Interior regularity criterion for incompressible Ericksen-Leslie system

Abstract

An interior regularity criterion of suitable weak solutions is formulated for the Ericksen-Leslie system of liquid crystals. Such a criterion is point-wise, with respect to some appropriate norm of velocity u and the gradient of d, and it can be viewed as a sort of simply sufficient condition on the local regularity of suitable weak solutions.

Introduction and main results

In this paper, we investigate the local regularity of weak solutions to the following 3D incompressible Ericksen-Leslie liquid crystal system:

$$\begin{aligned} &\partial_{t}u+(u\cdot\nabla)u-\Delta u+\nabla P=-\nabla\cdot(\nabla d\odot\nabla d), \end{aligned}$$

(1.1a)

$$\begin{aligned} & \nabla\cdot u=0, \end{aligned}$$

(1.1b)

$$\begin{aligned} &\partial_{t}d+(u\cdot\nabla) d =\Delta d-f(d), \end{aligned}$$

(1.1c)

with the initial boundary conditions

$$\begin{aligned} \begin{aligned} &(u, d) (x,t)|_{t=0}= \bigl(u_{0}(x), d_{0}(x) \bigr),\qquad \nabla\cdot u_{0}=0, \quad x\in\Omega, \\ &(u,d) (x,t)|_{x\in\partial\Omega}= \bigl(0, d_{0}(x) \bigr),\qquad u_{0}(x)\in H_{0}^{1}(\Omega ),\qquad d_{0}(x)\in H_{0}^{2}(\Omega), \end{aligned} \end{aligned}$$

(1.2)

where $u, d, P$ denote the velocity of the fluid, the uniaxial molecular direction, and the pressure, respectively, the $i,j$th element of $\nabla d\odot\nabla d$ is $\partial_{i}d^{k}\partial_{j}d^{k}$, $d_{0}(x)$ is a unit vector, $\Omega\subset \mathbb {R}^{3}$ is a smooth domain. Additionally, $f(d)=\nabla F(d)$, and $F(d)=\frac{1}{\zeta^{2}}( \vert d \vert ^{2}-1)^{2}, \zeta$ is a small number, formally speaking, as $\zeta\to0, d$ tends to a unit vector.

The dynamic flows of liquid crystals have been successfully described by the Ericksen-Leslie theory [1–4]. System (1.1a)-(1.1c) is a coupled system of the Navier-Stokes equations with a parabolic system. It is Leray [5] and Hopf [6] that established the global existence of weak solutions to the 3D Navier-Stokes; however, the regularity of the weak solutions is still an open problem. Since the regularity of weak solutions to the 3D Navier-Stokes equations is hard to get, some related conditions or criteria for the regularity of the weak solutions are considered, such as the well-known Serrin type criterion [7] and the Beale-Kato-Majda type criterion [8]. Furthermore, based on the suitable weak solutions, some point-wise sufficient regularity criteria were imposed in [9–12].

The global existence of suitable weak solutions to system (1.1a)-(1.1c) was established in [13, 14] by Lin and Liu; however, noticing that system (1.1a)-(1.1c) contains the 3D Navier-Stokes equations as a subsystem, the uniqueness and regularity of these weak solutions are not known. In this paper, we would extend some point-wise sufficient conditions, which guarantee the local regularity of weak solutions for 3D Navier-Stokes equations, to the Ericksen-Leslie system (1.1a)-(1.1c). We would like to mention that when $f(d)$ in system (1.1a)-(1.1c) is replaced by $- \vert \nabla d \vert ^{2}d$, the global existence of weak solutions to the resulting system in three dimensions has only been known under the additional assumption that $d_{3}\geq0$ or small initial data (see [15, 16]). Without these conditions, the general existence of weak solutions is still open. However, the Serrin type criterion and the Beale-Kato-Majda type criterion still hold true even for a weak solution (if it exists) (see [17, 18]).

The suitable weak solution established in [14] can be stated as below.

Definition 1.1

Suitable weak solutions in $\Omega\times(0, T)\subset\mathbb {R}^{3}\times(0,\infty)$

A pair $(u, d)$ is called a suitable weak solution to system (1.1a)-(1.1c) and (1.2) in an open set $\mathcal {O}\subset\mathbb {R}^{3}\times(0,\infty)$ (we set $\mathcal {O}_{t}=\mathcal {O}\cap(\mathbb {R}^{3}\times\{t\} )$), if it satisfies the following properties:

• $(u, d)$ is a weak solution in the sense of distribution;

• $u\in L^{\infty}(0,T;L^{2}(\Omega ))\cap L^{2}(0,T;H^{1}(\Omega)), d\in L^{\infty}(0,T;H^{1}(\Omega))\cap L^{2}(0,T;H^{2}(\Omega))$, or generally, there exist constants $E_{1}, E_{2}$, such that

  $$\begin{aligned} &\int_{\mathcal {O}_{t}} \bigl[ \vert u \vert ^{2}+ \vert \nabla d \vert ^{2}+F(d) \bigr]\,\mathrm{ d}x< E_{1},\\ &\int\!\int_{\mathcal {O}} \bigl[ \vert \nabla u \vert ^{2}+ \bigl\vert \Delta d-f(d) \bigr\vert ^{2}+F(d) \bigr]\,\mathrm{ d}x\,\mathrm{ d}t< E_{2}; \end{aligned}$$

• for any $\varphi\in C_{c}^{\infty}(\mathcal {O})$, more specifically, for any $\varphi\in C_{c}^{\infty}(B(x_{0}, R)\times(t_{0}-R^{2}, t_{0}))$, the following generalized energy inequality holds

  $$\begin{aligned} &\int_{B(x_{0}, R)} \bigl( \vert u \vert ^{2}+ \vert \nabla d \vert ^{2} \bigr) \varphi\,\mathrm{ d}x+2 \int _{t_{0}-R^{2}}^{t} \int_{B(x_{0}, R)} \bigl( \vert \nabla u \vert ^{2}+ \bigl\vert \nabla^{2} d \bigr\vert ^{2} \bigr)\varphi\,\mathrm{ d}x \,\mathrm{ d}\tau \\ &\quad\leq \int_{t_{0}-R^{2}}^{t} \int_{B(x_{0}, R)} \bigl\{ \bigl( \vert u \vert ^{2}+ \vert \nabla d \vert ^{2} \bigr) (\varphi _{t}+\Delta \varphi)+ \bigl( \vert u \vert ^{2}+ \vert \nabla d \vert ^{2}+2P \bigr)u\cdot\nabla\varphi \bigr\} \,\mathrm{ d}x\,\mathrm{ d}\tau \\ &\qquad{}+2 \int_{t_{0}-R^{2}}^{t} \int_{B(x_{0}, R)} \bigl((u\cdot\nabla) d\nabla d\nabla \varphi-\nabla f(d):\nabla d\varphi \bigr)\,\mathrm{ d}x\,\mathrm{ d}\tau. \end{aligned}$$

  (1.3)

In the following, we can take $Q((x_{0}, t_{0}), R)\equiv B(x_{0}, R)\times(t_{0}-R^{2}, t_{0})$, $B(x_{0}, R)\equiv\{y\in\mathbb {R}^{3}| \vert y-x_{0} \vert < R\}, z_{0}\equiv(x_{0}, t_{0})$ for simplicity.

We now state our main result of this paper.

Theorem 1.2

Let $(u, d)$ be a suitable weak solution to liquid crystal system (1.1a)-(1.1c) in $Q(z_{0}, R)$. The real numbers $l\geq1$ and $s\geq1$ satisfy

$$ \frac{1}{2}\geq\frac{3}{s}+\frac{2}{l}-\frac{3}{2}>\max \biggl\{ \frac{1}{2l}, \frac{1}{2}-\frac{1}{s}, \frac{1}{s}- \frac{1}{6} \biggr\} . $$

Then there is a positive number $\varepsilon=\varepsilon(s,l)$, such that if

$$ M^{s, l}(z_{0}, R)=\frac{1}{R^{\kappa}} \int_{t_{0}-R^{2}}^{t_{0}} \biggl( \int_{B(x_{0}, R)} \vert u \vert ^{s}+ \vert \nabla d \vert ^{s}\,\mathrm{ d}x \biggr)^{\frac{l}{s}}\,\mathrm{ d}t< \varepsilon,\quad \kappa=\frac{3l}{s}+2-l, $$

then $z_{0}$ is a regular point of $(u,\nabla d)$, i.e. $(u,\nabla d)$ is Hölder continuous in $Q(z_{0},r)$, for some $r\in(0, R]$.

Throughout this paper, we use c to denote a generic positive constant which can be different from line to line.

Preliminaries

As the preparation for proving Theorem 1.2, we first give two auxiliary lemmas.

Lemma 2.1

We have

$$ D(z_{0}, r; p)\leq c \biggl[\frac{r}{\rho}D(z_{0}, \rho; p)+ \biggl(\frac{\rho}{r} \biggr)^{2}C(z_{0}, \rho; u, \nabla d) \biggr], $$

(2.1)

where

$$ C(z_{0},r;u,\nabla d)=\frac{1}{r^{2}} \int_{Q(z_{0},r)} \bigl( \vert u \vert ^{3}+ \vert \nabla d \vert ^{3} \bigr) \,\mathrm{ d}z, \qquad D(z_{0},r;p)= \frac{1}{r^{2}} \int_{Q(z_{0},r)} \vert p \vert ^{\frac{3}{2}}\,\mathrm{ d}z. $$

Proof

Step 1. For (1.1a), we choose the test function $w=\chi\nabla q$, for any $\chi\in C_{c}^{\infty}((t_{0}-\rho^{2}, t_{0})), q\in C_{c}^{\infty}(B(x_{0}, \rho))$, then it yields

$$ \int_{Q(z_{0}, \rho)}-u\cdot\partial_{t}\chi\nabla q-(u\otimes u+ \nabla d\odot\nabla d):\chi\nabla^{2} q-u\cdot\chi\nabla\Delta q\,\mathrm{ d}z= \int _{Q(z_{0}, \rho)}p\chi\Delta q \,\mathrm{d}z. $$

It follows from $\nabla\cdot u=0$ that

$$\begin{aligned} - \int_{Q(z_{0}, \rho)}p\chi\Delta q \,\mathrm{d}z= \int_{Q(z_{0}, \rho)}\chi (u\otimes u+\nabla d\odot\nabla d): \nabla^{2} q\,\mathrm{ d}z. \end{aligned}$$

Therefore, for a.e. $t\in(t_{0}-\rho^{2}, t_{0})$, we have

$$ - \int_{B(x_{0}, \rho)}p\Delta q \,\mathrm{d}x= \int_{B(x_{0}, \rho)}(u\otimes u+\nabla d\odot\nabla d): \nabla^{2} q\,\mathrm{ d}x,\quad \forall q\in C_{c}^{\infty}\bigl(B(x_{0}, \rho) \bigr). $$

(2.2)

Step 2. Approximate p with $p_{1}$ by confining q in $W^{2, 3}(B(x_{0}, \rho))$.

Set $p_{1}\in L^{\frac{3}{2}}(Q(z_{0}, \rho))$ such that, for a.e. $t\in(t_{0}-\rho ^{2}, t_{0})$,

$$ - \int_{B(x_{0}, \rho)}p_{1}\Delta q \,\mathrm{d}x= \int_{B(x_{0}, \rho )}(u\otimes u+\nabla d\odot\nabla d): \nabla^{2} q\,\mathrm{ d}x, $$

(2.3)

for any $q(\cdot, t)\in W^{2, 3}(B(x_{0}, \rho))$, and $q(\cdot, t)=0 \text{ on } \partial B(x_{0}, \rho)$. The existence of $p_{1}$ is established due to the Lax-Milgram theorem with appropriate approximating process on u and d (see [11]).

Next, choose $q_{0}(\cdot, t)\in W^{2, 3}(B(x_{0}, \rho))$, such that, for a.e. $t\in(t_{0}-\rho^{2}, t_{0})$,

$$\begin{aligned} \Delta q_{0}(\cdot, t)=- \bigl\vert p_{1}(\cdot, t) \bigr\vert ^{\frac{1}{2}} \operatorname{sgn}p_{1}(\cdot, t),\quad \text{in } B(x_{0}, \rho),\qquad q_{0}(\cdot, t)=0, \quad\text{on } \partial B(x_{0}, \rho). \end{aligned}$$

Then, by the Calderon-Zygmund inequality, it yields

$$\begin{aligned} \biggl( \int_{B(x_{0}, \rho)} \bigl\vert \nabla^{2}q_{0}( \cdot, t) \bigr\vert ^{3}\,\mathrm{ d}x \biggr)^{\frac{1}{3}}\leq c \biggl( \int_{B(x_{0}, \rho)} \bigl\vert p_{1}(\cdot, t) \bigr\vert ^{\frac{3}{2}}\,\mathrm{ d}x \biggr)^{\frac{1}{3}}, \quad\mbox{a.e. }t\in \bigl(t_{0}- \rho^{2}, t_{0} \bigr). \end{aligned}$$

Therefore, it follows from (2.3) and the Hölder inequality that

$$\begin{aligned} \int_{B(x_{0}, \rho)} \bigl\vert p_{1}(\cdot, t) \bigr\vert ^{\frac{3}{2}}\,\mathrm{ d}x &\leq c \biggl( \int_{B(x_{0}, \rho)} \vert u \vert ^{3}+ \vert \nabla d \vert ^{3}\,\mathrm{ d}x \biggr)^{\frac{2}{3}} \biggl( \int_{B(x_{0}, \rho)} \bigl\vert \nabla^{2}q \bigr\vert ^{3}\,\mathrm{ d}x \biggr)^{\frac{1}{3}} \\ &\leq c \biggl( \int_{B(x_{0}, \rho)} \vert u \vert ^{3}+ \vert \nabla d \vert ^{3}\,\mathrm{ d}x \biggr)^{\frac{2}{3}} \biggl( \int_{B(x_{0}, \rho)} \vert p_{1} \vert ^{\frac{3}{2}} \,\mathrm{ d}x \biggr)^{\frac{1}{3}}, \end{aligned}$$

which yields $\int_{Q(z_{0}, \rho)} \vert p_{1}(\cdot, t) \vert ^{\frac{3}{2}}\,\mathrm{ d}z\leq c\rho^{2}C(z_{0}, \rho; u, \nabla d)$.

Step 3. Estimates for the remainder $p-p_{1}$.

For a.e. $t\in(t_{0}-\rho^{2}, t_{0})$, let $p_{2}=p-p_{1}$, then from (2.2)-(2.3) one infers that

$$\begin{aligned} \Delta p_{2}(\cdot, t)=0,\quad \text{in } B(x_{0}, \rho). \end{aligned}$$

By the harmonic property, one can get

$$\begin{aligned} \frac{1}{r^{3}} \int_{Q(z_{0}, r)} \vert p_{2} \vert ^{\frac{3}{2}} \,\mathrm{ d}z\leq \frac{c}{\rho ^{3}} \int_{Q(z_{0}, \rho)} \vert p_{2} \vert ^{\frac{3}{2}} \,\mathrm{ d}z,\quad \forall r< \rho, \end{aligned}$$

while

$$ \int_{Q(z_{0}, \rho)} \vert p_{2} \vert ^{\frac{3}{2}} \,\mathrm{ d}z \leq \int_{Q(z_{0}, \rho)} \bigl( \vert p \vert ^{\frac{3}{2}}+ \vert p_{1} \vert ^{\frac{3}{2}} \bigr)\,\mathrm{ d}z\leq c\rho ^{2} \bigl(D(z_{0}, \rho; p)+C(z_{0}, \rho; u, \nabla d) \bigr). $$

Step 4. Estimates for p.

We have

$$\begin{aligned} D(z_{0}, r; p)&\leq c \biggl(\frac{1}{r^{2}} \int_{Q(z_{0}, r)} \vert p_{1} \vert ^{\frac{3}{2}} \,\mathrm{ d}z+ \frac{r}{\rho^{3}} \int_{Q(z_{0}, \rho)} \vert p_{2} \vert ^{\frac{3}{2}} \,\mathrm{ d}z \biggr) \\ &\leq c \biggl(\frac{\rho^{2}}{r^{2}}\frac{1}{\rho^{2}} \int_{Q(z_{0}, r)} \vert p_{1} \vert ^{\frac{3}{2}} \,\mathrm{ d}z+ \frac{r}{\rho}\frac{1}{\rho^{2}} \int_{Q(z_{0}, \rho)} \vert p_{2} \vert ^{\frac{3}{2}} \,\mathrm{ d}z \biggr) \\ &\leq c \biggl[\frac{\rho^{2}}{r^{2}}C(z_{0}, \rho; u, \nabla d)+ \frac{r}{\rho } \bigl(D(z_{0}, \rho; p)+C(z_{0}, \rho; u, \nabla d) \bigr) \biggr] \\ &\leq c \biggl[\frac{r}{\rho}D(z_{0}, \rho; p)+ \biggl( \frac{\rho}{r} \biggr)^{2}C(z_{0}, \rho; u, \nabla d) \biggr]. \end{aligned}$$

 □

We denote

$$\begin{aligned} &A(\rho)=\operatorname{ess}\sup_{{t_{0}}-\rho ^{2}< t< {t_{0}}}\frac{1}{\rho}\int_{B(x_{0},\rho )} \bigl( \bigl\vert u(t) \bigr\vert ^{2}+ \bigl\vert \nabla d(t) \bigr\vert ^{2} \bigr)\,\mathrm{ d}x, \\ & E(\rho)=\frac{1}{\rho}\int_{Q(z_{0},\rho)} \bigl( \vert \nabla u \vert ^{2}+ \bigl\vert \nabla^{2} d \bigr\vert ^{2} \bigr)\,\mathrm{ d}z,\qquad H( \rho)=\frac{1}{\rho^{3}} \int_{Q(z_{0},\rho )} \bigl( \vert u \vert ^{2}+ \vert \nabla d \vert ^{2} \bigr) \,\mathrm{ d}z. \end{aligned}$$

Lemma 2.2

Under the assumptions of Theorem 1.2, we have

$$C(\rho)\leq c\epsilon^{\frac{1}{q}} \bigl(E(\rho)+A(\rho)+1 \bigr), $$

where $q=2l(\frac{3}{s}+\frac{2}{l}-\frac{3}{2})$, and $q'=\frac{q}{q-1}$.

Proof

With the help of the Hölder and Sobolev embedding inequalities, one gets

$$\begin{aligned} \int_{B(x_{0}, \rho)} \vert v \vert ^{3}\,\mathrm{ d}x={}& \int_{B(x_{0}, \rho)} \vert v \vert ^{\lambda s+2\mu+6\gamma}\,\mathrm{ d}x \\ \leq {}&\biggl( \int_{B(x_{0}, \rho)} \vert v \vert ^{2}\,\mathrm{ d}x \biggr)^{\mu}\biggl( \int _{B(x_{0}, \rho)} \vert v \vert ^{s}\,\mathrm{ d}x \biggr)^{\lambda}\biggl( \int_{B(x_{0}, \rho)} \vert v \vert ^{6}\,\mathrm{ d}x \biggr)^{\gamma}\\ \leq{}& \frac{c}{2}\rho^{\mu}\biggl(\operatorname{ess}\sup_{t_{0}-\rho^{2}< t< t_{0}} \frac{1}{\rho}\int _{B(x_{0},\rho)} \vert v \vert ^{2}\,\mathrm{ d}x \biggr)^{\mu}\biggl( \int_{B(x_{0}, \rho)} \vert v \vert ^{s}\,\mathrm{ d}x \biggr)^{\lambda}\\ &{}\times \biggl( \int_{B(x_{0}, \rho)} \vert \nabla v \vert ^{2}+ \frac{1}{\rho ^{2}} \vert v \vert ^{2} \,\mathrm{ d}x \biggr)^{3\gamma}, \end{aligned}$$

where $\lambda s+2\mu+6\gamma=3, \lambda+\mu+\gamma=1$. Substituting v by u and ∇d, respectively, then one can get the summation

$$\begin{aligned} \int_{B(x_{0}, \rho)} \vert u \vert ^{3}+ \vert \nabla d \vert ^{3}\,\mathrm{ d}x \leq{}&c\rho^{\mu}A^{\mu}( \rho) \biggl( \int_{B(x_{0}, \rho)} \bigl( \vert u \vert ^{s}+ \vert \nabla d \vert ^{s} \bigr) \,\mathrm{ d}x \biggr)^{\lambda}\\ &{}\times \biggl( \int_{B(x_{0}, \rho)} \bigl( \vert \nabla u \vert ^{2}+ \bigl\vert \nabla^{2} d \bigr\vert ^{2} \bigr)+ \frac{1}{\rho^{2}} \bigl( \vert u \vert ^{2}+ \vert \nabla d \vert ^{2} \bigr)\,\mathrm{ d}x \biggr)^{3\gamma}. \end{aligned}$$

Therefore, by choosing appropriate parameters $\lambda=\frac{1}{2s(\frac{3}{s}+\frac{2}{l}-\frac{3}{2})}$, $\mu=\frac {\frac{3}{s}+\frac{3}{l}-2}{2(\frac{3}{s}+\frac{2}{l}-\frac{3}{2})}$, $\gamma= \frac{\frac{2}{s}+\frac{1}{l}-1}{2(\frac{3}{s}+\frac{2}{l}-\frac{3}{2})}$, and integrating from $t_{0}-\rho^{2}$ to $t_{0}$ with the variable t, it follows from the Hölder and Young inequalities that

$$\begin{aligned} C(\rho)\leq{}&c\rho^{\mu-2} A^{\mu}(\rho){ \biggl( \int_{Q(z_{0}, \rho )} \bigl( \vert \nabla u \vert ^{2}+ \bigl\vert \nabla^{2} d \bigr\vert ^{2} \bigr)+ \frac{1}{\rho^{2}} \bigl( \vert u \vert ^{2}+ \vert \nabla d \vert ^{2} \bigr)\,\mathrm{ d}z \biggr)}^{\frac{1}{q'}} \\ &{}\times \biggl[ \int_{t_{0}-\rho^{2}}^{t_{0}} \biggl( \int_{B(x_{0}, \rho )} \bigl( \vert u \vert ^{s}+ \vert \nabla d \vert ^{s} \bigr) \,\mathrm{ d}x \biggr)^{\frac{l}{s}} \,\mathrm{ d}t \biggr]^{\frac{1}{q}} \\ \leq{}&c\rho^{\mu-2} A^{\mu}(\rho)\rho^{\frac{1}{q'}} \bigl(E(\rho)+H( \rho) \bigr)^{\frac{1}{q'}} \bigl(\rho^{\kappa}M^{s,l}(\rho) \bigr)^{\frac{1}{q}} \\ \leq{}&c A^{\mu}(\rho) \bigl(E(\rho)+H(\rho) \bigr)^{\frac{1}{q'}} \bigl(M^{s,l}(\rho) \bigr)^{\frac{1}{q}} \\ \leq{}&c \epsilon^{\frac{1}{q}} A^{\mu}(\rho) \bigl(E(\rho)+H(\rho) \bigr)^{\frac{1}{q'}} \\ \leq{}&c \epsilon^{\frac{1}{q}} \bigl(A^{\mu q}(\rho)+E(\rho)+H(\rho) \bigr) \\ \leq{}&c \epsilon^{\frac{1}{q}} \bigl(E(\rho)+A(\rho)+1 \bigr), \end{aligned}$$

where $\kappa=\frac{3l}{s}+2-l$ as in Theorem 1.2, and in the last step, we used the fact that $\mu q\leq1, H(\rho)\leq A(\rho)$. □

Proof of Theorem 1.2

Due to the induction argument as Proposition 2.6 in [10] or Lemma 2.2 in [19] (the parabolic version of the Campanato criterion), to get the desired consequence, it suffices to prove $C(\theta^{k})+D(\theta^{k})< \epsilon_{0}$ for some small $\epsilon_{0}$. Here θ is a small number, which will be chosen later.

From the generalized energy inequality, it is easy to check that, for $\rho\in(0, R]$,

$$ A \biggl(\frac{\rho}{2} \biggr)+E \biggl(\frac{\rho}{2} \biggr)\leq c \bigl[C^{\frac{2}{3}}(\rho)+C(\rho)+D(\rho) \bigr]. $$

Denoting $G(\rho)=A(\rho)+E(\rho)+D(\rho)$, due to Lemmas 2.1-2.2, and the fact that $C(2\theta\rho)\leq\frac{1}{4\theta^{2}}C(\rho)$, we can get

$$\begin{aligned} G(\theta\rho)\leq{}& c \biggl[C^{\frac{2}{3}}(2\theta\rho)+C(2\theta \rho)+D(2 \theta \rho)+\theta D(\rho)+\frac{1}{\theta^{2}}C(\rho) \biggr] \\ \leq{}& c \biggl[\frac{1}{\theta^{\frac{4}{3}}}C^{\frac{2}{3}}(\rho)+ \frac{1}{\theta^{2}}C( \rho )+\theta D(\rho) \biggr] \\ \leq{}& c \biggl[\frac{\epsilon^{\frac{2}{3q}}}{\theta^{\frac{4}{3}}} \bigl(G(\rho)+1 \bigr)^{\frac{2}{3}}+ \frac{\epsilon^{\frac{1}{q}}}{\theta^{2}} \bigl(G(\rho)+1 \bigr)+\theta G(\rho) \biggr] \\ \leq{}& c \biggl[ \biggl(\theta+\frac{\epsilon^{\frac{1}{2q}}}{\theta^{2}} \biggr)G(\rho)+ \frac {\epsilon^{\frac{1}{2q}}}{\theta^{2}} \biggr], \end{aligned}$$

where in the last step we have used $\frac{\epsilon^{\frac{2}{3q}}}{\theta^{\frac{4}{3}}}(G(\rho)+1)^{\frac{2}{3}}\leq c[\epsilon^{\frac{1}{q}}+\frac{\epsilon^{\frac{1}{2q}}}{\theta^{2}}(G(\rho)+1)]$. Now choosing θ and ϵ such that $c\theta<\frac{1}{4}$ and $c\frac{\epsilon^{\frac{1}{2q}}}{\theta^{2}}<\frac{1}{4}$, then it yields $G(\theta\rho)\leq\frac{1}{2}G(\rho)+c\frac{\epsilon^{\frac{1}{2q}}}{\theta^{2}}$. Iterating the above process, we obtain $G(\theta^{k} \rho)\leq\frac{1}{2^{k}}G(\rho)+c\frac{\epsilon^{\frac{1}{2q}}}{\theta^{2}}$, therefore,

$$ D \bigl(\theta^{k}\rho \bigr)\leq\frac{1}{2^{k}}G( \rho)+c\frac{\epsilon^{\frac{1}{2q}}}{\theta^{2}}. $$

(3.1)

For $C(\theta^{k}\rho)$, by Lemma 2.2, we have

$$\begin{aligned} C \bigl(\theta^{k} \rho \bigr)\leq c \epsilon^{\frac{1}{q}} \bigl[ G \bigl(\theta^{k} \rho \bigr)+1 \bigr]\leq c \epsilon^{\frac{1}{q}} \biggl[\frac{1}{2^{k}}G(\rho)+ \frac{\epsilon^{\frac{1}{2q}}}{\theta ^{2}}+1 \biggr]\leq c \biggl[\frac{1}{2^{k}}G(\rho)+ \frac{\epsilon^{\frac{1}{2q}}}{\theta^{2}} \biggr], \end{aligned}$$

(3.2)

where in the last step we use the fact that $\epsilon^{\frac{1}{q}}\leq \frac{\epsilon^{\frac{1}{2q}}}{\theta^{2}}$ for ϵ small enough. With these inequalities in hand, for fixed ρ and $\epsilon_{0}$, we can choose $k_{0}$ large enough such that $c\frac{1}{2^{k_{0}}}G(\rho)<\frac{\epsilon_{0}}{4}$, and choose ϵ small enough, such that $c\frac{\epsilon^{\frac{1}{2q}}}{\theta^{2}}<\frac{\epsilon_{0}}{4}$. With these prerequisites and (3.1)-(3.2), it follows that $D(\theta^{k}\rho)+C(\theta^{k}\rho)<\epsilon_{0}$.