On the Striated Regularity for the 2D Anisotropic Boussinesq System

Abstract

In this paper, we investigate the global existence and uniqueness of strong solutions to the 2D anisotropic Boussinesq system for rough initial data with striated regularity. We prove the global well-posedness of the Boussinesq system with anisotropic thermal diffusion with initial vorticity being discontinuous across some smooth interface. In the case of an anisotropic horizontal viscosity, we study the propagation of the striated regularity for the smooth temperature patches problem. The proofs rely on the idea of Chemin to solve the 2-D vortex patch problem for ideal fluids, namely the striated regularity can help to bound the gradient of the velocity.

A. Appendix

The first two parts of this appendix are to give the proof of Lemmas 2.5 and 2.8. Then, we show the proof of the results given in Remark 2 in the last of this appendix.

Proof of Lemma 2.5

The proof of (2.6) can be found in Danchin and Paicu (2011) which used the standard Bony’s decomposition (see Chemin 1998; Bahouri et al. 2011). Here we focus on proving (2.7) using the anisotropic idea. Firstly, we divide the first term of (2.7) into two terms,

$$\begin{aligned} \begin{aligned} -\int _{{{\mathbb {R}}}^2}\Delta _q(u\cdot \nabla f)\Delta _q f~{\text {d}}x&=-\int _{{{\mathbb {R}}}^2}\Delta _q(u^1\partial _1 f)\Delta _q f~{\text {d}}x -\int _{{{\mathbb {R}}}^2}\Delta _q(u^2\partial _2 f)\Delta _q f~{\text {d}}x\\&\triangleq P+Q. \end{aligned} \end{aligned}$$

For P, by Bony’s decomposition, we can divide it into the following three terms,

$$\begin{aligned} \begin{aligned}&-\int _{{{\mathbb {R}}}^2}\Delta _q(u^1 \partial _1 f)\Delta _q f~{\text {d}}x\\&\quad =-\sum _{|k-q|\le 2}\int _{{{\mathbb {R}}}^2}\Delta _q(S_{k-1}u^1 \Delta _k\partial _1 f)\Delta _q f~{\text {d}}x\\&\qquad -\sum _{|k-q|\le 2}\int _{{{\mathbb {R}}}^2}\Delta _q(\Delta _ku^1 S_{k-1}\partial _1 f)\Delta _q f~{\text {d}}x\\&\qquad -\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{{{\mathbb {R}}}^2}\Delta _q(\Delta _ku^1 \Delta _l\partial _1 f)\Delta _q f~{\text {d}}x\\&\quad \triangleq P_1+P_2+P_3. \end{aligned} \end{aligned}$$

(A.1)

For $P_1$, we can rewrite it as

$$\begin{aligned} P_1= & {} -\sum _{|q-k|\le 2}\int _{{{\mathbb {R}}}^2}\Delta _q(S_{k-1}u^1\partial _1\Delta _k f)\Delta _q f~{\text {d}}x\\= & {} -\sum _{|q-k|\le 2}\int _{{{\mathbb {R}}}^2} [\Delta _q, S_{k-1}u^1\partial _1]\Delta _k f \Delta _q f~{\text {d}}x\\&\quad -\sum _{|q-k|\le 2}\int _{{{\mathbb {R}}}^2}S_{k-1}u^1\partial _1\Delta _q\Delta _k f \Delta _q f~{\text {d}}x\\= & {} -\sum _{|q-k|\le 2}\int _{{{\mathbb {R}}}^2}[\Delta _q, S_{k-1}u^1\partial _1]\Delta _k f \Delta _q f~{\text {d}}x\\&\quad -\sum _{|q-k|\le 2}\int _{{{\mathbb {R}}}^2}(S_{k-1}u^1-S_qu^1)\partial _1\Delta _q \Delta _kf\Delta _q f~{\text {d}}x\\&\quad -\int _{{{\mathbb {R}}}^2}{S}_{q}u^1\partial _1\Delta _q f\Delta _q f~{\text {d}}x\\&\triangleq P_{11}+P_{12}+P_{13}, \end{aligned}$$

where we have used the fact $\sum _{|q-k|\le 2}\partial _1\Delta _q\Delta _k f={\partial _1}\Delta _q f$. For $P_{11}$, by Hölder inequality,

$$\begin{aligned} \begin{aligned} |P_{11}|&\le \sum _{|q-k|\le 2}\bigg |\int _{{{\mathbb {R}}}^2}[\Delta _q, S_{k-1}u^1\partial _1]\Delta _k f\Delta _k f~{\text {d}}x\bigg |\\&\le C\sum _{|q-k|\le 2}\Vert [\Delta _q, S_{k-1}u^1\partial _1]\Delta _k f\Vert _{L^2}\Vert \Delta _qf\Vert _{L^2}. \end{aligned} \end{aligned}$$

According to the definition of $\Delta _q$,

$$\begin{aligned} \begin{aligned} \left[ \Delta _q, S_{k-1}u^1\partial _1\right] \Delta _k f&=\int _{{{\mathbb {R}}}^2}\phi _q(x-y)(S_{k-1}u^1(y)\partial _1\Delta _k f(y))~dy\\&\quad -S_{k-1}u^1(x)\int _{{{\mathbb {R}}}^d}\phi _q(x-y)\partial _1\Delta _k f(y)~dy\\&=\int _{{{\mathbb {R}}}^2}\phi _q(x-y)(S_{k-1}u^1(y)-S_{k-1}u^1(x))\partial _1\Delta _k f(y)~dy\\&=\int _{{{\mathbb {R}}}^2}\phi _q(x-y)\int _0^1(y-x)\cdot \nabla S_{k-1}u^1(sy+(1-s)x)~{\text {d}}s\partial _1\Delta _k f(y)~{\text {d}}y\\&=\int _{{{\mathbb {R}}}^2}\int _0^1\phi _q(z)z\cdot \nabla S_{k-1}u^1(x-sz)\partial _1\Delta _k f(x-z)~{\text {d}}s {\text {d}}z,\\ \end{aligned} \end{aligned}$$

where $\phi _j(x)\triangleq 2^{jd}{\mathcal {F}}^{-1}({\varphi })(2^jx)$. Thus, we have by Hölder inequality and Bernstein inequality,

$$\begin{aligned} \begin{aligned}&\Vert [\Delta _q, S_{k-1}u^1\partial _1]\Delta _k f\Vert _{L^2}\\&\quad =\bigg \Vert \int _{{{\mathbb {R}}}^2}\int _0^1\phi _q(z)z\cdot \nabla S_{k-1}u^1(x-sz)\partial _1\Delta _k f(x-z)~{\text {d}}s {\text {d}}z\bigg \Vert _{L^2}\\&\quad \le C\int _{{{\mathbb {R}}}^2}\big |\phi _q(z)\big ||z|~dz\Vert \nabla {S}_{k-1}u^1(x-sz)\Vert _{L^\infty } \Vert \partial _1\Delta _k f(x-z)\Vert _{L^2}\\&\quad \le C\int _{{{\mathbb {R}}}^2}\big |\phi _q(z)\big ||z|~dz\Vert \nabla S_{k-1}u^1\Vert _{L^\infty }\Vert \partial _1\Delta _k f\Vert _{L^2}\\&\quad \le C2^{-q}2^k\Vert \nabla S_{k-1}u^1\Vert _{L^2}\Vert \partial _1\Delta _k f\Vert _{L^2}\\&\quad \le C2^{k-q}\Vert \omega \Vert _{L^2}\Vert \partial _1\Delta _k f\Vert _{L^2}. \end{aligned} \end{aligned}$$

Then, we obtain

$$\begin{aligned} \begin{aligned} |P_{11}|&\le C\sum _{|q-k|\le 2}\Vert [\Delta _q, S_{k-1}u^1\partial _1]\Delta _k f\Vert _{L^2}\Vert \Delta _qf\Vert _{L^2}\\&\le C\sum _{|q-k|\le 2}2^{k-q}\Vert \omega \Vert _{L^2}\Vert \partial _1\Delta _k f\Vert _{L^2}\Vert \Delta _q f\Vert _{L^2}\\&\le Cb_q 2^{{-2qs}}\Vert \omega \Vert _{L^2}\Vert f\Vert _{H^s}\Vert \partial _1f\Vert _{H^s}. \end{aligned} \end{aligned}$$

For $P_{12}$, by Hölder inequality and Bernstein inequality,

$$\begin{aligned} \begin{aligned} |P_{12}|&=\sum _{|q-k|\le 2}\bigg |\int _{{{\mathbb {R}}}^2} ((S_{k-1}u^1-S_{q}u^1)\partial _1\Delta _q\Delta _k f)\Delta _q f~{\text {d}}x\bigg |\\&\le C\sum _{|q-k|\le 2}\Vert (S_{k-1}u^1-S_{q}u^1)\partial _1\Delta _q \Delta _kf\Vert _{L^1}\Vert \Delta _qf\Vert _{L^\infty }\\&\le C\sum _{|q-k|\le 2}\Vert \Delta _k u^1\Vert _{L^2}\Vert \Delta _q \Delta _k\partial _1 f\Vert _{L^2}2^q\Vert \Delta _qf\Vert _{L^2}. \end{aligned} \end{aligned}$$

For the case $k=-1$, by Bernstein inequality,

$$\begin{aligned} \begin{aligned} |P_{12}|&\le C\Vert \Delta _{-1} u^1\Vert _{L^2}2^{-1}\Vert \Delta _q \Delta _{-1} f\Vert _{L^2}2^{-1}\Vert \Delta _qf\Vert _{L^2}\\&\le Cb_q2^{{-2qs}}\Vert u^1\Vert _{L^2}\Vert f\Vert _{H^s}^2. \end{aligned} \end{aligned}$$

For the case $k\ge 0$, by Bernstein inequality,

$$\begin{aligned} \begin{aligned} |P_{12}|&\le C\sum _{|q-k|\le 2}2^{-k}\Vert \nabla \Delta _k u^1\Vert _{L^2}\Vert \Delta _q \Delta _k\partial _1 f\Vert _{L^2}2^{q}\Vert \Delta _qf\Vert _{L^2}\\&\le C\sum _{|q-k|\le 2}2^{-k}\Vert \omega \Vert _{L^2}2^q\Vert \Delta _q\partial _1 f\Vert _{L^2}\Vert \Delta _qf\Vert _{L^2}\\&\le Cb_q2^{{-2qs}}\Vert \omega \Vert _{L^2}\Vert f\Vert _{H^s}\Vert \partial _1f\Vert _{H^s}. \end{aligned} \end{aligned}$$

From the above, it follows that

$$\begin{aligned} \begin{aligned} |P_{1}|\le Cb_q2^{{-2qs}}(\Vert u\Vert _{L^2}+\Vert \omega \Vert _{L^2})(\Vert f\Vert _{H^s}^2+\Vert f\Vert _{H^s}\Vert \partial _1f\Vert _{H^s}). \end{aligned} \end{aligned}$$

(A.2)

For $P_2$, we can bound it by Hölder inequality that

$$\begin{aligned} \begin{aligned} |P_2|\le C\sum _{|q-k|\le 2}\Vert \Delta _k u^1\Vert _{L^2}\Vert \partial _1 S_{k-1} f\Vert _{L^\infty }\Vert \Delta _qf\Vert _{L^2}. \end{aligned} \end{aligned}$$

Applying Bernstein inequality, similar as $P_{12}$,

$$\begin{aligned} \begin{aligned} |P_2|&\le C\sum _{|q-k|\le 2}\Vert \Delta _k u^1\Vert _{L^2}\sum _{m\le k-2}2^m\Vert \Delta _m \partial _1f\Vert _{L^2}\Vert \Delta _qf\Vert _{L^2}\\&\le C \sum _{|q-k|\le 2}2^q\Vert \Delta _k u^1\Vert _{L^2}\sum _{m\le q-2}2^{m-q}\Vert \Delta _m \partial _1f\Vert _{L^2}\Vert \Delta _qf\Vert _{L^2}\\&\le C 2^{-qs}\sum _{|q-k|\le 2}2^{q-k}2^k\Vert \Delta _k u^1\Vert _{L^2}\sum _{m\le q-2}2^{(m-q)(1-s)}2^{ms}\Vert \Delta _m\partial _1 f\Vert _{L^2}\Vert \Delta _qf\Vert _{L^2}\\&\le C b_q 2^{-2qs} (\Vert u^1\Vert _{L^2}+\Vert \omega \Vert _{L^2})\Vert f\Vert _{H^s}\Vert \partial _1f\Vert _{H^s},\\ \end{aligned} \end{aligned}$$

(A.3)

where we have used discrete Young’s inequality in the last step.

Next we estimate $P_3$. By Hölder inequality and Bernstein inequality,

$$\begin{aligned} \begin{aligned} |P_{3}|&\le \bigg |\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{{{\mathbb {R}}}^2} \Delta _q(\Delta _ku^1\partial _1\Delta _q f)\Delta _qf~{\text {d}}x\bigg |\\&\le C \sum _{k\ge q-1}\sum _{|k-l|\le 1}\Vert \Delta _q(\Delta _ku^1\Delta _l\partial _1 f)\Vert _{L^1}\Vert \Delta _qf\Vert _{L^\infty }\\&\le C 2^q\sum _{k\ge q-1}2^{-k}2^k\Vert \Delta _ku^1\Vert _{L^2}\Vert \Delta _k\partial _1f\Vert _{L^2}\Vert \Delta _qf\Vert _{L^2}\\&\le C2^{-qs} \sum _{k\ge q-1} 2^{(q-k)(1+s)}2^{ks}\Vert \Delta _k\partial _1f\Vert _{L^2}(\Vert u^1\Vert _{L^2}+\Vert \omega \Vert _{L^2})\Vert \Delta _qf\Vert _{L^2}\\&\le C b_q 2^{-2qs}(\Vert u^1\Vert _{L^2}+\Vert \omega \Vert _{L^2})\Vert f\Vert _{H^s}\Vert \partial _1f\Vert _{H^s},\\ \end{aligned} \end{aligned}$$

(A.4)

where discrete Young’s inequality has been used in the last two lines.

For Q, we can also divide it into three parts,

$$\begin{aligned} \begin{aligned} -\int _{{{\mathbb {R}}}^2}\Delta _q(u^2 \partial _2 f)\Delta _q f~{\text {d}}x =Q_1+Q_2+Q_3, \end{aligned} \end{aligned}$$

(A.5)

with

$$\begin{aligned} Q_1= & {} -\sum _{|k-q|\le 2}\int _{{{\mathbb {R}}}^2}\Delta _q(S_{k-1}u^2 \Delta _k\partial _2 f)\Delta _q f~{\text {d}}x,\\ Q_2= & {} -\sum _{|k-q|\le 2}\int _{{{\mathbb {R}}}^2}\Delta _q(\Delta _ku^2 S_{k-1}\partial _2 f)\Delta _q f~{\text {d}}x \end{aligned}$$

and

$$\begin{aligned} Q_3=-\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{{{\mathbb {R}}}^2}\Delta _q(\Delta _ku^2 \Delta _l\partial _2 f)\Delta _q f~{\text {d}}x. \end{aligned}$$

Similar as $P_1$, we can rewrite $Q_1$ as

$$\begin{aligned} \begin{aligned} Q_1&=-\sum _{|q-k|\le 2}\int _{{{\mathbb {R}}}^2}[\Delta _q, S_{k-1}u^2\partial _2]\Delta _k f \Delta _q f~{\text {d}}x\\&\quad -\sum _{|q-k|\le 2}\int _{{{\mathbb {R}}}^2}({ S_{k-1}u^2-S_qu^2})\partial _2\Delta _q \Delta _kf\Delta _q f~{\text {d}}x\\&\quad -\int _{{{\mathbb {R}}}^2}{S}_{q}u^2\partial _2\Delta _q f\Delta _q f~{\text {d}}x\\&\triangleq Q_{11}+Q_{12}+Q_{13}. \end{aligned} \end{aligned}$$

Here, we should notice that $P_{13}+Q_{13}=0$ because of the divergence-free condition of u, so we do not need to estimate these two terms.

For $Q_{11}$, by Hölder inequality,

$$\begin{aligned} \begin{aligned} |Q_{11}|&\le \sum _{|q-k|\le 2}\bigg |\int _{{{\mathbb {R}}}^2}[\Delta _q, S_{k-1}{ u^2\partial _2}]\Delta _k f\Delta _k f~{\text {d}}x\bigg |\\&\le C\sum _{|q-k|\le 2}\Vert [\Delta _q, S_{k-1}{u^2\partial _2}]\Delta _k f\Vert _{L^2}\Vert \Delta _qf\Vert _{L^2}. \end{aligned} \end{aligned}$$

According to the definition of $\Delta _q$ and similar as $P_{11}$,

$$\begin{aligned} \begin{aligned} \big [\Delta _q, S_{k-1}{u^2\partial _2}\big ]\Delta _k f&=\int _{{{\mathbb {R}}}^2}\int _0^1\phi _q(z)z\cdot \nabla S_{k-1}u^2(x-sz)\partial _2\Delta _k f(x-z)~{\text {d}}s {\text {d}}z.\\ \end{aligned} \end{aligned}$$

Making use of the anisotropic Hölder inequality and Bernstein inequality,

$$\begin{aligned} \begin{aligned}&\Vert [\Delta _q, S_{k-1}u^2\partial _2]\Delta _k f\Vert _{L^2}\\&\quad =\bigg \Vert \int _{{{\mathbb {R}}}^2}\int _0^1\varphi _q(z)z\cdot \nabla S_{k-1}u^2(x-sz)\partial _2\Delta _k f(x-z)~{\text {d}}s {\text {d}}z\bigg \Vert _{L^2}\\&\quad \le C\int _{{{\mathbb {R}}}^2}\big |\varphi _q(z)\big ||z|~dz\Vert \nabla {S}_{k-1}u^2(x-sz)\Vert _{L^\infty _{x_2}(L^2_{x_1})}\Vert \partial _2\Delta _k f(x-z)\Vert _{L^2_{x_2}(L^\infty _{x_1})}\\&\quad \le C2^{-q}\Vert \nabla S_{k-1}u^2\Vert _{L^2}^{\frac{1}{2}}\Vert \partial _2\nabla S_{k-1}u^2\Vert _{L^2}^{\frac{1}{2}}\Vert \partial _2\Delta _k f\Vert _{L^2}^{\frac{1}{2}} \Vert \partial _1\partial _2\Delta _k f\Vert _{L^2}^{\frac{1}{2}}. \end{aligned} \end{aligned}$$

Noticing that by Biot–Savart law $u^2=\partial _1\Delta ^{-1}\omega$, and combining with the boundedness of Riesz transform in $L^2$,

$$\begin{aligned} \begin{aligned} \Vert [\Delta _q, S_{k-1}u^2\partial _2]\Delta _k f\Vert _{L^2}&\le C2^{k-q}\Vert \omega \Vert _{L^2}^{\frac{1}{2}}\Vert \partial _2\nabla \partial _1\Delta ^{-1}\omega \Vert _{L^2}^{\frac{1}{2}}\Vert \Delta _k f\Vert _{L^2}^{\frac{1}{2}} \Vert \partial _1\Delta _k f\Vert _{L^2}^{\frac{1}{2}}\\&\le C2^{k-q}\Vert \omega \Vert _{L^2}^{\frac{1}{2}}\Vert \partial _1\omega \Vert _{L^2}^{\frac{1}{2}} \Vert \Delta _k f\Vert _{L^2}^{\frac{1}{2}} \Vert \partial _1\Delta _k f\Vert _{L^2}^{\frac{1}{2}}. \end{aligned} \end{aligned}$$

Then, $Q_{11}$ is bounded by

$$\begin{aligned} \begin{aligned} |Q_{11}|&\le C\sum _{|q-k|\le 2}\Vert [\Delta _q, S_{k-1}u^2\partial _2]\Delta _k f\Vert _{L^2}\Vert \Delta _qf\Vert _{L^2}\\&\le C\sum _{|q-k|\le 2}2^{k-q}\Vert \omega \Vert _{L^2}^{\frac{1}{2}}\Vert \partial _1\omega \Vert _{L^2}^{\frac{1}{2}} \Vert \Delta _k f\Vert _{L^2}^{\frac{1}{2}} \Vert \partial _1\Delta _k f\Vert _{L^2}^{\frac{1}{2}}\Vert \Delta _qf\Vert _{L^2}\\&\le Cb_q2^{-2qs}\Vert \omega \Vert _{L^2}^{\frac{1}{2}}\Vert \partial _1\omega \Vert _{L^2}^{\frac{1}{2}}\Vert f\Vert _{H^s}^{\frac{3}{2}}\Vert \partial _1f\Vert _{H^s}^{\frac{1}{2}}.\\ \end{aligned} \end{aligned}$$

For $Q_{12}$, by the anisotropic Hölder inequality and interpolation inequality,

$$\begin{aligned} \begin{aligned} |Q_{12}|&=\sum _{|q-k|\le 2}\bigg |\int _{{{\mathbb {R}}}^2} ((S_{k-1}u^2-S_{q}u^2)\partial _2\Delta _q\Delta _k f)\Delta _q f~{\text {d}}x\bigg |\\&\le C\sum _{|q-k|\le 2}\Vert (S_{k-1}u^2-S_{q}u^2)\partial _2\Delta _q \Delta _kf\Vert _{L^2}\Vert \Delta _qf\Vert _{L^2}\\&\le C\sum _{|q-k|\le 2}\Vert \Delta _k u^2\Vert _{L^\infty _{x_2}(L^2_{x_1})}\Vert \Delta _q \Delta _k\partial _2 f\Vert _{L^2_{x_2}(L^\infty _{x_1})}\Vert \Delta _qf\Vert _{L^2}\\&\le C\sum _{|q-k|\le 2}\Vert \Delta _k u^2\Vert _{L^2}^{\frac{1}{2}}\Vert \Delta _k \partial _2u^2\Vert _{L^2}^{\frac{1}{2}}\Vert \Delta _q \Delta _k\partial _2 f\Vert _{L^2}^{\frac{1}{2}}\Vert \Delta _q \Delta _k\partial _1\partial _2 f\Vert _{L^2}^{\frac{1}{2}}\Vert \Delta _qf\Vert _{L^2}\\ \end{aligned} \end{aligned}$$

For the case $k=-1$, by Bernstein inequality,

$$\begin{aligned} \begin{aligned} |Q_{12}|&\le C\Vert \Delta _{-1} u^2\Vert _{L^2}\Vert \Delta _q \Delta _{-1} f\Vert _{L^2}\Vert \Delta _qf\Vert _{L^2}\\&\le Cb_q2^{-2qs}\Vert u^2\Vert _{L^2}\Vert f\Vert _{H^s}^2. \end{aligned} \end{aligned}$$

For the case $k\ge 0$, by Bernstein inequality and the relation $u^2=\partial _1\Delta ^{-1}\omega$,

$$\begin{aligned} \begin{aligned} |Q_{12}|&\le C\sum _{|q-k|\le 2}2^{q-k}\Vert \nabla \Delta _k u^2\Vert _{L^2}^\frac{1}{2}\Vert \nabla \Delta _k \partial _1\Delta ^{-1}\omega \Vert _{L^2}^\frac{1}{2}\Vert \Delta _q \partial _1 f\Vert _{L^2}^{\frac{1}{2}}\Vert \Delta _qf\Vert _{L^2}^{\frac{3}{2}}\\&\le C\sum _{|q-k|\le 2}2^{q-k}\Vert \omega \Vert _{L^2}^\frac{1}{2}\Vert \partial _1\omega \Vert _{L^2}^\frac{1}{2} \Vert \Delta _q\partial _1 f\Vert _{L^2}^\frac{1}{2}\Vert \Delta _qf\Vert _{L^2}^\frac{3}{2}\\&\le Cb_q2^{-2qs}\Vert \omega \Vert _{L^2}^\frac{1}{2}\Vert \partial _1\omega \Vert _{L^2}^\frac{1}{2}\Vert f\Vert _{H^s}\Vert \partial _1f\Vert _{H^s}. \end{aligned} \end{aligned}$$

Then, it follows that

$$\begin{aligned} \begin{aligned} |Q_{1}|\le Cb_q2^{-2qs}(\Vert u\Vert _{L^2}+\Vert \omega \Vert _{L^2}^\frac{1}{2}\Vert \partial _1\omega \Vert _{L^2}^\frac{1}{2})(\Vert f\Vert _{H^s}^2+\Vert f\Vert _{H^s}\Vert \partial _1f\Vert _{H^s}). \end{aligned} \end{aligned}$$

(A.6)

Similar as $Q_{12}$, applying anisotropic Hölder inequality and Bernstein inequality, $Q_{2}$ can be bounded by

$$\begin{aligned} \begin{aligned} |Q_2|&\le C\sum _{|q-k|\le 2}\Vert \Delta _k u^2\Vert _{L^\infty _{x_2}(L^2_{x_1})}\Vert \partial _2 S_{k-1} f\Vert _{L^2_{x_2}(L^\infty _{x_1})}\Vert \Delta _qf\Vert _{L^2}\\&\le C\sum _{|q-k|\le 2}\Vert \Delta _k u^2\Vert _{L^2}^\frac{1}{2} \Vert \Delta _k \partial _2u^2\Vert _{L^2}^\frac{1}{2} \Vert \partial _2 S_{k-1} f\Vert _{L^2}^\frac{1}{2}\Vert \partial _1\partial _2 S_{k-1} f\Vert _{L^2}^\frac{1}{2}\Vert \Delta _qf\Vert _{L^2}\\&\le Cb_q2^{-2qs}\Vert u\Vert _{L^2}\Vert f\Vert _{H^s}^2+C\Vert \omega \Vert _{L^2}^\frac{1}{2}\Vert \partial _1\omega \Vert _{L^2}^\frac{1}{2}\bigg (\sum _{m\le q-2}2^{m-q}\Vert \Delta _m f\Vert _{L^2}\bigg )^{\frac{1}{2}}\\ {}&\quad \quad \times \bigg (\sum _{n\le q-2}2^{n-q}\Vert \Delta _n \partial _1f\Vert _{L^2}\bigg )^{\frac{1}{2}}\Vert \Delta _qf\Vert _{L^2}\\&\le C b_q2^{-2qs} (\Vert u\Vert _{L^2}+\Vert \omega \Vert _{L^2}^\frac{1}{2}\Vert \partial _1\omega \Vert _{L^2}^\frac{1}{2})(\Vert f\Vert _{H^s}^2+\Vert f\Vert _{H^s}^\frac{3}{2}\Vert \partial _1f\Vert _{H^s}^\frac{1}{2}).\\ \end{aligned} \end{aligned}$$

(A.7)

Finally, we estimate $Q_3$. By Hölder inequality and Bernstein inequality,

$$\begin{aligned} \begin{aligned} |Q_{3}|&\le \bigg |\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{{{\mathbb {R}}}^2} \Delta _q(\Delta _ku^2\partial _2\Delta _q f)\Delta _qf~{\text {d}}x\bigg |\\&\le C \sum _{k\ge q-1}\sum _{|k-l|\le 1}\Vert \Delta _q(\Delta _ku^2\Delta _l\partial _2 f)\Vert _{L^1}\Vert \Delta _qf\Vert _{L^\infty }\\&\le C 2^q\sum _{k\ge q-1}\Vert \Delta _ku^2\Vert _{L^2}\Vert \Delta _k\partial _2f\Vert _{L^2}\Vert \Delta _qf\Vert _{L^2}\\&\le C 2^q\sum _{k\ge q-1}(\Vert u\Vert _{L^2}+\Vert \partial _1\omega \Vert _{L^2})2^{-2k}2^k\Vert \Delta _kf\Vert _{L^2}\Vert \Delta _qf\Vert _{L^2}\\&\le C b_q2^{-2qs}(\Vert u\Vert _{L^2}+\Vert \partial _1\omega \Vert _{L^2})\Vert f\Vert _{H^s}^2.\\ \end{aligned} \end{aligned}$$

(A.8)

Taking all these estimates into account, we can obtain

$$\begin{aligned} \begin{aligned} -\int _{{{\mathbb {R}}}^2}\Delta _q(u\cdot \nabla f)\Delta _q f~{\text {d}}x&\le C b_q2^{-2qs}(\Vert u\Vert _{L^2}+\Vert \omega \Vert _{L^2}+\Vert \partial _1\omega \Vert _{L^2})\\&\quad \times (\Vert f\Vert _{H^s}^2+\Vert f\Vert _{H^s}^\frac{1}{2}\Vert \partial _1f\Vert _{H^s}^\frac{1}{2}+\Vert f\Vert _{H^s}^\frac{3}{2}\Vert \partial _1f\Vert _{H^s}^\frac{1}{2}), \end{aligned} \end{aligned}$$

which completes the proof of this lemma. $\square$

Lemma A.1

(Losing regularity estimate for transport equation) Let $\rho$ satisfy the transport equation

$$\begin{aligned} \left\{ \begin{array}{cc} \begin{aligned} &{}\partial _t \rho +u\cdot \nabla \rho =f,\\ &{}\rho (0,x)=\rho _0(x), \end{aligned} \end{array} \right. \end{aligned}$$

(A.9)

where $\rho _0\in B^s_{2,r}$, $f\in L^1([0,T];B^s_{2,r})$ with $r\in [1,\infty ]$. Here, $u \in L^2$ is a divergence-free vector field and for some $V(t)\in L^1([0,T])$, v satisfies

$$\begin{aligned} \sup _{N\ge 0}\frac{\Vert \nabla S_N u(t)\Vert _{L^{\infty }}}{\sqrt{1+N}}\le V(t). \end{aligned}$$

Then for all $s>0$, $\varepsilon \in (0,s)$ and $t\in [0,T]$, we have the following estimate,

$$\begin{aligned} \begin{aligned} \Vert \rho (t)\Vert _{B^{s-\varepsilon }_{2,r}}&\le C \bigg (\Vert \rho _0\Vert _{B^{s}_{2,r}}+\int _0^T\Vert f(\tau )\Vert _{B^{s}_{2,r}}~{\text {d}}\tau \bigg )e^{\frac{C}{\varepsilon }\big (\int _0^TV(\tau )~{\text {d}}\tau \big )^2}, \end{aligned} \end{aligned}$$

where C is a constant independent of T and $\varepsilon$.

Proof

The case $r=\infty$ has been shown in Danchin and Paicu (2011), here we just discuss $1\le r<\infty$. Applying $\Delta _q$ to (2.8), we obtain

$$\begin{aligned} \partial _t\Delta _q\rho +\Delta _q(u\cdot \nabla \rho )=\Delta _qf. \end{aligned}$$

(A.10)

Taking $L^2$ inner product with $\Delta _q\rho$,

$$\begin{aligned} \begin{aligned} \frac{1}{2}\frac{{\text {d}}}{{\text {d}}t}\Vert \Delta _q\rho \Vert _{L^2}^2=-\int _{{{\mathbb {R}}}^2}\Delta _q(u\cdot \nabla \rho )\Delta _q\rho ~{\text {d}}x+\int _{{{\mathbb {R}}}^2}\Delta _qf\Delta _q\rho ~{\text {d}}x \triangleq I+II. \end{aligned} \end{aligned}$$

(A.11)

For II, by Hölder inequality,

$$\begin{aligned} \begin{aligned} II=\int _{{{\mathbb {R}}}^2}\Delta _qf\Delta _q\rho \le \Vert \Delta _qf\Vert _{L^2}\Vert \Delta _q\rho \Vert _{L^2}. \end{aligned} \end{aligned}$$

(A.12)

For I, along a similar argument as Lemma 2.5, we can divide it as

$$\begin{aligned} \begin{aligned} I&=-\int _{{{\mathbb {R}}}^2}\Delta _q(u\cdot \nabla \rho )\Delta _q \rho ~{\text {d}}x\\&=-\sum _{|k-q|\le 2}\int _{{{\mathbb {R}}}^2}\Delta _q(S_{k-1}u\cdot \Delta _k\nabla \rho )\Delta _q \rho ~{\text {d}}x\\&\quad -\sum _{|k-q|\le 2}\int _{{{\mathbb {R}}}^2}\Delta _q(\Delta _ku\cdot \nabla S_{k-1} \rho )\Delta _q \rho ~{\text {d}}x\\&\quad -\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{{{\mathbb {R}}}^2}\Delta _q(\Delta _ku\cdot \nabla \Delta _l \rho )\Delta _q \rho ~{\text {d}}x\\&\quad \triangleq L_1+L_2+L_3. \end{aligned} \end{aligned}$$

For $L_1$, we can rewrite it as

$$\begin{aligned} \begin{aligned} L_1&=-\sum _{|q-k|\le 2}\int _{{{\mathbb {R}}}^2}[\Delta _q, S_{k-1}u\cdot \nabla ]\Delta _k \rho \Delta _q \rho ~{\text {d}}x\\&\quad -\sum _{|q-k|\le 2}\int _{{{\mathbb {R}}}^2}(S_{k-1}u-S_qu)\cdot \nabla \Delta _q \Delta _k\rho \Delta _q \rho ~{\text {d}}x\\&\quad -\int _{{{\mathbb {R}}}^2}{S}_{q}u\cdot \nabla \Delta _q \rho \Delta _q f~{\text {d}}x\\&\triangleq L_{11}+L_{12}+L_{13}, \end{aligned} \end{aligned}$$

According to divergence-free condition of u, it is not difficult to find that $L_{13}=0$. For $L_{11}$, by Hölder inequality,

$$\begin{aligned} \begin{aligned} |L_{11}|&\le \sum _{|q-k|\le 2}\bigg |\int _{{{\mathbb {R}}}^2}[\Delta _q, S_{k-1}u\cdot \nabla ]\Delta _k \rho \Delta _k \rho ~{\text {d}}x\bigg |\\&\le C\sum _{|q-k|\le 2}\Vert [\Delta _q, S_{k-1}u\cdot \nabla ]\Delta _k \rho \Vert _{L^2}\Vert \Delta _q\rho \Vert _{L^2}. \end{aligned} \end{aligned}$$

According to the definition of $\Delta _q$,

$$\begin{aligned} \begin{aligned} \left[ \Delta _q, S_{k-1}u\cdot \nabla \right] \Delta _k\rho&=\int _{{{\mathbb {R}}}^2}\phi _q(x-y)(S_{k-1}u(y)\cdot \nabla \Delta _k \rho (y))~dy\\&\quad -S_{k-1}u(x)\cdot \int _{{{\mathbb {R}}}^d}\phi _q(x-y)\nabla \Delta _k \rho (y)~dy\\&=\int _{{{\mathbb {R}}}^2}\phi _q(x-y)(S_{k-1}u(y)-S_{k-1}u(x))\cdot \nabla \Delta _k \rho (y)~dy\\&=\int _{{{\mathbb {R}}}^2}\phi _q(x-y)\int _0^1(y-x)\cdot \nabla S_{k-1}u(sy+(1-s)x)~{\text {d}}s\cdot \nabla \Delta _k \rho (y)~{\text {d}}y\\&=\int _{{{\mathbb {R}}}^2}\int _0^1\phi _q(z)z\cdot \nabla S_{k-1}u(x-sz)\cdot \nabla \Delta _k \rho (x-z)~{\text {d}}s {\text {d}}z.\\ \end{aligned} \end{aligned}$$

Thus, we have

$$\begin{aligned} \begin{aligned}&\Vert [\Delta _q, S_{k-1}u\cdot \nabla ]\Delta _k \rho \Vert _{L^2}\\&\quad =\bigg \Vert \int _{{{\mathbb {R}}}^2}\int _0^1\phi _q(z)z\cdot \nabla S_{k-1}u(x-sz)\cdot \nabla \Delta _k \rho (x-z)~{\text {d}}s {\text {d}}z\bigg \Vert _{L^2}\\&\quad \le C\int _{{{\mathbb {R}}}^2}\big |\phi _q(z)\big ||z|~dz\Vert \nabla {S}_{k-1}u(x-sz)\Vert _{L^\infty }\Vert \nabla \Delta _k \rho (x-z)\Vert _{L^2}\\&\quad \le C2^{-q}\int _{{{\mathbb {R}}}^2}\big |\phi _q(z)\big ||z|~dz\Vert \nabla S_{k-1}u\Vert _{L^\infty }\Vert \nabla \Delta _k \rho \Vert _{L^2}\\&\quad \le C2^{-q}\Vert \nabla S_{k-1}u\Vert _{L^\infty }2^k\Vert \Delta _k \rho \Vert _{L^2}. \end{aligned} \end{aligned}$$

Then, we obtain

$$\begin{aligned} \begin{aligned} |L_{11}|&\le C\sum _{|q-k|\le 2}\Vert [\Delta _q, S_{k-1}u\cdot \nabla ]\Delta _k \rho \Vert _{L^2}\Vert \Delta _q\rho \Vert _{L^2}\\&\le C\sum _{|q-k|\le 2}2^{k-q}\Vert \nabla S_{k-1}u\Vert _{L^\infty }\Vert \Delta _q \rho \Vert _{L^2}^2\\&\le C\sqrt{q}V(t)\Vert \Delta _q \rho \Vert _{L^2}^2\\&\le Cd_q2^{-\sigma q}\sqrt{q}V(t)\Vert \rho \Vert _{B^{\sigma }_{2,r}}\Vert \Delta _q \rho \Vert _{L^2}, \end{aligned} \end{aligned}$$

where $d_q\in \ell ^r$.

For $L_{12}$, by Hölder inequality,

$$\begin{aligned} \begin{aligned} |L_{12}|&=\sum _{|q-k|\le 2}\bigg |\int _{{{\mathbb {R}}}^2} ((S_{k-1}u-S_{q}u)\cdot \nabla \Delta _q\Delta _k \rho )\Delta _q \rho ~{\text {d}}x\bigg |\\&\le C\sum _{|q-k|\le 2}2^{q-k}\Vert \nabla \Delta _k u\Vert _{L^\infty }\Vert \Delta _q\rho \Vert _{L^2}^2+\Vert u\Vert _{L^2}\Vert \Delta _q\rho \Vert _{L^2}^2\\&\le C(\sqrt{q+2}V(t)+\Vert u\Vert _{L^2})\Vert \Delta _q\rho \Vert _{L^2}^2\\&\le Cd_q2^{-\sigma q}(\sqrt{q+2}V(t)+\Vert u\Vert _{L^2})\Vert \rho \Vert _{B^{\sigma }_{2,r}}\Vert \Delta _q \rho \Vert _{L^2}. \end{aligned} \end{aligned}$$

For $L_2$, we can bound it by Hölder inequality that

$$\begin{aligned} \begin{aligned} |L_2|\le C\sum _{|q-k|\le 2}\Vert \Delta _k u\Vert _{L^\infty }\Vert \nabla S_{k-1} \rho \Vert _{L^2}\Vert \Delta _q\rho \Vert _{L^2}. \end{aligned} \end{aligned}$$

According to Bernstein inequality,

$$\begin{aligned} \begin{aligned} |L_2|&\le C \sum _{|q-k|\le 2}\Vert \Delta _k u\Vert _{L^\infty }\sum _{m\le q-2}2^{m}\Vert \Delta _m \rho \Vert _{L^2}\Vert \Delta _q\rho \Vert _{L^2}\\&\le C \sum _{|q-k|\le 2}2^q\Vert \Delta _k u\Vert _{L^\infty }\sum _{m\le q-2}2^{m-q}\Vert \Delta _m \rho \Vert _{L^2}\Vert \Delta _q\rho \Vert _{L^2}\\&\le C \sum _{|q-k|\le 2}2^q\Vert \Delta _k u\Vert _{L^\infty }\sum _{m\le q-2}2^{m-q}\Vert \Delta _m \rho \Vert _{L^2}\Vert \Delta _q\rho \Vert _{L^2}\\&\le C (\sqrt{q+2}V(t)+\Vert u\Vert _{L^2})\sum _{m\le q-2}2^{m-q}\Vert \Delta _m \rho \Vert _{L^2}\Vert \Delta _q\rho \Vert _{L^2}\\&\le C d_q2^{-\sigma q}(\sqrt{q+2}V(t)+\Vert u\Vert _{L^2}) \Vert \rho \Vert _{B^{\sigma }_{2,r}}\Vert \Delta _q\rho \Vert _{L^2}. \end{aligned} \end{aligned}$$

Then, we bound $L_3$. By Hölder inequality and Bernstein inequality,

$$\begin{aligned} \begin{aligned} |L_{3}|&\le \bigg |\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{{{\mathbb {R}}}^2} \Delta _q(\Delta _ku\cdot \nabla \Delta _q \rho )\Vert \Delta _q\rho \Vert _{L^2}~{\text {d}}x\bigg |\\&\le C \sum _{k\ge q-1}\sum _{|k-l|\le 1}\Vert \Delta _q\nabla \cdot (\Delta _ku\Delta _l \rho )\Vert _{L^2}\Vert \Delta _q\rho \Vert _{L^2}\\&\le C 2^q\sum _{k\ge q-1}\Vert \Delta _ku\Vert _{L^\infty }\Vert \Delta _k\rho \Vert _{L^2}\Vert \Delta _q\rho \Vert _{L^2}\\&\le C (\sqrt{q+2}V(t)+\Vert u\Vert _{L^2})\sum _{k\ge q-1}2^{q-k}\Vert \Delta _k\rho \Vert _{L^2}\Vert \Delta _q\rho \Vert _{L^2}\\&\le C d_q2^{-\sigma q}(\sqrt{q+2}V(t)+\Vert u\Vert _{L^2})\Vert \rho \Vert _{B^{\sigma }_{2,r}}\Vert \Delta _q\rho \Vert _{L^2}.\\ \end{aligned} \end{aligned}$$

Thus, we obtain I can be bounded by

$$\begin{aligned} \begin{aligned} I\le Cd_q2^{-\sigma q}(\sqrt{q+2}V(t)+1)\Vert \rho \Vert _{B^{\sigma }_{2,r}}\Vert \Delta _q\rho \Vert _{L^2}. \end{aligned} \end{aligned}$$

(A.13)

Inserting (A.12) and (A.13) into (A.11), one can obtain

$$\begin{aligned} \begin{aligned} \frac{{\text {d}}}{{\text {d}}t}\Vert \Delta _q\rho (t)\Vert _{L^2}\le \Vert \Delta _qf\Vert _{L^2}+ Cd_q2^{-\sigma q}(\sqrt{q+2}V(t)+1)\Vert \rho \Vert _{B^{\sigma }_{2,r}}. \end{aligned} \end{aligned}$$

(A.14)

Denoting $s_t\triangleq s-\eta \int _0^tV(\tau )~{\text {d}}\tau$ for $t\in [0,T]$ with $\eta =\varepsilon \big (\int _0^TV(\tau )~{\text {d}}\tau \big )^{-1}$. Choosing $\sigma =s_t$ and integrating (A.14) from 0 to t with respect to time variable and then multiplying by $2^{s_tq}$,

$$\begin{aligned} \begin{aligned} 2^{s_tq}\Vert \Delta _q\rho (t)\Vert _{L^2}&\le d_q \Vert \rho _0\Vert _{B^{s_t}_{2,r}}+d_q\int _0^t\Vert f(\tau )\Vert _{B^{s_t}_{2,1}}~{\text {d}}\tau \\ {}&\quad + Cd_q\int _0^t2^{\big (-\eta \int _{\tau }^{t}V(s){\text {d}}s\big ) q}(\sqrt{q+2}V(\tau )+1)\Vert \rho \Vert _{B^{s_\tau }_{2,r}}~{\text {d}}\tau . \end{aligned} \end{aligned}$$

(A.15)

Choosing $q_0>0$ is the smallest integer such that

$$\begin{aligned} \frac{4C^2\Vert d_q\Vert _{\ell ^r}^2}{(\log 2)^2\eta ^2}\le q_0+2. \end{aligned}$$

Then for $q\ge q_0$, we have

$$\begin{aligned} C\int _0^t2^{\big (-\eta \int _{\tau }^{t}V(s){\text {d}}s\big ) q}\sqrt{q+2}V(\tau )~{\text {d}}\tau \le \frac{1}{2\Vert b_q\Vert _{\ell ^r}}. \end{aligned}$$

(A.16)

Inserting these results into (A.15) and taking $\ell ^r$ norm of q, on can deduce

$$\begin{aligned} \begin{aligned} \Vert \rho (t)\Vert _{B^{s_t}_{2,r}}&\le C \Vert \rho _0\Vert _{B^{s}_{2,r}}+C\int _0^t\Vert f(\tau )\Vert _{B^{s}_{2,r}}~{\text {d}}\tau \\ {}&\quad + C\bigg (\sum _{q\ge q_0}\bigg (d_q\int _0^t2^{\big (-\eta \int _{\tau }^{t}V(s){\text {d}}s\big ) q}\sqrt{q+2}V(\tau )\Vert \rho \Vert _{B^{s_\tau }_{2,r}}~{\text {d}}\tau \bigg )^r\bigg )^{\frac{1}{r}}\\&\quad + C\bigg (\sum _{1\le q<q_0 }\bigg (d_q\int _0^t2^{\big (-\eta \int _{\tau }^{t}V(s){\text {d}}s\big ) q}\sqrt{q+2}V(\tau )\Vert \rho \Vert _{B^{s_\tau }_{2,r}}~{\text {d}}\tau \bigg )^r\bigg )^{\frac{1}{r}}\\&\le C \Vert \rho _0\Vert _{B^{s}_{2,r}}+C\int _0^t\Vert f(\tau )\Vert _{B^{s}_{2,r}}~{\text {d}}\tau \\ {}&\quad + \frac{1}{2}\sup _{t\in [0,T]}\Vert \rho \Vert _{B^{s_t}_{2,r}}+ C\sqrt{q_0+1}\int _0^tV(\tau )\Vert \rho \Vert _{B^{s_\tau }_{2,r}}~{\text {d}}\tau .\\ \end{aligned} \end{aligned}$$

(A.17)

Taking supremum of time t from 0 to T and applying the Grönwall’s Lemma, it follows that

$$\begin{aligned} \begin{aligned} \sup _{t\in [0,T]}\Vert \rho (t)\Vert _{B^{s_t}_{2,r}}&\le {C} \bigg (\Vert \rho _0\Vert _{B^{s}_{2,r}}+\int _0^T\Vert f(\tau )\Vert _{B^{s}_{2,r}}~{\text {d}}\tau \bigg )e^{\sqrt{q_0+1}\int _0^TV(\tau )~{\text {d}}\tau }. \end{aligned} \end{aligned}$$

According to the definition of $q_0$, finally we conclude that

$$\begin{aligned} \begin{aligned} \sup _{t\in [0,T]}\Vert \rho (t)\Vert _{B^{s_t}_{2,r}}&\le {C} \bigg (\Vert \rho _0\Vert _{B^{s}_{2,r}}+\int _0^T\Vert f(\tau )\Vert _{B^{s}_{2,r}}~{\text {d}}\tau \bigg )e^{\frac{C}{\varepsilon }\big (\int _0^TV(\tau )~{\text {d}}\tau \big )^2}, \end{aligned} \end{aligned}$$

which entails the desired inequality given that $s\ge s_t\ge s-\varepsilon$ for all $t\in [0,T]$.

$\square$

The next proposition gives the estimate for Lipschitz norm of the velocity with anisotropic initial data $\omega _0\in {\mathcal {B}}^{0,\frac{1}{2}}$ for system (1.2), which shows the proof of Remark 2.

Proposition A.1

Assume $u_0$ is a divergence-free vector in $H^1$, $\omega _0\in \sqrt{L}\cap {\mathcal {B}}^{0,\frac{1}{2}}$ and $\theta _0\in L^\infty \cap H^{s}$ with $s\in (\frac{1}{2},1]$, then the solution u of Theorem 4.1 satisfies $\nabla u\in L^2_{loc}({{\mathbb {R}}}_+;L^\infty )$.

Proof

Applying $\Delta _q^v$ to (1.5), and taking $L^2$ inner product with $\Delta _q^v\omega$, we have

$$\begin{aligned} \frac{1}{2}\frac{{\text {d}}}{{\text {d}}t}\Vert \Delta _q^v\omega (t)\Vert _{L^2}^2+\Vert \partial _1\Delta _q^v\omega \Vert _{L^2}^2 =-\int _{{{\mathbb {R}}}^2}\Delta _q^v\theta \partial _1\Delta _q^v\omega ~{\text {d}}x-\int _{{{\mathbb {R}}}^2}\Delta _q^v(u\cdot \nabla \omega )\Delta _q^v\omega ~{\text {d}}x. \end{aligned}$$

(A.18)

For the first term in the right-hand side of (A.18), by Hölder inequality, Young’s inequality and the definition of the space ${\mathcal {B}}^{0,\frac{1}{2}}$, we obtain

$$\begin{aligned} \begin{aligned} -\int _{{{\mathbb {R}}}^2}\Delta _q^v\theta \partial _1\Delta _q^v\omega ~{\text {d}}x&\le \Vert \Delta _q^v\theta \Vert _{L^2}\Vert \partial _1\Delta _q^v\omega \Vert _{L^2}\\&\le \frac{1}{2}\Vert \Delta _q^v\theta \Vert _{L^2}^2+\frac{1}{2}\Vert \partial _1\Delta _q^v\omega \Vert _{L^2}^2\\&\le \frac{1}{2}2^{-q}(2^{{\frac{q}{2}}}\Vert \Delta _q^v\theta \Vert _{L^2})^2+\frac{1}{2}\Vert \partial _1\Delta _q^v\omega \Vert _{L^2}^2\\&\le C2^{-q}a_q\Vert \theta \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^2+\frac{1}{2}\Vert \partial _1\Delta _q^v\omega \Vert _{L^2}^2. \end{aligned} \end{aligned}$$

(A.19)

with $a_{q}=\frac{(2^{q/2}\Vert \Delta _q^v\theta \Vert _{L^2})^2}{\Vert \theta \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^2}\in \ell ^{\frac{1}{2}}$. For the last term of (A.18), we divide it as,

$$\begin{aligned} \begin{aligned} -\int _{{{\mathbb {R}}}^2}\Delta _q^v(u\cdot \nabla \omega )\Delta _q^v\omega ~{\text {d}}x&=-\int _{{{\mathbb {R}}}^2}\Delta _q^v(u^1\partial _1 \omega )\Delta _q^v \omega ~{\text {d}}x -\int _{{{\mathbb {R}}}^2}\Delta _q^v(u^2\partial _2 \omega )\Delta _q^v \omega ~{\text {d}}x\\&\triangleq Y_1+Y_2. \end{aligned} \end{aligned}$$

For $Y_1$, by Bony’s decomposition, we can divide it into the following three terms,

$$\begin{aligned} \begin{aligned}&-\int _{{{\mathbb {R}}}^2}\Delta _q^v(u^1 \partial _1 \omega )\Delta _q^v \omega ~{\text {d}}x\\&\quad =-\sum _{|k-q|\le 2}\int _{{{\mathbb {R}}}^2}\Delta _q^v(S^v_{k-1}u^1 \Delta _k^v\partial _1 \omega )\Delta _q^v \omega ~{\text {d}}x\\&\qquad -\sum _{|k-q|\le 2}\int _{{{\mathbb {R}}}^2}\Delta _q^v(\Delta _k^vu^1 S^v_{k-1}\partial _1 \omega )\Delta _q^v \omega ~{\text {d}}x\\&\qquad -\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{{{\mathbb {R}}}^2}\Delta _q^v(\Delta _k^vu^1 \Delta _l^v\partial _1 \omega )\Delta _q^v \omega ~{\text {d}}x\\&\quad \triangleq Y_{11}+Y_{12}+Y_{13}. \end{aligned} \end{aligned}$$

(A.20)

For $Y_{11}$, we can rewrite it as

$$\begin{aligned} Y_{11}= & {} -\sum _{|q-k|\le 2}\int _{{{\mathbb {R}}}^2}\Delta _q^v(S^v_{k-1}u^1\partial _1\Delta _k^v \omega )\Delta _q^v \omega ~{\text {d}}x\\= & {} -\sum _{|q-k|\le 2}\int _{{{\mathbb {R}}}^2}[\Delta _q^v, S^v_{k-1}u^1\partial _1]\Delta _k^v \omega \Delta _q^v \omega ~{\text {d}}x\\&\quad -\sum _{|q-k|\le 2}\int _{{{\mathbb {R}}}^2}(S^v_{k-1}u^1-S_q^vu^1)\partial _1\Delta _q^v \Delta _k^v\omega \Delta _q^v \omega ~{\text {d}}x\\&\quad -\int _{{{\mathbb {R}}}^2}{S}^v_{q}u^1\partial _1\Delta ^v_q \omega \Delta _q^v \omega ~{\text {d}}x\\&\triangleq Y_{111}+Y_{112}+Y_{113}, \end{aligned}$$

where we have used the fact $\sum _{|q-k|\le 2}\partial _1\Delta _q^v\Delta _k^v \omega =\partial _1\Delta _q \omega$. For $Y_{111}$, the commutator can be written as,

$$\begin{aligned} \begin{aligned} {\left[ \Delta _q^v, S^v_{k-1}u^1\partial _1\right] }\Delta _k^v \omega&=\int _{{{\mathbb {R}}}}\phi _q(x_1,x_2-y)(S^v_{k-1}u^1(x_1,y)\partial _1\Delta _k^v \omega (x_1,y))~dy\\&\quad -S^v_{k-1}u^1(x_1,x_2)\int _{{{\mathbb {R}}}}\phi _q(x_1,x_2-y)\partial _1\Delta ^v_k \omega (x_1,y)~dy\\&=\int _{{{\mathbb {R}}}}\phi _q(x_1,x_2-y)(S^v_{k-1}u^1(x_1,y)-S^v_{k-1}u^1(x_1,x_2))\partial _1\Delta ^v_k \omega (x_1,y)~dy\\&=\int _{{{\mathbb {R}}}}\phi _q(x_1,x_2-y)\int _0^1(y-x_2)\partial _2 S^v_{k-1}u^1(sy+(1-s)x_2)~{\text {d}}s\partial _1\Delta ^v_k \omega (x_1,y)~dy. \end{aligned} \end{aligned}$$

where $\phi _j(x)\triangleq 2^{2j}{\mathcal {F}}^{-1}({\varphi })(2^jx)$. By anisotropic Hölder inequality and Bernstein inequality, we can bound $Y_{111}$ by

$$\begin{aligned} \begin{aligned} |Y_{111}|&\le C\sum _{|q-k|\le 2}\Vert [\Delta _q^v, S^v_{k-1}u^1\partial _1]\Delta _k^v \omega \Vert _{L^2}\Vert \Delta _q^v\omega \Vert _{L^2}\\&\le C\sum _{|q-k|\le 2}2^{-q}\Vert \partial _2S_{k-1}^vu_1\Vert _{L^2_{x_2}L^\infty _{x_1}}\Vert \partial _1\Delta _k^v\omega \Vert _{L^\infty _{x_2}L^2_{x_1}}\Vert \Delta _q^v\omega \Vert _{L^2}\\&\le C\sum _{|q-k|\le 2}2^{-q+\frac{q}{2}}\Vert \partial _2S_{k-1}^vu_1\Vert _{L^2}^{\frac{1}{2}}\Vert \partial _1\partial _2S_{k-1}^vu_1\Vert _{L^2}^{\frac{1}{2}}\Vert \partial _1\Delta _q^v\omega \Vert _{L^2}\Vert \Delta _q^v\omega \Vert _{L^2}\\&\le C\sum _{|q-k|\le 2}2^{\frac{k-q}{2}}\Vert \omega \Vert _{L^2}\Vert \partial _1\Delta _q^v\omega \Vert _{L^2}\Vert \Delta _q^v\omega \Vert _{L^2}\\&\le C\sum _{|q-k|\le 2}2^{-q}a_q\Vert \omega \Vert _{L^2}\Vert \partial _1\omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}\Vert \omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}. \end{aligned} \end{aligned}$$

For $Y_{112}$, by anisotropic Hölder inequality and Bernstein inequality,

$$\begin{aligned} Y_{112}= & {} -\sum _{|q-k|\le 2}\int _{{{\mathbb {R}}}^2} ((S^v_{k-1}u^1-S^v_{q}u^1)\partial _1\Delta ^v_q\Delta _k^v \omega )\Delta _q^v \omega ~{\text {d}}x\\\le & {} C\sum _{|q-k|\le 2}\Vert \Delta _k^vu^1\Vert _{L^2_{x_2}L^\infty _{x_1}}\Vert \partial _1\Delta _k^v\Delta _q^v\omega \Vert _{L^2}\Vert \Delta _q^v\omega \Vert _{L^\infty _{x_2}L^2_{x_1}}\\\le & {} C\sum _{|q-k|\le 2}\Vert \Delta _k^v u^1\Vert _{L^2}^{\frac{1}{2}}\Vert \partial _1\Delta _k^v u^1\Vert _{L^2}^{\frac{1}{2}}\Vert \partial _1\Delta _q^v \omega \Vert _{L^2}2^{\frac{q}{2}}\Vert \Delta _q^v\omega \Vert _{L^2}\\\le & {} C \sum _{|q-k|\le 2}2^{\frac{q-k}{2}} \Vert \partial _2\Delta _k^v u^1\Vert _{L^2}^{\frac{1}{2}}\Vert \partial _1\Delta _k^v u^1\Vert _{L^2}^{\frac{1}{2}}\Vert \partial _1\Delta _q^v \omega \Vert _{L^2}\Vert \Delta _q^v\omega \Vert _{L^2}\\\le & {} C2^{-q}a_q\Vert \omega \Vert _{L^2} \Vert \omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}} \Vert \partial _1\omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}. \end{aligned}$$

Next we estimate $Y_{12}$, using anisotropic Hölder inequality and Bernstein inequality,

$$\begin{aligned} \begin{aligned} Y_{12}&=-\sum _{|k-q|\le 2}\int _{{{\mathbb {R}}}^2}\Delta _q^v(\Delta _k^vu^1 S^v_{k-1}\partial _1 \omega )\Delta _q^v \omega ~{\text {d}}x\\&\le C\sum _{|q-k|\le 2}\Vert \Delta _k^v u^1\Vert _{L^\infty _{x_2}L^2_{x_1}}\Vert S_{k-1}^v \partial _1\omega \Vert _{L^2}\Vert \Delta _q^v\omega \Vert _{L^2_{x_2}L^\infty _{x_1}}\\&\le C\sum _{|q-k|\le 2}2^{\frac{k}{2}}\Vert \Delta _k^v u^1\Vert _{L^2}\Vert S_{k-1}^v \partial _1\omega \Vert _{L^2}\Vert \Delta _q^v\omega \Vert _{L^2}^{\frac{1}{2}}\Vert \Delta _q^v\partial _1\omega \Vert _{L^2}^{\frac{1}{2}}\\&\le C\sum _{|q-k|\le 2}2^{\frac{q-k}{2}}\Vert \partial _2\Delta _k^v u^1\Vert _{L^2}\Vert \partial _1\omega \Vert _{L^2}2^{-\frac{q}{2}}\Vert \Delta _q^v\omega \Vert _{L^2}^{\frac{1}{2}}\Vert \Delta _q^v\partial _1\omega \Vert _{L^2}^{\frac{1}{2}}\\&\quad +C \Vert \Delta _{-1}^vu^1\Vert _{L^2}\Vert \Delta _{-1}^v\partial _1\omega \Vert _{L^2}\Vert \partial _1\Delta _q^v \omega \Vert _{L^2}\Vert \Delta _q^v\omega \Vert _{L^2}\\&\le C2^{-q}a_q\big (\Vert \partial _1\omega \Vert _{L^2}\Vert \omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^{\frac{3}{2}} \Vert \partial _1\omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^{\frac{1}{2}}+\Vert u\Vert _{L^2}\Vert \partial _1\omega \Vert _{L^2}\Vert \omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}\Vert \partial _1\omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}\big ).\\ \end{aligned} \end{aligned}$$

In the same manner, $Y_{13}$ can be handled by,

$$\begin{aligned} \begin{aligned} Y_{13}&=-\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{{{\mathbb {R}}}^2}\Delta _q^v(\Delta _k^vu^1 \Delta _l^v\partial _1 \omega )\Delta _q^v \omega ~{\text {d}}x\\&\le C \sum _{k\ge q-1}\Vert \Delta _k^vu^1\Vert _{L^\infty _{x_1}L^2_{x_2}}\Vert \Delta _k^v\partial _1\omega \Vert _{L^2}\Vert \Delta _q^v\omega \Vert _{L^2_{x_1}L^\infty _{x_2}}\\&\le C\sum _{k\ge q-1} \Vert \Delta _k^v u^1\Vert _{L^2}^{\frac{1}{2}}\Vert \partial _1\Delta _k^v u^1\Vert _{L^2}^{\frac{1}{2}}\Vert \Delta _k^v\partial _1\omega \Vert _{L^2}2^{\frac{q}{2}}\Vert \Delta _q^v\omega \Vert _{L^2}\\&\le C \sum _{|q-k|\le 2}2^{\frac{q-k}{2}} \Vert \partial _2\Delta _k^v u^1\Vert _{L^2}^{\frac{1}{2}}\Vert \partial _1\Delta _k^v u^1\Vert _{L^2}^{\frac{1}{2}}\Vert \partial _1\Delta _q^v \omega \Vert _{L^2}\Vert \Delta _q^v\omega \Vert _{L^2}\\&\quad +C \Vert \Delta _{-1}^vu^1\Vert _{L^2}^{\frac{1}{2}}\Vert \partial _1\Delta _{-1}^vu^1\Vert _{L^2}^{\frac{1}{2}}\Vert \partial _1\Delta _q^v \omega \Vert _{L^2}\Vert \Delta _q^v\omega \Vert _{L^2}\\&\le C(\Vert u\Vert _{L^2}+\Vert \omega \Vert _{L^2})2^{-q}a_q \Vert \omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}} \Vert \partial _1\omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}. \end{aligned} \end{aligned}$$

(A.21)

where discrete Young’s inequality has been used in the last two lines.

Then, we estimate $Y_2$. Similar as $Y_1$, we can also divide it into the following three terms by the Bony’s decomposition,

$$\begin{aligned} \begin{aligned} Y_2&=-\int _{{{\mathbb {R}}}^2}\Delta _q^v(u^2 \partial _2 \omega )\Delta _q^v \omega ~{\text {d}}x\\&=-\sum _{|k-q|\le 2}\int _{{{\mathbb {R}}}^2}\Delta _q^v(S^v_{k-1}u^2 \Delta _k^v\partial _2 \omega )\Delta _q^v \omega ~{\text {d}}x\\&\quad -\sum _{|k-q|\le 2}\int _{{{\mathbb {R}}}^2}\Delta _q^v(\Delta _k^vu^2 S^v_{k-1}\partial _2 \omega )\Delta _q^v \omega ~{\text {d}}x\\&\quad -\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{{{\mathbb {R}}}^2}\Delta _q^v(\Delta _k^vu^2 \Delta _l^v\partial _2 \omega )\Delta _q^v \omega ~{\text {d}}x\\&\triangleq Y_{21}+Y_{22}+Y_{23}. \end{aligned} \end{aligned}$$

(A.22)

For $Y_{21}$, we can write it as,

$$\begin{aligned} \begin{aligned} Y_{21}&=-\sum _{|q-k|\le 2}\int _{{{\mathbb {R}}}^2}\Delta _q^v(S^v_{k-1}u^2\partial _2\Delta _k^v \omega )\Delta _q^v \omega ~{\text {d}}x\\&=-\sum _{|q-k|\le 2}\int _{{{\mathbb {R}}}^2}[\Delta _q^v, S^v_{k-1}u^2\partial _2]\Delta _k^v \omega \Delta _q^v \omega ~{\text {d}}x\\&\quad -\sum _{|q-k|\le 2}\int _{{{\mathbb {R}}}^2}(S^v_{k-1}u^2-S_q^vu^2)\partial _2\Delta _q^v \Delta _k^v\omega \Delta _q^v \omega ~{\text {d}}x\\&\quad -\int _{{{\mathbb {R}}}^2}{S}^v_{q}u^2\partial _2\Delta ^v_q \omega \Delta _q^v \omega ~{\text {d}}x\\&\triangleq Y_{211}+Y_{212}+Y_{213}, \end{aligned} \end{aligned}$$

Here, we should notice that $Y_{113}+Y_{213}=0$ because of the divergence-free condition of u, so we do not need to estimate these two terms.

For $Y_{211}$, similar as $Y_{111}$, the commutator can be written as,

$$\begin{aligned} \begin{aligned} \left[ \Delta _q^v, S^v_{k-1}u^2\partial _2\right] \Delta _k^v \omega =\int _{{{\mathbb {R}}}}\phi _q(x_1,x_2-y)\int _0^1(y-x_2)\partial _2 S^v_{k-1}u^2(sy+(1-s)x_2)~{\text {d}}s\partial _2\Delta ^v_k \omega (x_1,y)~dy. \end{aligned} \end{aligned}$$

Thus by anisotropic Hölder inequality and Biot–Savart law $u^2=\partial _1\Delta ^{-1}\omega$,

$$\begin{aligned} \begin{aligned} |Y_{211}|&\le C\sum _{|q-k|\le 2}2^{-q}\Vert \partial _2S^v_{k-1}u^2\Vert _{L^\infty _{x_2}L^2_{x_1}}\Vert \Delta ^v_{k}\partial _2\omega \Vert _{L^2_{x_2}L^\infty _{x_1}}\Vert \Delta _q^v\omega \Vert _{L^2}\\&\le C\sum _{|q-k|\le 2}2^{k-q}\Vert \partial _2S^v_{k-1}u^2\Vert _{L^2}^{\frac{1}{2}}\Vert \partial _2\partial _2S^v_{k-1}u^2\Vert _{L^2}^{\frac{1}{2}} \Vert \Delta _k^v \omega \Vert _{L^2}^{\frac{1}{2}}\Vert \Delta _k^v\partial _1\omega \Vert _{L^2}^{\frac{1}{2}}\Vert \Delta _q^v \omega \Vert _{L^2}\\&\le C2^{-q}a_q(\Vert \omega \Vert _{L^2}+\Vert \partial _1\omega \Vert _{L^2})\Vert \omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^{\frac{3}{2}} \Vert \partial _1\omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^{\frac{1}{2}}.\\ \end{aligned} \end{aligned}$$

Similarly, $Y_{212}$ can be bounded by

$$\begin{aligned} \begin{aligned} |Y_{212}|\le C2^{-q}a_q(\Vert u\Vert _{L^2}+\Vert \omega \Vert _{L^2}+\Vert \partial _1\omega \Vert _{L^2})\Vert \omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^{\frac{3}{2}} \Vert \partial _1\omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^{\frac{1}{2}}.\\ \end{aligned} \end{aligned}$$

Next we estimate $Y_{22}$, by the anisotropic Hölder inequality and discrete Young’s inequality,

$$\begin{aligned} \begin{aligned} Y_{22}&=-\sum _{|k-q|\le 2}\int _{{{\mathbb {R}}}^2}\Delta _q^v(\Delta _k^vu^2 S^v_{k-1}\partial _2 \omega )\Delta _q^v \omega ~{\text {d}}x\\&\le C\sum _{|q-k|\le 2}\Vert \Delta ^v_ku^2\Vert _{L^\infty _{x_2}L^2_{x_1}}\Vert S^v_{k-1}\partial _2 \omega \Vert _{L^2_{x_2}L^\infty _{x_1}}\Vert \Delta ^v_q\omega \Vert _{L^2}\\&\le C\sum _{|q-k|\le 2}\Vert \Delta ^v_ku^2\Vert _{L^2}^{\frac{1}{2}}\Vert \partial _2\Delta ^v_ku^2\Vert _{L^2}^{\frac{1}{2}}\Vert S^v_{k-1}\partial _2 \omega \Vert _{L^2}^{\frac{1}{2}}\Vert \partial _1S^v_{k-1}\partial _2 \omega \Vert _{L^2}^{\frac{1}{2}}\Vert \Delta ^v_q\omega \Vert _{L^2}\\&\le C\sum _{|q-k|\le 2}2^{\frac{q}{2}}\Vert \Delta ^v_ku^2\Vert _{L^2}^{\frac{1}{2}}\Vert \partial _2\Delta ^v_ku^2\Vert _{L^2}^{\frac{1}{2}}\big (\sum _{m\le q-2}2^{\frac{m-q}{2}}2^\frac{m}{2}\Vert \Delta _m^v\omega \Vert _{L^2}\big )^{\frac{1}{2}}\\&\quad \times \big (\sum _{n\le q-2}2^{\frac{n-q}{2}}2^\frac{n}{2}\Vert \Delta _n^v\partial _1\omega \Vert _{L^2}\big )^{\frac{1}{2}}\Vert \Delta ^v_q\omega \Vert _{L^2}\\&\le C2^{-q}a_q(\Vert u\Vert _{L^2}+\Vert \omega \Vert _{L^2}+\Vert \partial _1\omega \Vert _{L^2})\Vert \omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^{\frac{3}{2}} \Vert \partial _1\omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^{\frac{1}{2}}. \end{aligned} \end{aligned}$$

The estimate of $Y_{23}$ is the same as $Y_{13}$; thus, we can obtain

$$\begin{aligned} \begin{aligned} |Y_{23}|\le C2^{-q}a_q(\Vert u\Vert _{L^2}+\Vert \omega \Vert _{L^2}+\Vert \partial _1\omega \Vert _{L^2})\Vert \omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^{2}.\\ \end{aligned} \end{aligned}$$

Summing up all these estimates above, noticing that u and $\omega$ are bounded in $L^2$, we deduce from (A.18) that

$$\begin{aligned} \begin{aligned} \frac{1}{2}\frac{{\text {d}}}{{\text {d}}t}\Vert \Delta _q^v\omega (t)\Vert _{L^2}^2+\frac{1}{2}\Vert \partial _1\Delta _q^v\omega \Vert _{L^2}^2&\le C2^{-q}a_q\Vert \theta \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^2+C2^{-q}a_q(1+\Vert \partial _1\omega \Vert _{L^2})\\&\quad \times \left( \Vert \omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}} \Vert \partial _1\omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}+\Vert \omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^{\frac{3}{2}} \Vert \partial _1\omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^{\frac{1}{2}}\right. \\&\quad \left. +\,\Vert \omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^2\right) . \end{aligned} \end{aligned}$$

Then integrating from 0 to t with respect to time variable, taking square root on both sides and summing up of q, by Minkowski inequality we obtain

$$\begin{aligned} \begin{aligned}&\Vert \omega (t)\Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}+\bigg (\int _0^t\Vert \partial _1\omega (\tau )\Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}~{\text {d}}\tau \bigg )^{\frac{1}{2}}\\&\quad \le \Vert \omega _0\Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}+ C\bigg (\int _0^t\Vert \theta \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^2~{\text {d}}\tau \bigg )^{\frac{1}{2}}+C\bigg (\int _0^t(1+\Vert \partial _1\omega \Vert _{L^2})\\&\qquad \times (\Vert \omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}} \Vert \partial _1\omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}+\Vert \omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^{\frac{3}{2}} \Vert \partial _1\omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^{\frac{1}{2}}+\Vert \omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^2)~{\text {d}}\tau \bigg )^{\frac{1}{2}}\\&\quad \le \Vert \omega _0\Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}+C(t)\Vert \theta \Vert _{L^\infty ({\mathcal {B}}^{0,\frac{1}{2}})}+C \bigg (\int _0^t(1+\Vert \partial _1\omega \Vert _{L^2})^2\Vert \omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^2~{\text {d}}\tau \bigg )^{\frac{1}{2}}\\&\qquad +\frac{1}{2}\bigg (\int _0^t\Vert \partial _1\omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^2~{\text {d}}\tau \bigg )^{\frac{1}{2}}. \end{aligned} \end{aligned}$$

According to Lemma 2.6, we have $\theta$ is bounded in $B^{\frac{1}{2}}_{2,1}$, then by the Besov embedding $B^{\frac{1}{2}}_{2,1}\hookrightarrow {\mathcal {B}}^{0,\frac{1}{2}}$, we get

$$\begin{aligned} \begin{aligned} \Vert \omega (t)\Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^2 \le C(t)+ C\int _0^t(1+\Vert \partial _1\omega \Vert _{L^2})^2\Vert \omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^2~{\text {d}}\tau . \end{aligned} \end{aligned}$$

Then, using Grönwall’s Lemma, We deduce

$$\begin{aligned} \begin{aligned} \Vert \omega (t)\Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^2+\int _0^t\Vert \partial _1\omega (\tau )\Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}^2~{\text {d}}\tau \le C(t). \end{aligned} \end{aligned}$$

(A.23)

According to Biot–Savart law (2.3), divergence-free condition of the velocity u and Besov embedding, it follows that

$$\begin{aligned} \begin{aligned} \int _0^t\Vert \partial _1u^1,\partial _1u^2,\partial _2u^2\Vert _{L^\infty }^2~{\text {d}}\tau \le C(t). \end{aligned} \end{aligned}$$

Also, using inequality $\Vert f\Vert _{L^\infty _{x_1}}^2\le C\Vert f\Vert _{L^2_{x_1}}\Vert \partial _1 f\Vert _{L^2_{x_1}}$ and estimate (A.23), we have

$$\begin{aligned} \begin{aligned} \int _0^t\Vert \omega \Vert _{L^\infty }^2~{\text {d}}\tau \le C\int _0^t\Vert \omega \Vert _{{\mathcal {B}}^{0,\frac{1}{2}}}\Vert \partial _1\omega \Vert _{\mathcal B^{0,\frac{1}{2}}}~{\text {d}}\tau \le C(t). \end{aligned} \end{aligned}$$

From the above, it follows that

$$\begin{aligned} \begin{aligned} \int _0^t\Vert \partial _2u^1\Vert _{L^\infty }^2~{\text {d}}\tau \le \int _0^t\Vert \omega \Vert _{L^\infty }^2~{\text {d}}\tau +\int _0^t\Vert \partial _1u^2\Vert _{L^\infty }^2~{\text {d}}\tau \le C(t). \end{aligned} \end{aligned}$$

Finally, we obtain the estimate

$$\begin{aligned} \begin{aligned} \int _0^t\Vert \nabla u\Vert _{L^\infty }^2~{\text {d}}\tau \le C(t), \end{aligned} \end{aligned}$$

which completes the proof of this proposition. $\square$