Large time behavior and convergence for the Camassa–Holm equations with fractional Laplacian viscosity

Abstract

In this paper, we consider the n-dimensional ($n=2,3$) Camassa–Holm equations with fractional Laplacian viscosity in the whole space. In contrast to the Camassa–Holm equations without any nonlocal effect, much less has been known on the large time behavior and convergences of solutions. Here we study first the large time behavior of solutions, then consider the relation between the equations under consideration and the imcompressible Navier–Stokes equations with fractional Laplacian viscosity (INSF). By applying the fractional Leibniz chain rule and the fractional Gagliardo–Nirenberg–Sobolev type estimates, the high and low frequency splitting method and the Fourier splitting method, we shall establish the large time non-uniform decays and algebraic rate decays of solutions. In the critical case $s=\dfrac{n}{4}$, the nonlocal version of Ladyzhenskaya’s inequality along with the smallness of initial data in suitable Sobolev spaces is needed. In addition, by estimates for the fractional heat kernels, we prove that the solutions to the Camassa–Holm equations with nonlocal viscosity converge strongly as the filter parameter $\alpha \rightarrow ~0$ to solutions of the equations INSF.

Appendices

Appendix A

We are now in the position to show (3.18). Note that $\mathbf {v}^{\varepsilon }$ is a solution of (1.1)–(1.2), multiplying the first equation for $\mathbf {v}^{\varepsilon }$ in (1.1) by $\Delta \mathbf {v}^{\varepsilon }$, then integrating by parts yields

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\left\| \nabla \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+\nu \left\| \Lambda ^{\beta }\nabla \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}=\left\langle \mathbf {u}^{\varepsilon }\cdot \nabla \mathbf {v}^{\varepsilon }, \Delta \mathbf {v}^{\varepsilon }\right\rangle +\left\langle \mathbf {v}^{\varepsilon }\cdot \nabla \mathbf {u}^{\varepsilon T}, \Delta \mathbf {v}^{\varepsilon }\right\rangle . \end{aligned}$$

(A.1)

We then deal with the two terms on the right hand side of (A.1) through two cases:

Case (I) $\dfrac{n}{4}< \beta < 1$ for $n=2,3$;

Case (II) $\beta =\dfrac{n}{4}$ for $n=2,3$.

We first consider Case (I)   $\dfrac{n}{4}< \beta < 1$ for $n=2,3$.

In this case, notice that $\left\langle \mathbf {u}^{\varepsilon }\cdot \nabla \mathbf {v}^{\varepsilon }, \mathbf {v}^{\varepsilon }\right\rangle =0$,    $\displaystyle \frac{\beta }{n}=\frac{1}{2}-\frac{n/2-\beta }{n}$,  $\displaystyle \frac{n}{2}-1<\frac{n}{2}-\beta <\frac{n}{4}$,  $\displaystyle \frac{n}{2}-1<\frac{n}{2}-2\beta +1<1$, Hölder’s inequality, Sobolev inequality and Cauchy-Schwartz inequality yield that

$$\begin{aligned} \displaystyle \left| \left\langle \mathbf {u}^{\varepsilon }\cdot \nabla \mathbf {v}^{\varepsilon }, \Delta \mathbf {v}^{\varepsilon }\right\rangle \right|= & {} \left| \left\langle \nabla \mathbf {u}^{\varepsilon } \cdot \nabla \mathbf {v}^{\varepsilon }, \nabla \mathbf {v}^{\varepsilon } \right\rangle \right| \nonumber \\\le & {} C\left\| \nabla \mathbf { v}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \nabla \mathbf {u}^{\varepsilon }\nabla v^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\\le & {} C\left\| \nabla \mathbf {v}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \nabla \mathbf {v}^{\varepsilon } \right\| _{L^{\frac{2n}{n-2\beta }}(\mathbb {R}^{n})} \left\| \nabla \mathbf {u}^{\varepsilon } \right\| _{L^{\frac{n}{\beta }}(\mathbb {R}^{n})}\nonumber \\\le & {} C\left\| \nabla \mathbf {v}^{\varepsilon }\right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\beta }\nabla \mathbf {v}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})}\left\| \Lambda ^{\frac{n}{2}-\beta } \nabla \mathbf {u}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\\le & {} C\left\| \nabla \mathbf {v}^{\varepsilon }\right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\beta }\nabla \mathbf {v}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \nabla \mathbf {u}^{\varepsilon } \right\| ^{1+\beta -\frac{n}{2}} _{L^{2}(\mathbb {R}^{n})} \left\| \Delta \mathbf {u}^{\varepsilon } \right\| ^{\frac{n}{2}-\beta }_{L^{2}(\mathbb {R}^{n})}\nonumber \\\le & {} C\left\| \nabla \mathbf {v}^{\varepsilon }\right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\beta }\nabla \mathbf {v}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})}\left\| \mathbf {v}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\\le & {} \frac{\nu }{4}\left\| \Lambda ^{\beta } \nabla \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+C\left\| \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\left\| \nabla \mathbf {v}^{\varepsilon }\right\| ^{2}_{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(A.2)

On the other hand, Lemmas 2.8 and 2.11 ensure that

$$\begin{aligned}&\left| \left\langle \mathbf {v}^{\varepsilon }\cdot \nabla \mathbf {u}^{\varepsilon T}, \Delta \mathbf {v}^{\varepsilon }\right\rangle \right| \nonumber \\&\qquad \le \left| \left\langle \Lambda ^{1-\beta } \left( \mathbf {v}^{\varepsilon }\cdot \nabla \mathbf {u}^{\varepsilon T}\right) , \Lambda ^{\beta }\nabla \mathbf {v}^{\varepsilon }\right\rangle \right| \nonumber \\&\qquad \le \left\| \Lambda ^{1-\beta } \left( \mathbf {v}^{\varepsilon }\cdot \nabla \mathbf {u}^{\varepsilon T}\right) \right\| _{L^{2}(\mathbb {R}^{n})}\left\| \Lambda ^{\beta }\nabla \mathbf {v}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad \displaystyle \le C\left( \left\| \Lambda ^{1-\beta } \mathbf {v}^{\varepsilon }\right\| _{L^{ \frac{2n}{n-4\beta +2} }(\mathbb {R}^{n})}\left\| \nabla \mathbf {u}^{\varepsilon } \right\| _{L^{\frac{2n}{4\beta -2} }(\mathbb {R}^{n})}\right. \nonumber \\&\qquad \left. +\left\| \mathbf {v}^{\varepsilon } \right\| _{L^{ \frac{2n}{n-2\beta } }(\mathbb {R}^{n})}\left\| \nabla \mathbf {u}^{\varepsilon } \right\| _{L^{ \frac{n}{\beta } }(\mathbb {R}^{n})}\right) \cdot \left\| \Lambda ^{\beta }\nabla \mathbf {v}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad \le \displaystyle \frac{\nu }{4}\left\| \Lambda ^{\beta } \nabla \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+C\left\| \Lambda ^{\beta }\mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\left\| \Lambda ^{ \frac{n}{2}-2\beta +1}\nabla \mathbf {u}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad \le \displaystyle \frac{\nu }{4}\left\| \Lambda ^{\beta } \nabla \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+C\left\| \Lambda ^{\beta }\mathbf {v}^{\varepsilon }\right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\Vert \mathbf {v}^{\varepsilon } \Vert ^{2}_{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad \le \displaystyle \frac{\nu }{4}\left\| \Lambda ^{\beta } \nabla \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+C\left( \left\| \mathbf {v}^{\varepsilon }\right\| _{L^{2}(\mathbb {R}^{n})}+\left\| \nabla \mathbf {v}^{\varepsilon }\right\| _{L^{2}(\mathbb {R}^{n})}\right) ^{2}\Vert \mathbf {v}^{\varepsilon } \Vert ^{2}_{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(A.3)

Combining (A.1) with (A.2) and (A.3) gives rise to

$$\begin{aligned} \frac{d}{dt}\left\| \nabla \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+\frac{\nu }{2}\left\| \Lambda ^{\beta }\nabla \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\le C\left\| \nabla \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\left\| \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(A.4)

We next consider Case (II)   $\beta =\dfrac{n}{4}$ for $n=2,3$.

In this case, thanks to $\left\langle \mathbf {u}^{\varepsilon }\cdot \nabla \mathbf {v}^{\varepsilon }, \mathbf {v}^{\varepsilon }\right\rangle =0$ and $\frac{1}{4}=\frac{1}{2}-\frac{\beta }{n}$, note that Lemma 2.12, applying Hölder’s inequality and Gagliardo–Nirenberg–Sobolev inequality imply

$$\begin{aligned} \left| \left\langle \mathbf {u}^{\varepsilon }\cdot \nabla \mathbf {v}^{\varepsilon }, \Delta \mathbf {v}^{\varepsilon }\right\rangle \right|\le & {} \left| \left\langle \nabla \mathbf {u}^{\varepsilon }\cdot \nabla \mathbf {v}^{\varepsilon }, \nabla \mathbf {v}^{\varepsilon }\right\rangle \right| \nonumber \\\le & {} \left\| \nabla \mathbf {u}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \nabla \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{4}(\mathbb {R}^{n})}\nonumber \\\le & {} C\left\| \nabla \mathbf {u}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})} \left( \left\| \nabla \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+\left\| \Lambda ^{\frac{n}{4}}\nabla \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\right) , \end{aligned}$$

(A.5)

and

$$\begin{aligned}&\left| \left\langle \mathbf {v}^{\varepsilon }\cdot \nabla \mathbf {u}^{\varepsilon T}, \Delta \mathbf {v}^{\varepsilon }\right\rangle \right| \nonumber \\&\quad \le \left\| \nabla \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{4}(\mathbb {R}^{n})}\left\| \nabla \mathbf {u}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})}+\left\| \Delta \mathbf {u}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})}\left\| \mathbf {v}^{\varepsilon }\nabla v^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le \left\| \nabla \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{4}(\mathbb {R}^{n})}\left\| \nabla \mathbf {u}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})}+\left\| \Delta \mathbf {u}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})}\left\| \mathbf {v}^{\varepsilon } \right\| _{L^{4}(\mathbb {R}^{n})}\left\| \nabla v^{\varepsilon } \right\| _{L^{4}(\mathbb {R}^{n})}\nonumber \\&\quad \le C\left\| \nabla \mathbf {u}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}}\nabla \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+\left\| \Delta \mathbf {u}^{\varepsilon }\right\| _{L^{2}(\mathbb {R}^{n})}\left\| \Lambda ^{\frac{n}{4}} \mathbf {v}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}} \nabla \mathbf {v}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C\left\| \nabla \mathbf {u}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}}\nabla \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})} +\left\| \Delta \mathbf {u}^{\varepsilon }\right\| _{L^{2}(\mathbb {R}^{n})}\left( \left\| \Lambda ^{\frac{n}{4}} \mathbf {v}^{\varepsilon } \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}+ \left\| \Lambda ^{\frac{n}{4}} \nabla \mathbf {v}^{\varepsilon } \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\right) \nonumber \\&\quad \lesssim \left( \left\| \nabla \mathbf {u}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})}+\left\| \Delta \mathbf {u}^{\varepsilon }\right\| _{L^{2}(\mathbb {R}^{n})}\right) \left\| \Lambda ^{\frac{n}{4}} \nabla \mathbf {v}^{\varepsilon } \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}+\left\| \Delta \mathbf {u}^{\varepsilon }\right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}} \mathbf {v}^{\varepsilon } \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \lesssim \left( \left\| \nabla \mathbf {u}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})}+\left\| \Delta \mathbf {u}^{\varepsilon }\right\| _{L^{2}(\mathbb {R}^{n})}\right) \left\| \Lambda ^{\frac{n}{4}} \nabla \mathbf {v}^{\varepsilon } \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad +\left\| \Delta \mathbf {u}^{\varepsilon }\right\| _{L^{2}(\mathbb {R}^{n})} \left( \left\| \mathbf {v}^{\varepsilon } \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}+\left\| \nabla \mathbf {v}^{\varepsilon } \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\right) . \end{aligned}$$

(A.6)

By Proposition 2.4, the assumptions in (III) of this theorem, once we choose $\left\| \nabla \mathbf {u}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})}+\left\| \Delta \mathbf {u}^{\varepsilon }\right\| _{L^{2}(\mathbb {R}^{n})}\lesssim \left\| \mathbf {v}^{\varepsilon } \right\| _{L^{2}(\mathbb {R}^{n})} \lesssim \Vert \mathbf {v}_{0}^{\varepsilon } \Vert _{L^{2}(\mathbb {R}^{n})}\lesssim \Vert \mathbf {v}_{0} \Vert _{L^{2}(\mathbb {R}^{n})}\le \varepsilon ^{*} \le \frac{\nu }{4}$, (A.1), (A.5) and (A.6) yield

$$\begin{aligned} \frac{d}{dt}\Vert \nabla \mathbf {v}^{\varepsilon } \Vert ^{2}_{L^{2}(\mathbb {R}^{n})}+\frac{\nu }{2}\left\| \Lambda ^{\frac{n}{4}}\nabla \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\lesssim \Vert \mathbf {v}^{\varepsilon } \Vert _{L^{2}(\mathbb {R}^{n})}\Vert \nabla \mathbf {v}^{\varepsilon } \Vert ^{2}_{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(A.7)

Using (A.4) and (A.7), Gronwall’s lemma yields that for $\displaystyle \frac{n}{4}\le \beta <1$ with $n=2,3$

$$\begin{aligned} \Vert \nabla \mathbf { v}^{\varepsilon } \Vert ^{2}_{L^{2}(\mathbb {R}^{n})}\le \Vert \nabla \mathbf {v}^{\varepsilon }_{0} \Vert ^{2}_{L^{2}(\mathbb {R}^{n})}e^{C\Vert \mathbf {v}^{\varepsilon }_{0} \Vert ^{2}_{L^{2}(\mathbb {R}^{n})}}\le C\varepsilon ^{2}. \end{aligned}$$

(A.8)

This gives

$$\begin{aligned}&\left\| \Lambda ^{\beta } \mathbf {u}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+\alpha ^{2}\left\| \Lambda ^{\beta } \nabla \mathbf {u}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})} \le \left\| \nabla \mathbf {u}^{\varepsilon }\right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+\alpha ^{2}\left\| \Delta \mathbf {u}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})} \nonumber \\&\le \left\| \nabla \mathbf {v}^{\varepsilon } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\le C\varepsilon ^{2}. \end{aligned}$$

(A.9)

It follows from (2.1) and (A.9) that

$$\begin{aligned} \frac{d}{dt}\left( \Vert \mathbf {u}^{\varepsilon } \Vert ^{2}_{L^{2}(\mathbb {R}^{n})}+\alpha ^{2}\Vert \nabla \mathbf {u}^{\varepsilon } \Vert ^{2}_{L^{2}(\mathbb {R}^{n})}\right) \ge -C\varepsilon ^{2}. \end{aligned}$$

This is the estimate (3.18). So far, we finish the proof of Theorem 3.1.

Appendix B

We shall prove (VI-1) of Theorem 4.1 here by using inductive argument. Due to the regularity of solutions (Proposition 2.4), we present the proof only formally. It should be pointed out that the key point of the proof is to establish an inequality in a form satisfing the conclusion in (IV) of this theorem. To achieve this, we shall divide the proof into the following three steps.

Step 1. For $m=0,1$, the inequality holds by (III-2) and (II), respectively. That is,

$$\begin{aligned} \Vert \mathbf {v} \Vert ^{2}_{L^{2}(\mathbb {R}^{n})}\le C(1+t)^{-\frac{n}{2\beta }}, ~~\Vert \nabla \mathbf {v} \Vert ^{2}_{L^{2}(\mathbb {R}^{n})}\le C(1+t)^{-\frac{1}{\beta }-\frac{n}{2\beta }}. \end{aligned}$$

Step 2. We now assume (inductive assumption) that the decay

$$\begin{aligned} \Vert \nabla ^{m} \mathbf {v} \Vert ^{2}_{L^{2}(\mathbb {R}^{n})}\le C(1+t)^{-\frac{m}{\beta }-\frac{n}{2\beta }} \end{aligned}$$

(B.1)

holds for all $m<M$. Here, m and M are both non-negative integers.

Step 3. We will verify that the inequality (B.1) is true for $m=M$.

Multiplying the first equation in (1.1) by $\Delta ^{M}\mathbf {v}$, and then integrating by parts the resulting equation gives rise to

$$\begin{aligned}&\displaystyle \frac{d}{dt}\left\| \nabla ^{M} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+2\nu \left\| \Lambda ^{\beta }\nabla ^{M} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le \left| \left\langle \mathbf {u} \cdot \nabla \mathbf {v}, \Delta ^{M} \mathbf {v}\right\rangle \right| + \left| \left\langle \mathbf {v} \cdot \nabla \mathbf {u}^{T}, \Delta ^{M} \mathbf {v} \right\rangle \right| \nonumber \\&\quad \triangleq ~~I_{M}+J_{M}. \end{aligned}$$

(B.2)

To bound $I_{M}$ and $J_{M}$, there are two cases to consider.

Case (I) $\displaystyle \frac{n}{4}< \beta < 1$ for $n=2,3$;

Case (II) $\displaystyle \beta =\frac{n}{4}$ for $n=2,3$.

We first consider Case (I) $\displaystyle \frac{n}{4}< \beta < 1$ for $n=2,3$.

In this case, it is easy to check that $\displaystyle \frac{n}{2}-\beta <\beta$ and $\frac{n}{\beta }=\frac{2n}{n-2(n/2-\beta )}$. Recall that (B.2) and $\left\langle \mathbf {u} \cdot \nabla \mathbf {v}, \mathbf {v} \right\rangle =0$, thanks to Cauchy’s inequality, Hölder’s inequality and Gagliardo–Nirenberg–Sobolev inequality, one deduces that

$$\begin{aligned} \displaystyle I_{M}= & {} \left| \left\langle \mathbf {u} \cdot \nabla \mathbf {v}, \Delta ^{M} \mathbf {v} \right\rangle \right| \nonumber \\= & {} \left| \sum \limits ^{M}_{m=1} \left( \begin{array}{l} M\\ m \end{array}\right) \left\langle \nabla ^{m}\mathbf {u}\cdot \nabla \nabla ^{M-m}\mathbf {v}, \nabla ^{M} \mathbf {v} \right\rangle \right| \nonumber \\\le & {} C\sum \limits ^{M}_{m=1} \left\| \nabla ^{M+1-m}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\left\| \nabla ^{m}\mathbf {u}\cdot \nabla ^{M }\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\\le & {} C\sum \limits ^{M}_{m=1} \left\| \nabla ^{M+1-m}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \nabla ^{m}\mathbf {u} \right\| _{L^{\frac{n}{\beta }}(\mathbb {R}^{n})} \left\| \nabla ^{M}\mathbf {v} \right\| _{L^{\frac{2n}{n-2\beta }}(\mathbb {R}^{n})}\nonumber \\\le & {} C\sum \limits ^{M}_{m=1} \left\| \nabla ^{M+1-m}\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\beta }\nabla ^{m}\mathbf {u} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\beta }\nabla ^{M}\mathbf {v } \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\\le & {} C\sum \limits ^{M}_{m=1} \left\| \nabla ^{M+1-m}\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{2}-\beta }\nabla ^{m}\mathbf {u} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\beta }\nabla ^{M}\mathbf {v } \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\\le & {} C\sum \limits ^{M}_{m=1} \left\| \nabla ^{M+1-m}\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\beta }\nabla ^{m-1}\mathbf {u} \right\| ^{2\beta -\frac{n}{2}} _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\beta }\nabla ^{m}\mathbf {u} \right\| ^{ \frac{n}{2}+1-2\beta } _{L^{2}(\mathbb {R}^{n})}\left\| \Lambda ^{\beta }\nabla ^{M}\mathbf {v } \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\\le & {} C\sum \limits ^{M}_{m=1} \left\| \nabla ^{M+1-m}\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\beta }\nabla ^{m-1}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\beta }\nabla ^{M}\mathbf {v } \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\\le & {} C\sum \limits ^{M}_{m=1} \left\| \nabla ^{M+1-m}\mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\beta }\nabla ^{m-1}\mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})} +\frac{\nu }{2}\left\| \Lambda ^{\beta }\nabla ^{M}\mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}, \end{aligned}$$

(B.3)

and

$$\begin{aligned} J_{M}= & {} \left| \left\langle \mathbf {v} \cdot \nabla \mathbf {u}^{T}, \Delta ^{M} \mathbf {v} \right\rangle \right| \nonumber \\= & {} \sum \limits ^{M}_{m=0} \left( \begin{array}{l} M\\ m \end{array}\right) \left| \left\langle \nabla ^{M}\mathbf {v}\cdot \nabla \nabla ^{m}\mathbf {u}, \nabla ^{M-m} \mathbf {v} \right\rangle \right| \nonumber \\\le & {} C \left| \left\langle \nabla ^{M}\mathbf {v}\cdot \nabla \mathbf {u}, \nabla ^{M } \mathbf {v} \right\rangle \right| +C\sum \limits ^{M}_{m=1}\left| \left\langle \nabla ^{M}\mathbf {v}\cdot \nabla ^{m+1} \mathbf {u}, \nabla ^{M -m} \mathbf {v} \right\rangle \right| . \end{aligned}$$

(B.4)

By Lemma 2.12, a similar computation to (B.3) shows that

$$\begin{aligned}&\left| \left\langle \nabla ^{M}\mathbf {v}\cdot \nabla \mathbf {u}, \nabla ^{M } \mathbf {v} \right\rangle \right| \nonumber \\&\quad \le \left\| \nabla ^{M}\mathbf {v} \right\| ^{2} _{L^{4}(\mathbb {R}^{n})}\left\| \nabla \mathbf {u} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le \left( \left\| \nabla ^{M}\mathbf {v} \right\| ^{1-\frac{n}{4\beta }} _{L^{2}(\mathbb {R}^{n})}\left\| \nabla ^{M}\mathbf {v} \right\| ^{\frac{n}{4\beta }} _{L^{2}(\mathbb {R}^{n})}\right) \left\| \nabla \mathbf {u} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le \left( C(\varepsilon )\left\| \nabla ^{M}\mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}+\varepsilon \left\| \Lambda ^{\beta } \nabla ^{M}\mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\right) \left\| \nabla \mathbf {u} \right\| _{L^{2}(\mathbb {R}^{n})}, \end{aligned}$$

(B.5)

and

$$\begin{aligned}&\sum \limits ^{M}_{m=1}\left| \left\langle \nabla ^{M}\mathbf {v}\cdot \nabla ^{m+1} \mathbf {u}, \nabla ^{M -m} \mathbf {v} \right\rangle \right| \nonumber \\&\quad \le \sum \limits ^{M}_{m=1}\left\| \nabla ^{M-m}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \nabla ^{m+1}\mathbf {u}\cdot \nabla ^{M}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le \sum \limits ^{M}_{m=1}\left\| \nabla ^{M-m}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \nabla ^{m+1}\mathbf {u} \right\| _{L^{\frac{n}{\beta }}(\mathbb {R}^{n})}\left\| \nabla ^{M}\mathbf {v} \right\| _{L^{\frac{2n}{n-2\beta }}(\mathbb {R}^{n})}\nonumber \\&\quad \qquad \le \sum \limits ^{M}_{m=1}\left\| \nabla ^{M-m}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{2}-\beta }\nabla ^{m+1}\mathbf {u} \right\| _{L^{2}(\mathbb {R}^{n})}\left\| \Lambda ^{\beta } \nabla ^{M}\mathbf {v } \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le \sum \limits ^{M}_{m=1}\left\| \nabla ^{M-m}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\beta }\nabla ^{m-1}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\left\| \Lambda ^{\beta } \nabla ^{M}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C\sum \limits ^{M}_{m=1}\left\| \nabla ^{M-m}\mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\beta }\nabla ^{m-1}\mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}+\frac{\nu }{4}\left\| \Lambda ^{\beta } \nabla ^{M}\mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(B.6)

We have used the relation $\mathbf {u}-\alpha ^{2}\Delta \mathbf {u}=\mathbf {v}$ in the estimates (B.3) and (B.6). Choosing $\varepsilon \le \dfrac{\nu }{4\Vert \nabla \mathbf {u}_{0} \Vert _{L^{2}(\mathbb {R}^{n})}}$, if follows from (B.2), (B.3), (B.4), (B.5) and (B.6) that

$$\begin{aligned}&\frac{d}{dt}\left\| \nabla ^{M} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+ \nu \left\| \Lambda ^{\beta }\nabla ^{M} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C\left\| \nabla ^{M} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\left\| \Lambda ^{\beta }\mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+C(\varepsilon )\left\| \nabla ^{M} \mathbf {v } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\left\| \nabla \mathbf {u} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad +\left\| \nabla ^{M-1} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\left\| \Lambda ^{\beta }\mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad +C \sum \limits ^{M}_{m=2}\left\| \nabla ^{M-m} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\left\| \Lambda ^{\beta }\nabla ^{m-1}\mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(B.7)

We next consider Case (II) $\displaystyle \beta =\frac{n}{4}$ for $n=2,3$.

In this case, thanks to Lemma 2.12, note that (B.2) and $\left\langle \mathbf {u}\cdot \nabla \mathbf {v}, \mathbf {v} \right\rangle =0$, we deduce the following two estimates:

$$\begin{aligned} I_{M}= & {} \left| \left\langle \mathbf {u} \cdot \nabla \mathbf {v}, \Delta ^{M} \mathbf {v} \right\rangle \right| \nonumber \\= & {} \left| \sum \limits ^{M}_{m=1} \left( \begin{array}{l} M\\ m \end{array}\right) \left\langle \nabla ^{m}\mathbf {u}\cdot \nabla \nabla ^{M-m}\mathbf {v}, \nabla ^{M} \mathbf {v} \right\rangle \right| \nonumber \\\le & {} C\sum \limits ^{M}_{m=1} \left\| \nabla ^{M+1-m}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \nabla ^{m}\mathbf {u}\cdot \nabla ^{M }\mathbf {v } \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\\le & {} C\sum \limits ^{M}_{m=1}\left\| \nabla ^{M+1-m}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\left\| \nabla ^{m}\mathbf {u} \right\| _{L^{4}(\mathbb {R}^{n})} \left\| \nabla ^{M}\mathbf {v} \right\| _{L^{4}(\mathbb {R}^{n})}\nonumber \\\le & {} C\sum \limits ^{M}_{m=1}\left\| \nabla ^{M+1-m}\mathbf {v }\right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}}\nabla ^{m}\mathbf {u} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}}\nabla ^{M}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\\le & {} C\sum \limits ^{M}_{m=1}\left\| \nabla ^{M+1-m}\mathbf {v }\right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}}\nabla ^{m-1}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}}\nabla ^{M}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(B.8)

and

$$\begin{aligned} J_{M}= & {} \left| \left\langle \mathbf {v} \cdot \nabla \mathbf {u}^{T}, \Delta ^{M} \mathbf {v} \right\rangle \right| \nonumber \\= & {} \sum \limits ^{M}_{m=0}\left( \begin{array}{l} M\\ m \end{array}\right) \left| \left\langle \nabla ^{M}\mathbf {v}\cdot \nabla \nabla ^{m}\mathbf {u}, \nabla ^{M-m} \mathbf {v } \right\rangle \right| \nonumber \\\le & {} C \sum \limits ^{M}_{m=0}\left\| \nabla ^{M-m}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \nabla ^{m+1}\mathbf {u}\cdot \nabla ^{M}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\\le & {} C \sum \limits ^{M}_{m=0}\left\| \nabla ^{M-m}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\left\| \nabla ^{m+1}\mathbf {u} \right\| _{L^{4}(\mathbb {R}^{n})} \left\| \nabla ^{M}\mathbf {v}\right\| _{L^{4}(\mathbb {R}^{n})}\nonumber \\\le & {} C \sum \limits ^{M}_{m=0}\left\| \nabla ^{M-m}\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}}\nabla ^{m+1}\mathbf {u} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}} \nabla ^{M}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\\le & {} C \sum \limits ^{M}_{m=0}\left\| \nabla ^{M-m}\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}}\nabla ^{m}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}} \nabla ^{M}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(B.9)

Combining (B.8) with (B.9) yields that

$$\begin{aligned} I_{M}+J_{M}\le & {} C \sum \limits ^{M}_{m=0} \left\| \nabla ^{M-m}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}} \nabla ^{m}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}} \nabla ^{M}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\\le & {} C \left\| \nabla ^{M}\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}} \mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}}\nabla ^{M} \mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&+C \sum \limits ^{M-1}_{m=1} \left\| \nabla ^{M-m}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}} \nabla ^{m}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}} \nabla ^{M}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&+C \left\| \mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}} \nabla ^{M}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\\le & {} C \left\| \nabla ^{M}\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})}\left\| \mathbf {v}\right\| ^{1-\frac{n}{4}} _{L^{2}(\mathbb {R}^{n})}\left\| \nabla \mathbf {v}\right\| ^{\frac{n}{4}} _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}}\nabla ^{M} \mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&+C \sum \limits ^{M-1}_{m=1} \left\| \nabla ^{M-m}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\left\| \nabla ^{m}\mathbf {v}\right\| ^{1-\frac{n}{4}} _{L^{2}(\mathbb {R}^{n})}\left\| \nabla ^{m+1}\mathbf {v}\right\| ^{\frac{n}{4}} _{L^{2}(\mathbb {R}^{n})} \left\| \Lambda ^{\frac{n}{4}}\nabla ^{M} \mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&+C\left\| \mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})}\left\| \Lambda ^{\frac{n}{4}}\nabla ^{M} \mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\\lesssim & {} \left\| \nabla ^{M}\mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\left\| \mathbf {v}\right\| ^{2-\frac{n}{2}} _{L^{2}(\mathbb {R}^{n})}\left\| \nabla \mathbf {v}\right\| ^{\frac{n}{2}} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&+\sum \limits ^{M-1}_{m=1}\left( \left\| \nabla ^{M-m}\mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}+\left\| \nabla ^{m}\mathbf {v}\right\| ^{(1-\frac{n}{4})\times 2} _{L^{2}(\mathbb {R}^{n})}\left\| \nabla ^{m+1}\mathbf {v}\right\| ^{\frac{n}{4}\times 2} _{L^{2}(\mathbb {R}^{n})}\right) \nonumber \\&+\left\| \mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})}\left\| \Lambda ^{\frac{n}{4}}\nabla ^{M} \mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}+\frac{\nu }{4} \left\| \Lambda ^{\frac{n}{4}}\nabla ^{M} \mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(B.10)

This together with (B.2) and the smallness assumption of the initial data for $\beta =\dfrac{n}{4}$ in (VI) with $M\le K$ ensures that

$$\begin{aligned}&\frac{d}{dt}\left\| \nabla ^{M} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+ \nu \left\| \Lambda ^{\frac{n}{4}}\nabla ^{M} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \lesssim \left\| \nabla ^{M} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\left\| \Lambda ^{\frac{n}{4}}\mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})} +\sum \limits ^{M-1}_{m=1} \left\| \nabla ^{M-m} \mathbf {v}\right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\left\| \Lambda ^\frac{n}{4}\nabla ^{m}\mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \lesssim \left\| \nabla ^{M} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\left\| \mathbf {v} \right\| ^{2-\frac{n}{2}}_{L^{2}(\mathbb {R}^{n})}\left\| \nabla \mathbf {v} \right\| ^{\frac{n}{2}}_{L^{2}(\mathbb {R}^{n})} +\sum \limits ^{M-1}_{m=1} \left\| \nabla ^{M-m} \mathbf {v}\right\| ^{2-\frac{n}{2}}_{L^{2}(\mathbb {R}^{n})}\left\| \nabla ^{m+1}\mathbf {v} \right\| ^{\frac{n}{2}}_{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \lesssim \left( \left\| \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+\left\| \nabla \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\right) \left\| \nabla ^{M} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad +\sum \limits ^{M-2}_{m=1}\left( \left\| \nabla ^{m} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+\left\| \nabla ^{m+1} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\right) \left\| \nabla ^{M-m} \mathbf {v}\right\| ^{2 }_{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad + \left( \left\| \nabla ^{M-1} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+\left\| \nabla ^{M} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\right) \left\| \nabla \mathbf {v}\right\| ^{2 }_{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \lesssim \left( (1+t)^{-\frac{n}{2\beta }}+(1+t)^{-\frac{1}{ \beta }-\frac{n}{2\beta }}\right) \left\| \nabla ^{M} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad +\sum \limits ^{M-2}_{m=1}\left( (1+t)^{-\frac{m}{ \beta }-\frac{n}{2\beta }}+(1+t)^{-\frac{m+1}{ \beta }-\frac{n}{2\beta }}\right) (1+t)^{-\frac{M-m}{ \beta }-\frac{n}{2\beta }}\nonumber \\&\qquad + (1+t)^{-\frac{M-1}{ \beta }-\frac{n}{2\beta }}(1+t)^{-\frac{1}{ \beta }-\frac{n}{2\beta }} +(1+t)^{-\frac{1}{ \beta }-\frac{n}{2\beta }} \left\| \nabla ^{M} \mathbf {v}\right\| ^{2 }_{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \lesssim (1+t)^{ -\frac{n}{2\beta }} \left\| \nabla ^{M} \mathbf {v}\right\| ^{2 }_{L^{2}(\mathbb {R}^{n})}+(1+t)^{-\frac{M}{ \beta }-\frac{n}{\beta }}. \end{aligned}$$

(B.11)

Note that $\left\| \nabla ^{M} \mathbf {v } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\le C$ by Proposition 2.4, (I), (II) of this theorem and the inductive assumption (B.1), applying interpolation inequality and a bootstrap argument, it follows from (B.7) and (B.11) that for $\displaystyle \frac{n}{4}\le \beta <1$ with $n=2,3$,

$$\begin{aligned}&\frac{d}{dt}\left\| \nabla ^{M} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+ \nu \left\| \Lambda ^{\beta }\nabla ^{M} \mathbf { v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C(1+t)^{-\frac{n}{2\beta } }\left\| \nabla ^{M} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+C(1+t)^{-\frac{M}{\beta } -\frac{n}{\beta } }\nonumber \\&\quad \le C(1+t)^{-\frac{n}{2\beta } }\left\| \nabla ^{M} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+C(1+t)^{-\frac{M}{\beta } -\frac{n}{2\beta }-1 }(1+t)^{ -\frac{n}{2\beta }+\frac{1}{\beta } }\nonumber \\&\quad \le C(1+t)^{-\frac{M}{\beta }-\frac{n}{2\beta }-1 }. \end{aligned}$$

(B.12)

Here, we applied the fact that for $n=2,3$,$-1+\frac{1}{\beta }>0$ and $-\frac{n}{2\beta }+\frac{1}{\beta } \le 0$. Since $\left| \mathcal {F}(\mathbf {v})\right| \le C$ by (III-1) of this theorem, applying (IV) of this theorem to estimate (B.12) gives rise to

$$\begin{aligned} \left\| \nabla ^{M} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\le C(1+t)^{-\frac{M}{\beta }-\frac{n}{2\beta }}. \end{aligned}$$

Combinig Step 1 with Step 2 and Step 3 finishes the proof of (VI-1).

Appendix C

We show (VI-2) of Theorem 4.1 here. We will adopt an inductive argument as above. The inductive assumption is as follows.

For $p\le \frac{K}{2\beta }$, the decay rate

$$\begin{aligned} \left\| \partial ^{p}_{t} \nabla ^{m} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\le C(1+t)^{-2p-\frac{m}{\beta }-\frac{n}{2\beta }} \end{aligned}$$

(C.1)

holds for all $p<P$ and m such that $2p\beta +m\le K$. Here, p, P and m are all non-negative integers.

In the following, based on the inductive assumption (C.1), we divided the proof into four steps. In Step 1, we show that for $\displaystyle |\xi |^{2\beta }\le \frac{b}{\nu (1+t)}$, $\displaystyle \left\| \partial ^{p}_{t} \hat{\mathbf {v}}(\xi )\right\| \le C(1+t)^{-P}$. In the second step, we verify that the decay rate (C.1) holds for $p=P$ and $m=0$ by an inductive argument on p. We will check the decay rate (C.1) holds for any $m>0$ by another inductive argument on m in the third step. In the fourth step, we conclude the expected result by a bootstrap argument.

We begin to show (VI-2) step by step in detail.

Step 1 We show for $\displaystyle |\xi |^{2\beta }\le \frac{b}{\nu (1+t)}$, $\displaystyle \left\| \partial ^{p}_{t} \hat{\mathbf {v}}(\xi )\right\| \le C(1+t)^{-P}$.

By (C.1) we get for all $p<P$ and $m=0,1$,

$$\begin{aligned} \left\| \partial ^{p}_{t} \nabla ^{m} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\le C(1+t)^{-2p-\frac{m}{\beta }-\frac{n}{2\beta }}. \end{aligned}$$

(C.2)

By the aid of (V) of theorem 4.1, (C.2) implies that for $\displaystyle |\xi |^{2\beta }\le \frac{b}{\nu (1+t)}$,

$$\begin{aligned} \left\| \partial ^{P}_{t} \hat{\mathbf {v}}(\xi ) \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\le C(1+t)^{-P}. \end{aligned}$$

(C.3)

Step 2 We now show that the decay rate (C.1) holds for $p=P$ and $m=0$ by an inductive argument on p.

Note that $\mathbf {v}\cdot \nabla \mathbf {u}^{T}=\nabla (\mathbf {u}\mathbf {v})-\mathbf {u}\cdot \nabla \mathbf {v}^{T}$ by (4.4) and $\text{ div } ~\mathbf {v}=0$, choosing P and M such that $M+2P\le K$, then applying $\partial ^{P}_{t}$ to the first equation in (1.1), multiplying the resulting equation by $\partial ^{P}_{t}\Delta ^{M}\mathbf {v}$ and integrating in space variable x yields, after some integration by parts,

$$\begin{aligned}&\frac{d}{dt}\left\| \partial ^{P}_{t} \nabla ^{M}\mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+2\nu \left\| \partial ^{P}_{t}\nabla ^{M}\Lambda ^{\beta } \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})} \nonumber \\&\le \left| \left\langle \partial ^{P}_{t}(\mathbf {u} \cdot \nabla \mathbf {v}), \partial ^{P}_{t}\Delta ^{M} \mathbf {v} \right\rangle \right| +\left| \left\langle \partial ^{P}_{t}\left( \mathbf {v} \cdot \nabla \mathbf {u}^{T}\right) , \partial ^{P}_{t}\Delta ^{M} \mathbf {v }\right\rangle \right| \nonumber \\&\quad =\left| \left\langle \partial ^{P}_{t}(\mathbf {u} \cdot \nabla \mathbf {v}), \partial ^{P}_{t}\Delta ^{M} \mathbf {v} \right\rangle \right| +\left| \left\langle \partial ^{P}_{t}\left( \mathbf {u} \cdot \nabla \mathbf {v}^{T}\right) , \partial ^{P}_{t}\Delta ^{M} \mathbf {v} \right\rangle \right| \nonumber \\&\quad \triangleq I_{M,P}+J_{M,P}. \end{aligned}$$

(C.4)

In the following, we deal with the two terms on the right hand side of (C.4) by considering two cases:

Case (1) $\displaystyle \frac{n}{4}< \beta < 1$ for $n=2,3$;

Case (2) $\displaystyle \beta =\frac{n}{4}$ for $n=2,3$.

$\heartsuit$ We first consider Case (1) $\displaystyle \frac{n}{4}<\beta <1$ for $n=2,3$.

In this case, a straightforward computation shows that

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \dfrac{n}{\beta }=\dfrac{2n}{n-2\cdot \frac{n-2\beta }{2}},~~\dfrac{1}{2}=\dfrac{n-2}{2}<\dfrac{n-2\beta }{2}<\dfrac{3}{4}~~ &{}\text{ for }~~n=3, \\ \displaystyle 0=\dfrac{n-2}{2}<\dfrac{n-2\beta }{2}< \dfrac{1}{2}~~ &{}\text{ for }~~n=2, \\ \displaystyle B=\dfrac{n-2\beta }{2}+1-\beta =\dfrac{n}{2}+1-2\beta ,&{} \\ \displaystyle \dfrac{n}{2}-1<B< 1,~~ &{}\text{ for }~~n=2,3. \end{array}\right. \end{aligned}$$

(C.5)

Note that (C.4), one attains

$$\begin{aligned} I_{M,P}= & {} \left| \left\langle \partial ^{P}_{t}(\mathbf {u} \cdot \nabla \mathbf {v}), \partial ^{P}_{t}\Delta ^{M} \mathbf {v} \right\rangle \right| \nonumber \\= & {} \sum \limits ^{P}_{p=0} \sum \limits ^{M-1}_{m=0} \left( \begin{array}{l} P\\ p \end{array}\right) \left( \begin{array}{c} M-1\\ m \end{array}\right) \left| \left\langle \Lambda ^{1-\beta }\left( \partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right) , \partial ^{P}_{t} \nabla ^{M }\Lambda ^{\beta } \mathbf {v} \right\rangle \right| \nonumber \\\le & {} \sum \limits ^{P}_{p=0} \sum \limits ^{M-1}_{m=0}\left( \begin{array}{l} P\\ p \end{array}\right) \left( \begin{array}{c} M-1\\ m \end{array}\right) \left\| \Lambda ^{1-\beta }\left( \partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right) \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\cdot \left\| \partial ^{P}_{t} \nabla ^{M }\Lambda ^{\beta } \mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(C.6)

Thanks to higher order fractional Leibniz’s rule [18], $\displaystyle \left\| \Lambda ^{1-\beta }\left( \partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right) \right\| _{L^{2}(\mathbb {R}^{n})}$ can be bounded as follows:

$$\begin{aligned}&\displaystyle \left\| \Lambda ^{1-\beta }\left( \partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right) \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le \left\| \Lambda ^{1-\beta }\left( \partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right) \right. \nonumber \\&\qquad -\partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\Lambda ^{1-\beta }\mathbf {v}\nonumber \\&\qquad \left. -\partial ^{p}_{t}\Lambda ^{1-\beta }\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad +\left\| \partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\Lambda ^{1-\beta }\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad +\left\| \partial ^{p}_{t}\Lambda ^{1-\beta }\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(C.7)

Due to Lemma 2.10, (I) of Lemma 2.11 and (C.5), for $\displaystyle \frac{1}{2}=\frac{1}{n/\beta }+\frac{1}{2n/(n-2\beta )}$ and $0<1-\beta<\beta <1$, the first term on the right hand side of (C.7) can be bounded by

$$\begin{aligned}&\left\| \Lambda ^{1-\beta }\left( \partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right) \right. \nonumber \\&\qquad -\partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\Lambda ^{1-\beta }\mathbf {v}\nonumber \\&\qquad \left. -\partial ^{p}_{t}\Lambda ^{1-\beta }\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C \left\| \partial ^{p}_{t}\Lambda ^{1-\beta } \nabla ^{m}\mathbf {u} \right\| _{L^{\frac{n}{\beta }}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M-m} \mathbf {v} \right\| _{L^{\frac{2n}{n-2\beta }}(\mathbb {R}^{n})}\nonumber \\&\quad \le C \left\| \partial ^{p}_{t}\Lambda ^{\frac{n}{2}+1-2\beta } \nabla ^{m}\mathbf {u} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M-m} \Lambda ^{ \beta }\mathbf { v }\right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C \left\| \partial ^{p}_{t} \nabla ^{m}\mathbf {u} \right\| ^{2\beta -\frac{n}{2}}_{L^{2}(\mathbb {R}^{n})}\left\| \partial ^{p}_{t} \nabla ^{m+1}\mathbf {u} \right\| ^{\frac{n}{2}+1-2\beta }_{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M-m} \Lambda ^{ \beta }\mathbf { v }\right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C \left\| \partial ^{p}_{t} \nabla ^{m-1}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M-m} \Lambda ^{ \beta } \mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(C.8)

On the other hand, note that Lemmas 2.8, 2.9 and 2.11, the second and the third terms on the right hand side of (C.7) can be bounded by

$$\begin{aligned}&\left\| \partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\Lambda ^{1-\beta }\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C\left\| \partial ^{p}_{t} \nabla ^{m}\mathbf {u} \right\| _{L^{\frac{n}{ 2\beta -1 }}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M-m} \Lambda ^{1-\beta }\mathbf {v} \right\| _{L^{\frac{2n}{n-2(2\beta -1)}}(\mathbb {R}^{n})}\nonumber \\&\quad \le C \left\| \partial ^{p}_{t} \nabla ^{m}\mathbf {u} \right\| _{L^{\frac{2n}{n-2\left( \frac{n}{2}+1-2\beta \right) }}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M-m} \Lambda ^{\beta }\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C \left\| \partial ^{p}_{t} \nabla ^{m}\Lambda ^{\frac{n}{2}+1-2\beta }\mathbf {u } \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M-m} \Lambda ^{\beta }\mathbf {v } \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C \left\| \partial ^{p}_{t} \nabla ^{m-1} \mathbf {v }\right\| _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M-m} \Lambda ^{\beta }\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}, \end{aligned}$$

(C.9)

and

$$\begin{aligned}&\left\| \partial ^{p}_{t}\Lambda ^{1-\beta }\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C \left\| \partial ^{p}_{t} \Lambda ^{1-\beta } \nabla ^{m}\mathbf {u} \right\| _{L^{\frac{n}{\beta }}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M-m} \mathbf { v} \right\| _{L^{\frac{2n}{n-2\beta }}(\mathbb {R}^{n})}\nonumber \\&\quad \le C \left\| \partial ^{p}_{t} \Lambda ^{\frac{n}{2}+1-2\beta } \nabla ^{m}\mathbf {u} \right\| _{L^{\frac{n}{\beta }}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M-m} \Lambda ^{ \beta } \mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C \left\| \partial ^{p}_{t} \nabla ^{m-1}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M-m} \Lambda ^{ \beta } \mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(C.10)

Hence, combining (C.6) with (C.7), (C.8), (C.9) and (C.10) gives rise to

$$\begin{aligned} I_{M,P}= & {} \left| \left\langle \partial ^{P}_{t}\left( \mathbf {u} \cdot \nabla \mathbf {v}\right) , \partial ^{P}_{t}\Delta ^{M} \mathbf {v} \right\rangle \right| \nonumber \\\le & {} C\sum \limits ^{P}_{p=0} \sum \limits ^{M-1}_{m=0} \left\| \partial ^{p}_{t} \nabla ^{m+1}\mathbf {u} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M-m} \Lambda ^{\beta } \mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\cdot \left\| \partial ^{P}_{t} \nabla ^{M}\Lambda ^{\beta }\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\\le & {} C\sum \limits ^{P}_{p=0} \left\| \partial ^{p}_{t}\mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M } \Lambda ^{\beta }\mathbf { v } \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&+C\sum \limits ^{P}_{p=0} \left\| \partial ^{p}_{t}\nabla \mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M -1} \Lambda ^{\beta } \mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&+C\sum \limits ^{P}_{p=0}\sum \limits ^{M-1}_{m=2} \left\| \partial ^{p}_{t}\nabla ^{m-1} \mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M -m} \Lambda ^{\beta } \mathbf {v } \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&+\frac{\nu }{4} \left\| \partial ^{P }_{t} \nabla ^{M } \Lambda ^{\beta } \mathbf {v } \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(C.11)

Similar estimates to that used in the estimates for $I_{M,P}$ are valid for $J_{M,P}$ in (C.4):

$$\begin{aligned} J_{M,P}= & {} \left| \left\langle \partial ^{P}_{t}\left( \mathbf {u} \cdot \nabla \mathbf {v}^{T}\right) , \partial ^{P}_{t}\Delta ^{M} \mathbf {v} \right\rangle \right| \nonumber \\\le & {} C\sum \limits ^{P}_{p=0} \left\| \partial ^{p}_{t}\mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\left\| \partial ^{P-p}_{t} \nabla ^{M } \Lambda ^{\beta } \mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&+ C\sum \limits ^{P}_{p=0} \left\| \partial ^{p}_{t}\nabla \mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\left\| \partial ^{P-p}_{t} \nabla ^{M-1 } \Lambda ^{\beta } \mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&+ C\sum \limits ^{P}_{p=0}\sum \limits ^{M-1}_{m=2} \left\| \partial ^{p}_{t}\nabla ^{m-1} \mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\left\| \partial ^{P-p}_{t} \nabla ^{M -m} \Lambda ^{\beta } \mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&+ \frac{\nu }{4} \left\| \partial ^{P }_{t} \nabla ^{M } \Lambda ^{\beta } \mathbf {v }\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(C.12)

Substituting the above estimates (C.11) and (C.12) into (C.4) leads to the following estimate under the case $\displaystyle \frac{n}{4}< \beta < 1$ with $n=2,3$:

$$\begin{aligned}&\frac{d}{dt} \left\| \partial ^{P}_{t}\nabla ^{M} \mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}+ \nu \left\| \partial ^{P }_{t} \nabla ^{M } \Lambda ^{\beta } \mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C\sum \limits ^{P}_{p=0} \left\| \partial ^{p}_{t}\mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\left\| \partial ^{P-p}_{t} \nabla ^{M } \Lambda ^{\beta } \mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad + C\sum \limits ^{P}_{p=0} \left\| \partial ^{p}_{t}\nabla \mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M-1 } \Lambda ^{\beta } \mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad +C\sum \limits ^{P}_{p=0}\sum \limits ^{M-1}_{m=2} \left\| \partial ^{p}_{t}\nabla ^{m-1}\mathbf { v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M -m} \Lambda ^{\beta } \mathbf {v} \right\| ^{2} _{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(C.13)

$\heartsuit$ We now tackle (C.4) under the Case (2) $\displaystyle \beta =\frac{n}{4}$ for $n=2,3$.

In this case, it is easy to check that $\displaystyle 1-\frac{n}{4}\le \frac{n}{4}$. Recall (C.4), we shall estimate $I_{M,P}$ and $J_{M,P}$, respectively. We first handle $I_{M,P}$.

By a similar proof to that for case (1), one deduces the following:

$$\begin{aligned} I_{M,P}= & {} \left| \left\langle \partial ^{P}_{t}(\mathbf {u} \cdot \nabla \mathbf {v}), \partial ^{P}_{t}\Delta ^{M} \mathbf {v} \right\rangle \right| \nonumber \\= & {} \sum \limits ^{P}_{p=0} \sum \limits ^{M-1}_{m=0}\left( \begin{array}{l} P\\ p \end{array}\right) \left( \begin{array}{c} M-1\\ m \end{array}\right) \left\langle \Lambda ^{1-\frac{n}{4}}\left( \partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right) , \partial ^{P}_{t} \nabla ^{M }\Lambda ^{\frac{n}{4}}\mathbf { v} \right\rangle \nonumber \\\le & {} \sum \limits ^{P}_{p=0} \sum \limits ^{M-1}_{m=0}\left( \begin{array}{l} P\\ p \end{array}\right) \left( \begin{array}{c} M-1\\ m \end{array}\right) \left\| \Lambda ^{1-\frac{n}{4}}\left( \partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right) \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\cdot \left\| \partial ^{P}_{t} \nabla ^{M }\Lambda ^{\frac{n}{4}} \mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(C.14)

However, direct calculation gives

$$\begin{aligned}&\left\| \Lambda ^{1-\frac{n}{4}}\left( \partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right) \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le \left\| \Lambda ^{1-\frac{n}{4}}\left( \partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right) \right. \nonumber \\&\qquad -\partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\Lambda ^{1-\frac{n}{4}}\mathbf {v}\nonumber \\&\qquad -\left. \partial ^{p}_{t}\Lambda ^{1-\frac{n}{4}}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad +\left\| \partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\Lambda ^{1-\frac{n}{4}}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad +\left\| \partial ^{p}_{t}\Lambda ^{1-\frac{n}{4}}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(C.15)

Due to Lemma 2.9 and Lemma 2.11, one deduces that for $\displaystyle 0<\beta _{1}<1-\frac{n}{4}$, the first term on the right hand side of (C.15) can be estimated by

$$\begin{aligned}&\left\| \Lambda ^{1-\frac{n}{4}}\left( \partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right) \right. \nonumber \\&\qquad -\partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\Lambda ^{1-\frac{n}{4}}\mathbf {v}\nonumber \\&\qquad \left. -\partial ^{p}_{t}\Lambda ^{1-\frac{n}{4}}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C \left\| \partial ^{p}_{t}\Lambda ^{1- \frac{n}{4}-\beta _{1}} \nabla ^{m}\mathbf {u} \right\| _{L^{\frac{2n}{n-2(n/4+\beta _{1})}}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M-m} \Lambda ^{ \beta _{1}} \mathbf {v} \right\| _{L^{\frac{2n}{n-2(n/4-\beta _{1}) }}(\mathbb {R}^{n})}\nonumber \\&\quad \le C \left\| \partial ^{p}_{t} \nabla ^{m}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M-m} \Lambda ^{\frac{n}{4}} \mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}, \end{aligned}$$

(C.16)

where $\displaystyle \beta _{1}\in \left( 0,\frac{1}{2}\right)$, $\displaystyle 1-\frac{n}{4}-\beta _{1}\in \left( 0,\frac{1}{2}\right)$, $\displaystyle \frac{1}{2} =\frac{1}{p_{1}}+\frac{1}{p_{2}}$ with $p_{1},p_{2}\in (1,\infty )$, $\displaystyle p_{1}=\frac{2n}{n-2(n/4+\beta _{1})}$, $\displaystyle p_{2}=\frac{2n}{n-2(n/4-\beta _{1})}$.

Let us turn to estimate the second and the third terms on the right hand side of (C.15). Thanks to Agmon’s inequality, interpolation inequality, Lemma 2.12 and the assumption of (VI) for $\beta =\dfrac{n}{4}$, we have

$$\begin{aligned}&\left\| \partial ^{p}_{t}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\Lambda ^{1-\frac{n}{4}}\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad + \left\| \partial ^{p}_{t}\Lambda ^{1-\frac{n}{4}}\nabla ^{m}\mathbf {u}\cdot \partial ^{P-p}_{t}\nabla ^{M-m}\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C \left\| \partial ^{p}_{t} \nabla ^{m}\mathbf {u} \right\| _{L^{\infty }(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M-m} \Lambda ^{ 1-\frac{n}{4}} \mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad +\left\| \partial ^{p}_{t} \Lambda ^{ 1-\frac{n}{4}} \nabla ^{m}\mathbf {u} \right\| _{L^{4}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M-m} \mathbf {v} \right\| _{L^{4}(\mathbb {R}^{n})}\nonumber \\&\quad \le C \left\| \partial ^{p}_{t} \nabla ^{m}\mathbf {u} \right\| ^{\frac{1}{2}} _{H^{1}(\mathbb {R}^{n})}\left\| \partial ^{p}_{t} \nabla ^{m}\mathbf {u} \right\| ^{\frac{1}{2}} _{H^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t} \nabla ^{M-m} \Lambda ^{\frac{n}{4}} \mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad +C \left\| \partial ^{p}_{t}\Lambda ^{ 1-\frac{n}{4}+\frac{n}{4}} \nabla ^{m}\mathbf {u} \right\| _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t}\Lambda ^{ \frac{n}{4}} \nabla ^{M-m}\mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C\left\| \partial ^{p}_{t} \nabla ^{m}\mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}\left\| \partial ^{P-p}_{t} \nabla ^{M-m}\Lambda ^{ \frac{n}{4}} \mathbf {v} \right\| _{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(C.17)

This together with (C.14), (C.15) and (C.16) gives rise to

$$\begin{aligned}&I_{M,P}= \left| \left\langle \partial ^{P}_{t}(\mathbf {u} \cdot \nabla \mathbf {v}), \partial ^{P}_{t}\Delta ^{M} \mathbf {v} \right\rangle \right| \nonumber \\&\quad \le C\sum \limits ^{P}_{p=0} \sum \limits ^{M-1}_{m=0}\left\| \partial ^{p}_{t}\nabla ^{m} \mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t}\Lambda ^{\frac{n}{4}}\nabla ^{M-m} \mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p }_{t}\Lambda ^{\frac{n}{4}}\nabla ^{M } \mathbf {v}\right\| _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C \sum \limits ^{P}_{p=0} \left\| \partial ^{p}_{t} \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t}\Lambda ^{\frac{n}{4}}\nabla ^{M } \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad + C \sum \limits ^{P}_{p=0} \left\| \partial ^{p}_{t} \nabla \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t}\Lambda ^{\frac{n}{4}}\nabla ^{M-1 } \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad + C \sum \limits ^{P}_{p=0} \sum \limits ^{M-1}_{m=2} \left\| \partial ^{p}_{t} \nabla ^{m} \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t}\Lambda ^{\frac{n}{4}}\nabla ^{M-m } \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad + \frac{\nu }{4}\left\| \partial ^{P }_{t}\Lambda ^{\frac{n}{4}}\nabla ^{M } \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(C.18)

Here we have used Cauchy–Schwarz inequality and Young’s inequality in the third inequality.

In the same manner, one may deduce the following estimate for $J_{M,P}$ in (C.4):

$$\begin{aligned} J_{M,P}= & {} \left| \left\langle \partial ^{P}_{t}\left( \mathbf {u} \cdot \nabla \mathbf {v}^{T}\right) , \partial ^{P}_{t}\Delta ^{M} \mathbf {v} \right\rangle \right| \nonumber \\\le & {} C \sum \limits ^{P}_{p=0} \left\| \partial ^{p}_{t} \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t}\Lambda ^{\frac{n}{4}}\nabla ^{M } \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&+ C \sum \limits ^{P}_{p=0} \left\| \partial ^{p}_{t} \nabla \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t}\Lambda ^{\frac{n}{4}}\nabla ^{M-1 }\mathbf { v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&+ C \sum \limits ^{P}_{p=0} \sum \limits ^{M-1}_{m=2} \left\| \partial ^{p}_{t} \nabla ^{m} \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t}\Lambda ^{\frac{n}{4}}\nabla ^{M-m } \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&+ \frac{\nu }{4}\left\| \partial ^{P }_{t}\Lambda ^{\frac{n}{4}}\nabla ^{M } \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(C.19)

Therefore, under the case of $\displaystyle \beta =\frac{n}{4}$ with $n=2,3$, substituting (C.18) and (C.19) into (C.4) yields that

$$\begin{aligned}&\frac{d}{dt} \left\| \partial ^{P}_{t}\nabla ^{M } \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+\nu \left\| \partial ^{P}_{t}\Lambda ^{\frac{n}{4}} \nabla ^{M } \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C \sum \limits ^{P}_{p=0} \left\| \partial ^{p}_{t} \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t}\Lambda ^{\frac{n}{4}}\nabla ^{M } \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad + C \sum \limits ^{P}_{p=0} \left\| \partial ^{p}_{t} \nabla \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t}\Lambda ^{\frac{n}{4}}\nabla ^{M-1 } \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad + C \sum \limits ^{P}_{p=0} \sum \limits ^{M-1}_{m=2} \left\| \partial ^{p}_{t} \nabla ^{m}\mathbf { v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t}\Lambda ^{\frac{n}{4}}\nabla ^{M-m } \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(C.20)

With (C.4), (C.13) and (C.20), for $\displaystyle \frac{n}{4}\le \beta < 1$ with $n=2,3$, we always have

$$\begin{aligned}&\frac{d}{dt}\left\| \partial ^{P}_{t}\nabla ^{M } \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+\nu \left\| \partial ^{P}_{t}\Lambda ^{\beta } \nabla ^{M } \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C \sum \limits ^{P}_{p=0} \left\| \partial ^{p}_{t} \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t}\Lambda ^{\beta }\nabla ^{M } \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad + C \sum \limits ^{P}_{p=0} \left\| \partial ^{p}_{t} \nabla \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t}\Lambda ^{\beta }\nabla ^{M-1 } \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad + C \sum \limits ^{P}_{p=0} \sum \limits ^{M}_{m=2} \left\| \partial ^{p}_{t} \nabla ^{m} \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})} \left\| \partial ^{P-p}_{t}\Lambda ^{\beta }\nabla ^{M-m } \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}. \end{aligned}$$

(C.21)

Note that the inductive assumption (C.1), one deduces that for $p=P$ and $m=0$

$$\begin{aligned}&\frac{d}{dt}\left\| \partial ^{P}_{t} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+\nu \left\| \partial ^{P}_{t}\Lambda ^{\beta } \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})} \\&\quad \le C (1+t)^{-\frac{n}{2\beta }}\left\| \partial ^{p}_{t}\Lambda ^{\beta } \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}+ C (1+t)^{-2p-\frac{n}{2\beta }} (1+t)^{-2(P-p) -\frac{\beta }{\beta }-\frac{n}{2\beta }} \\&\quad \le C (1+t)^{-\frac{n}{2\beta }}\left\| \partial ^{p}_{t} \Lambda ^{\beta } \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}+ C (1+t)^{-2P-1-\frac{n}{\beta }}. \end{aligned}$$

Takeing t large enough such that $\displaystyle C(1+t)^{- \frac{n}{ 2 \beta } }\le \frac{\nu }{2}$, the above inequality then implies that

$$\begin{aligned} \frac{d}{dt}\left\| \partial ^{P}_{t} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+\frac{\nu }{2}\left\| \partial ^{P}_{t}\Lambda ^{\beta } \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\le C(1+t)^{-2P-1 -\frac{ n}{ \beta }}. \end{aligned}$$

(C.22)

This together with (IV) of this theorem ensures that (C.1) and (C.3) hold for $p=P$ and $m=0$.

So far we have shown that for $m=0$, and $\forall P\le \frac{K}{2}$, there holds

$$\begin{aligned} \frac{d}{dt}\left\| \partial ^{P}_{t} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+\frac{\nu }{2}\left\| \partial ^{P}_{t}\Lambda ^{\beta } \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\le C(1+t)^{-2P-1 -\frac{ n}{ \beta }}. \end{aligned}$$

This deduces by Gronwall’s inequality that

$$\begin{aligned} \frac{d}{dt}\left\| \partial ^{P}_{t} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})} \le C(1+t)^{-2P-\frac{ n}{ \beta }}. \end{aligned}$$

(C.23)

Step 3 We show that the decay rate (C.1) holds for any $m\le M+1$ for $p<P$, and $m<M$ for $p=P$. That is,

$$\begin{aligned} \left\| \partial ^{p}_{t}\nabla ^{m} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\le C(1+t)^{-2p-\frac{m}{\beta }-\frac{n}{2\beta }}. \end{aligned}$$

(C.24)

The base case is (C.23) where (C.24) holds for $p=P$ and $m=0$. In the following, based on the inductive assumption (C.24), we will show that the decay rate (C.24) holds for $m=M$ and $p=P$.

Recall (I) and (II) of this theorem, applying the inductive assumption (C.21)–(C.24), one deduces that

$$\begin{aligned}&\frac{d}{dt}\left\| \partial ^{P}_{t}\nabla ^{M} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}+\nu \left\| \partial ^{P}_{t}\Lambda ^{\beta } \nabla ^{M}\mathbf { v } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})}\nonumber \\&\quad \le C (1+t)^{-\frac{n}{2\beta }}\left\| \partial ^{P}_{t} \Lambda ^{\beta }\nabla ^{M} \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad + C (1+t)^{ -\frac{1}{ \beta }-\frac{n}{2\beta }}\left\| \partial ^{P}_{t} \Lambda ^{\beta }\nabla ^{M-1} \mathbf {v}\right\| ^{2} _{L^{2}(\mathbb {R}^{n})}\nonumber \\&\qquad + C (1+t)^{-2P-\frac{M}{ \beta }-\frac{n}{ \beta }-1}. \end{aligned}$$

(C.25)

Taking t large enough such that $\displaystyle C(1+t)^{ -\frac{n}{ 2\beta }}\le \frac{\nu }{2}$, thanks to (C.3), using (IV) once again deduces that

$$\begin{aligned} \left\| \partial ^{P}_{t}\nabla ^{M} \mathbf {v} \right\| ^{2}_{L^{2}(\mathbb {R}^{n})} \le C(1+t)^{-2P -\frac{ M}{ \beta }-\frac{ n}{\beta }}\le C(1+t)^{-2P -\frac{ M}{ \beta }-\frac{ n}{2\beta }}. \end{aligned}$$

(C.26)

This implies that the inductive assumption (C.24) holds for $m=M$ and $p=P$. By another bootstrap argument, we obtain for all $m+2p\beta \le K$, the following optimal decay holds:

$$\begin{aligned} \left\| \partial ^{p}_{t}\nabla ^{m} \mathbf {v } \right\| ^{2}_{L^{2}(\mathbb {R}^{n})} \le C(1+t)^{-2p -\frac{m}{ \beta }-\frac{ n}{2\beta }}. \end{aligned}$$

This completes the proof of (VI-2).