations for ideal gases without heat conduction in the half space or outside a fixed ball in $\mathbb R^N$, with $N\geq1$. We prove that any classical solutions $(\rho, u, \theta)$, in the class $C^1([0,T]; H^m(\Omega))$, $m>[\frac N2]+2$, with bounded from below initial entropy and compactly supported initial density, which allows to touch the physical boundary, must blow-up in finite time, as long as the initial mass is positive. This paper extends the classical reault by Xin [CPAM, 1998], in which the Cauchy probelm is considered, to the case that with physical boundary.

Equations

$${\partial }_t\rho +\text{div}(\rho u)=0,$$

$${\partial }_t(\rho u)+\text{div}(\rho u\otimes u)-\text{div}S+\nabla p=0,$$

$${\partial }_t(\rho E)+\text{div}(u(\rho E+p))=\text{div}q+\text{div}(S\cdot u),$$

$$S=\mu (\nabla u+(\nabla u{)}^T)+\lambda \text{div}uI,$$

$$\mu \ge 0,2\mu +N\lambda \ge 0.$$

$$e=c_v\theta ,p=R\rho \theta ,p=A{e}^{s/c_v}{\rho }^{\gamma },$$

$${\partial }_t\rho +\text{div}(\rho u)=0,$$

$${\partial }_t(\rho u)+\text{div}(\rho u\otimes u)-\text{div}S+\nabla p=0,$$

$${\partial }_t(\rho E)+\text{div}(u(\rho E+p))=\text{div}(S\cdot u).$$

$$\mu >0,2\mu +N\lambda >0.$$

$$u{|}_{\partial \Omega }=0,(u(x,t),\theta (x,t))\to 0,\text{mbox}asx\to \infty ,$$

$$(\rho ,u,\theta ){|}_{t=0}=({\rho }_0,u_0,{\theta }_0)\in H^m(\Omega ),$$

$${\theta }_0(x)>0,\text{mbox}foranyx\in {\mathcal{O}}_{+}:=\{x\in \Omega |{\rho }_0(x)>0\}.$$

$$s_0(x)=c_v\log \left(\frac{R}{A}{\theta }_0(x){\rho }_0^{1-\gamma }(x)\right),\text{mbox}forx\in {\mathcal{O}}_{+}.$$

$$supp{\rho }_0\subseteq B_{R_0},{\int }_{\Omega }{\rho }_0(x)dx>0,$$

$$\underset{x\in {\mathcal{O}}_{+}}{\inf }{s}_0(x)\ge {\underline{s}}_0,$$

$$(\rho ,p)\in C^1([0.T];H^m(\Omega )),u\in C^1([0,T];{\dot{H}}^m(\Omega )),$$

$$\left\{\begin{array}[]{l}\frac{d}{dt}X(t;x)=u(X(t;x),t),\quad t\in(0,T),\\ X(0;x)=x.\end{array}\right.$$

$$X(t;K):=\{y=X(t;x)|x\in K\}.$$

$$[X(t;K){]}^c\cap \Omega =X(t;K^c\cap \Omega ),X(t;K_1)\cup X(t;K_2)=X(t;K_1\cup K_2),$$

$$S:\nabla u=$$

$$=$$

$$S:\nabla u\ge \frac{\mu }{2}|\nabla u+(\nabla u{)}^T{|}^2\ge 0,$$

$$S:\nabla u=$$

$$-(N-1)\lambda \sum _{i=1}^N({\partial }_i{u}_i{)}^2+\lambda \sum _{i\ne j}{\partial }_i{u}_i{\partial }_j{u}_j$$

$$=$$

$$S:\nabla u\ge \frac{\mu }{2}\sum _{i\ne j}({\partial }_i{u}_j+{\partial }_j{u}_i{)}^2+(2\mu +N\lambda )\sum _{i=1}^N({\partial }_i{u}_i{)}^2\ge 0.$$

$$supp{\rho }_0\subseteq B_{R_0},$$

$$supp\rho (\cdot ,t)\subseteq B_{R_0},suppu(\cdot ,t)\subseteq B_{R_0}.$$

$$\frac{d}{dt}\rho (X(t;x),t)=$$

$$\rho (X(t;x),t)=$$

$$\rho (\cdot ,t)\equiv 0,\text{mbox}onX(t;B_{R_0}^c\cap \Omega )=[X(t;B_{R_0}\cap \Omega ){]}^c.$$

$$\text{div}S=\text{div}(S\cdot u)=0,\text{mbox}on[X(t;B_{R_0}\cap \Omega ){]}^c\times \{t\},$$

$$S:\nabla u=\text{div}(S\cdot u)-\text{div}S\cdot u=0,\text{mbox}on[X(t;B_{R_0}\cap \Omega ){]}^c\times \{t\}.$$

Full text

1. Introduction

The compressible Navier-Stokes equations for idea gases on a domain $\Omega\subseteq\mathbb{R}^{N}$, $N\geq 1$, read as

[TABLE]

where the unknowns are the density $\rho\geq 0$, the velocity $u\in\mathbb{R}^{N}$, and the specific total energy $E\geq 0$, with $E=\frac{|u|^{2}}{2}+e$, and $e$ the specific internal energy. The stress tensor $S$ is given by

[TABLE]

with two constant Lamé viscosity coefficients $\mu$ and $\lambda$ satisfying

[TABLE]

The heat flux $q$ is given by $q=\kappa\nabla\theta,$ for some nonnegative constant coefficient $k$. Recalling that we consider the ideal gases, the state equations are

[TABLE]

where $s$ is the entropy, $c_{v}$, $R$, $A$ and $\gamma>1$ are positive constants, with $c_{v}=\frac{R}{\gamma-1}$.

In the absence of vacuum, i.e. the density is away from zero, local well-posedness of classical solutions in the Hölder spaces to the compressible Navier-Stokes equations was established by Itaya [9] and Tani [18], while the global well-posedness of classical solutions in the Sobolev spaces was firstly established by Matsummura and Nishida [14, 15], under the condition that $\|(\rho_{0}-\bar{\rho},u_{0},\theta_{0}-\bar{\theta})\|_{H^{m}}$ is suitably small, where $m>[\frac{N}{2}]+2$, and $\bar{\rho}$ and $\bar{\theta}$ are two positive constants. In the presence of vacuum, it was first proved by Lions [12] the global existence of weak solutions to the isentropic compressible Navier-Stokes equations (i.e. system (1.1)–(1.2) by setting $p=A\rho^{\gamma}$), with $\gamma\geq\frac{3N}{N+2}$, $N=2,3$. His result was later extended by Feireisl, Novotny and Pezeltova [7] to the case $\gamma>\frac{3}{2}$, and by Jiang and Zhang [11] to the case $\gamma>1$ for axisymmetric solutions. Concerning the full compressible Navier-Stokes equations, only the global existence of the so called variational solutions was proven by Feireisl [5, 6], where the energy equation was satisfied only in the sense of inequality. Local well-posedness of strong solutions to the full compressible Navier-Stokes equations was established by Cho and Kim [3]; however, it should be noted that the strong solutions established in [3] have no information on the entropy, and in particular it is not known if the corresponding entropy is bounded or not.

A natural question is whether the classical solutions to the compressible Navier-Stokes equations exist globally or not, when the initial vacuum is allowed. It was first proved by Xin [19] that smooth solutions, with nontrivial and compactly supported initial density, to any dimensional full compressible Navier-Stokes equations without heat conduction or one dimensional isentropic compressible Navier-Stokes equations, must blow up in finite time. Xin’s blow up result was later generalized by Cho and Jin [1], and Tan and Wang [17] to the case with heat conduction, and by Rozanova [16] to the case of rapidly decreasing solutions. Moreover, it was shown in a recent paper by Xin and Yan [20] that the blow up result may still hold without the assumptions of compactly supported initial density or rapidly decreasing of the solutions; they proved that the blow up for classical solutions occurs in finite time, as long as the initial density is not identically equal to zero, on a bounded open set surrounded by vacuum region. Finally, if we focus on the radially symmetric solutions, then the finite time blow up result also holds for the two dimensional isentropic or isothermal compressible Navier-Stokes equations, see Luo [13], and Du, Li and Zhang [4]. However, there is a somewhat surprising result by Huang, Li and Xin [8], where they proved the global well-posedness of classical solutions to the three dimensional isentropic compressible Navier-Stokes equations, with initial data of small energy but allowed to have vacuum and even compactly supported initial density.

Note that in all the papers [19, 17, 1, 16, 13, 4], concerning the finite time blow up of classical solutions to the compressible Navier-Stokes equations, the Cauchy problem was considered, in other words, the domain under consideration has no physical boundary. In [20], the initial-boundary value problem was also considered, and thus the physical boundary was allowed; however, since the additional assumption imposed on the initial data in [20] prevent the isolated mass group from touching the boundary, it was essentially reduced to the case without any physical boundary. In view of the finite time blow up results in the above mentioned papers, the remaining question is if the classical solutions to the compressible Naiver-Stokes equations still blow up in finite time in the presence of physical boundary. We will partially answer this question. Precisely, we will prove that if the domain $\Omega$ under consideration is either the half space $\mathbb{R}^{N}_{+}$ or the exterior domain $\mathbb{R}^{N}\setminus B_{r_{0}}$, then classical solutions to the compressible Navier-Stokes equations must blow up in finite time, as long as the initial mass is positive and the initial density is compactly supported in $\overline{\Omega}$.

In this paper, we consider the compressible Navier-Stokes equations without heat conduction, in other words, we consider the following system

[TABLE]

We always suppose that the viscosity coefficients $\mu$ and $\lambda$ satisfy

[TABLE]

We consider the initial-boundary value problems to system (1.5)–(1.7), on the half space or outside a fixed ball (without loss of generality, we can suppose that the fixed ball is centered at the origin). Hence, the domain $\Omega$ under consideration is taken as one of the following two cases:

(i) $\Omega=\mathbb{R}^{N}_{+}=\{(x_{1},\cdots,x_{N})|x_{i}\in\mathbb{R},1\leq i\leq N-1,x_{N}>0\}$,

(ii) $\Omega=\mathbb{R}^{N}\setminus\overline{B}_{r_{0}}=\{x\in\mathbb{R}^{N}||x|>r_{0}\}$, for some positive number $r_{0}$.

We complement system (1.5)–(1.7) with the following boundary condition

[TABLE]

while the initial condition reads as

[TABLE]

for some integer $m>[\frac{N}{2}]+2$.

Recalling the state equation $p=Ae^{s/c_{v}}\rho^{\gamma}$ in (1.4), it is natural for us to assume the following compatibility condition on the initial data

[TABLE]

As a result, in the non-vacuum region, by the state equations in (1.4), the initial data of the entropy $s$ on $\mathcal{O}_{+}$ is well-defined as

[TABLE]

We have the following theorem on the blow up of classical solutions to system (1.5)–(1.7), subject to (1.9)–(1.10).

Theorem 1.1**.**

Suppose that $(\rho_{0},u_{0},\theta_{0})\in H^{m}(\Omega)$, for some $m>[\frac{N}{2}]+2$, satisfying the compatibility condition (1.11), and

[TABLE]

for some positive number $R_{0}$ (with $R_{0}>r_{0}$, for the case $\Omega=\mathbb{R}^{N}\setminus\overline{B}_{r_{0}}$). Let $s_{0}$ be the function defined by (1.12) on $\mathcal{O}_{+}$, satisfying

[TABLE]

for some constant $\underline{s}_{0}$.

Then, the compressible Navier-Stokes equations (1.5)–(1.7), subject to (1.9)–(1.10), either do not have any local classical solution $(\rho,u,\theta)$ in the class $C^{1}([0,T];H^{m}(\Omega))$, or otherwise this local solution must blow up in finite time.

For a special case, the assumption (1.13) in the above theorem can be removed, and in fact we have the following:

Theorem 1.2**.**

If $N=1$, then the result in Theorem 1.1 still holds without the assumption (1.13).

Remark 1.1**.**

It should be pointed out that, same as in [19, 20], the existence of local solution $(\rho,u,\theta)$ in the class $C^{1}([0,T];H^{m}(\Omega))$ to system (1.5)–(1.7), subject to (1.9)–(1.10), is still open. If following the arguments in [2, 3], one can obtain a unique solution $(\rho,u,p)$ ($p$ is chosen as an unknown) in the class

[TABLE]

here $\dot{H}^{m}(\Omega):=\{f\in L_{loc}(\Omega)|\nabla^{\alpha}f\in L^{2}(\Omega),1\leq|\alpha|\leq m\}$, which unfortunately does not meet the requirements on $(\rho,u,\theta)$ in Theorem 1.1 and Theorem 1.2. The existence of solution $(\rho,u,\theta)$ in the class $C([0,T];H^{1}(\mathbb{R}))$ (which can be further strengthened in $C^{1}([0,T];H^{m}(\mathbb{R}))$ if putting more regularity assumptions and compatibility conditions on the initial data) to the Cauchy problem of (1.5)–(1.7), in the presence of vacuum at the far field only, has recently been proved by the second author [10]. Unfortunately, since the crucial assumption (1.12) there does not hold if $\rho_{0}$ is compactly supported, the argument in [10] does not leads to the desired existence of solution $(\rho,u,\theta)$ required in Theorem 1.1 and Theorem 1.2 either.

Some comments on the proofs of Theorem 1.1 and Theorem 1.2 are stated as follows. Note that the main ingredients of the proofs in [19, 17, 1, 16, 13, 4, 20] are multiplying the transport equation by $|x|^{2}$, and correspondingly multiplying the momentum equations by $x$, where the key observation is that the term $\int_{\Omega}\text{div}S\cdot xdx$ vanishes, if either the domain under consideration has no physical boundary or the isolated mass group never touches the boundary. Unfortunately, it is not the case when there is some physical boundary of the domain, and one can not expect that the isolated mass group will never touch the boundary, even if it is initially away from the boundary. Therefore, we will always encounter some boundary integrals coming from $\int_{\Omega}\text{div}S\cdot xdx$ after integration by parts. To overcome this difficulty, taking the case $\Omega=\mathbb{R}^{N}\setminus B_{r_{0}}$ as an example, we multiply the transport equation by some positive function $f(|x|)$, rather than the very special function $|x|^{2}$, and correspondingly multiply the momentum equation by $f^{\prime}(|x|)\frac{x}{|x|}$, and encounter the term $\int_{\Omega}\text{div}Sf^{\prime}(|x|)\frac{x}{|x|}dx$. To ensure that this last term vanishes, by integration by parts, it suffices to ask for $f^{\prime}(|x|)|_{\partial\Omega}=0$ and $\Delta(f^{\prime}(|x|)\frac{x}{|x|})=0$ on $\Omega$. The existence of such an auxiliary function $f$ can be easily verified, and consequently one can obtain the finite time blow up results.

2. Proofs of the theorems

Given a velocity field $u\in C^{1}([0,T];H^{m}(\Omega))$, for some $m>[\frac{N}{2}]+2$, with $u=0$ on $\partial\Omega$. Denote by $X(t;x)$ the particle path, which goes along the velocity field $u$ and starts from $x\in\Omega$ at time zero:

[TABLE]

For any subset $K\subseteq\Omega$, for simplicity we denote

[TABLE]

By the Sobolev embedding, one has $u\in C^{1}([0,T];C^{2}(\overline{\Omega}))$, and thus, by the standard existence and uniqueness results for ordinary differential equations, the particle pathes are well-defined, and different particle pathes never meet each other. Moreover, at each time, any point $y\in\Omega$ can be reached by some particle path, in other words, one has $X(t;\Omega)=\Omega$. Using these facts, one can easily verify that

[TABLE]

for any subsets $K$, $K_{1}$ and $K_{2}$ of $\Omega$. These facts will be used later without any further mentions.

Recalling the expression of the stress tensor $S$, we have

[TABLE]

which simply implies that

[TABLE]

if $\lambda\geq 0$. While if $\lambda<0$, by transforming $S:\nabla u$ as

[TABLE]

and recalling (1.8), we still have

[TABLE]

Some preparations are required before proving our main results, that is the following two propositions.

Proposition 2.1**.**

Let $(\rho,u,\theta)\in C^{1}([0,T];H^{m}(\Omega))$, with $m>[\frac{N}{2}]+2$, be a classical solution to system (1.5)–(1.7), subject to (1.9)–(1.10). Suppose that

[TABLE]

for some positive number $R_{0}$ (with $R_{0}>r_{0}$, if $\Omega=\mathbb{R}^{N}\setminus\overline{B}_{r_{0}}$). Then, we have

[TABLE]

Proof.

By the definition of $X(t;x)$, and using (1.5), we deduce

[TABLE]

from which, by assumption, and recalling $u\in C^{1}([0,T];C^{2}(\overline{\Omega}))$, we have

[TABLE]

for any $x\in B_{R_{0}}^{c}\cap\Omega$. Hence, one has

[TABLE]

Thanks to this, it follows from equations (1.6) and (1.7) that

[TABLE]

and thus

[TABLE]

As a result, it follows from (2.15) and (2.16) that

[TABLE]

This and the assumption $u\in C^{1}([0,T];H^{m}(\Omega))$ imply

[TABLE]

Thus, for any $x\in B_{R_{0}}^{c}\cap\Omega$, one has

[TABLE]

which implies $X(t;x)=x$, for any $x\in B_{R_{0}}^{c}\cap\Omega$. Therefore, we have

[TABLE]

and consequently, the conclusion follows from (2.18) and (2.19). ∎

Proposition 2.2**.**

Let $(\rho,u,\theta)\in C^{1}([0,T];H^{m}(\Omega))$, with $m>[\frac{N}{2}]+2$, be a classical solution to system (1.5)–(1.7), subject to (1.9)–(1.10). Suppose that the compatibility condition (1.11) holds. Then, we have

[TABLE]

for any $x\in\mathcal{O}_{+}(t):=X(t;\mathcal{O}_{+})$, and

[TABLE]

Proof.

For any $x\in\mathcal{O}_{+}$, by equation (2.17), we have $\rho(X(t;x),t)>0$. Hence, $\rho(x,t)>0$, for any $x\in\mathcal{O}_{+}(t)$. To prove the positivity of $\theta$ on $\mathcal{O}_{+}(t)$, we need to derive the equation for $\theta$. Using equation (1.5) and the state equation $e=c_{v}\theta$, it follows from equation (1.7) that

[TABLE]

Multiplying equation (1.6) by $u$, and using equation (1.5) yields

[TABLE]

Subtracting the previous two equations, recalling the state equation $p=R\rho\theta$, one obtains

[TABLE]

Hence, recalling the nonnegativity of $S:\nabla u$, see (2.15) and (2.16), we deduce

[TABLE]

from which, recalling that $u\in C^{1}([0,T];C^{2}(\overline{\Omega}))$, one obtains $\theta(X(t;x),t)>0$, for any $x\in\mathcal{O}_{+}$. Therefore, we have $\theta(x,t)>0$, for any $x\in\mathcal{O}_{+}(t)$. This proves the first conclusion.

Now, let us prove the second conclusion. Take arbitrary $x\in\mathcal{O}_{+}$, then $\rho(X(t;x),t)>0$ and $\theta(X(t;x),t)>0$. Set $l(t)=(\theta\rho^{1-\gamma})|_{(X(t;x),t)}$, then it follows from equations (1.5) and (2.20) that

[TABLE]

Hence, one has

[TABLE]

from which, by taking the logarithm to both sides of the above inequality yields

[TABLE]

Therefore, we have

[TABLE]

proving the second conclusion. ∎

We are now ready to prove the main results.

Proof of Theorem 1.1.

Case I: $\Omega=\mathbb{R}^{N}\setminus B_{r_{0}}$. Define two radially symmetric functions $f$ and $g$ on $\mathbb{R}^{N}\setminus\{0\}$ as

[TABLE]

and

[TABLE]

Then, one can check that

[TABLE]

Define $m_{0},m_{1}$ and $m_{2}$ as

[TABLE]

Then, by assumption, we have $m_{0}>0$. Note that, using Proposition 2.1 and the boundary condition (1.9), we have

[TABLE]

Using equation (1.5) and the above boundary conditions, it follows from integration by parts that

[TABLE]

Noticing that $g|_{\partial B_{r_{0}}}=0$, and recalling the boundary conditions (2.21), it follows from equation (1.6) and integration by parts that

[TABLE]

Recalling (2.21), it follows from equation (1.5) and integration by parts that

[TABLE]

which provides

[TABLE]

Applying Proposition 2.2, for any $x\in\mathcal{O}_{+}(t):=X(t;\mathcal{O}_{+})$, we have $\rho(x,t)>0$, $\theta(x,t)>0$, and

[TABLE]

Thanks to the above estimate, and noticing that $\mathcal{O}_{+}(t)=\{x\in\Omega|\rho(x,t)>0\},$ we deduce

[TABLE]

By the Hölder inequality, we have

[TABLE]

and thus, recalling (2.24), we have

[TABLE]

where $\omega_{N}$ is the volume of the unit ball in $\mathbb{R}^{N}$. Substituting the above estimate into (2.25) yields

[TABLE]

Thanks to (2.26), it follows from (2.23) that

[TABLE]

With the aid of the above estimate, it follows from (2.22) that

[TABLE]

On the other hand, by (2.24), one has

[TABLE]

Combining the above two estimates, we then obtain

[TABLE]

which, recalling that $m_{0}>0$, implies $t\leq T_{*}$, for some finite time $T_{*}$. Therefore, $(\rho,u,\theta)$ can not exist for all time. This completes the proof of Case I.

Case II: $\Omega=\mathbb{R}^{N}_{+}$. Define $M_{0},M_{1}$ and $M_{2}$ as

[TABLE]

Then, by assumption, one has $M_{0}>0$. By Proposition 2.1 and the boundary condition (1.9), we have

[TABLE]

Using equation (1.5), it follows from integration by parts that

[TABLE]

and

[TABLE]

Recalling the boundary conditions (2.27), and using equation (1.6), it follows from integration by parts that

[TABLE]

By (2.28) and the assumption, we have

[TABLE]

Following the same argument as that for (2.26), one can obtain

[TABLE]

and thus, it follows from (2.30) that

[TABLE]

which, substituted into (2), yields

[TABLE]

On the other hand, recalling (2.32), we have

[TABLE]

Combining the above two estimates, and recalling that $M_{0}>0$, one obtains $t\leq T_{**}$, for some positive time $T_{**}$. This completes the proof of Case II. ∎

Proof of Theorem 1.2.

Note that for $N=1$, the cases $\Omega=\mathbb{R}^{N}\setminus\overline{B}_{r_{0}}$ and $\Omega=\mathbb{R}^{N}_{+}$ are essentially the same, because the domain $\mathbb{R}\setminus[-r_{0},r_{0}]$ breaks into two half lines, and system (1.5)–(1.7) on these two half lines does not effect each other. Therefore, we only need to consider the case $\Omega=\mathbb{R}_{+}$. Denote

[TABLE]

Then, by assumption, we have $K_{0}>0$. By Proposition 2.1 and the boundary condition (1.9), we have

[TABLE]

Following the arguments in Case II of the proof of Theorem 1.1, we have

[TABLE]

and

[TABLE]

Integrating equation (1.7) over $(0,R_{0})\times(0,t)$ yields

[TABLE]

Note that, by the assumption $\int_{0}^{\infty}\rho_{0}dx>0$ and using the compatibility condition (1.11), one has $\mathcal{E}_{0}>0$.

If $R\geq 2c_{v}$, i.e. $\gamma\geq 3$, then

[TABLE]

and if $R<2c_{v}$, i.e. $\gamma\in(1,3)$, then

[TABLE]

Therefore, for any $\gamma\in(1,\infty)$, we have

[TABLE]

Thanks to the above, and recalling (2.36), we then obtain

[TABLE]

Substituting this into (2.35), and integrating in $t$ yields

[TABLE]

With the aid of the above estimate, it following from (2.34) that

[TABLE]

On the other hand, recalling (2.33), we have

[TABLE]

Combing the above two estimates, and recalling that $\mathcal{E}_{0}>0$, one obtains $t\leq T_{***}$, for some positive time $T_{***}$. This proves the conclusion. ∎

Acknowledgments

D.Bian was partially supported by NSFC under the contracts 11501028 and 11871005. J.Li was partly supported by start-up fund 550-8S0315 of the South China Normal University, NSFC 11771156, and the Hong Kong RGC Grant CUHK-14302917.