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 \(\mu q\leq1, H(\rho)\leq A(\rho)\). □