Global Smooth Solutions to the n-Dimensional Damped Models of Incompressible Fluid Mechanics with Small Initial Datum

Abstract

In this paper, we consider the $n$-dimensional ($n\ge 2$) damped models of incompressible fluid mechanics in Besov spaces and establish the global (in time) regularity of classical solutions provided that the initial data are suitable small.

Appendix: Local Existence and Uniqueness Theory to Damped MHD

For sake of completeness, we provide the local existence and uniqueness to the system (1.4) with initial data $(u_{0},\,b_{0})\in H^{s}(\mathbb {R}^{n})\times H^{s}(\mathbb {R}^{n})$ with $s>1+\frac{n}{2}$. The local existence and uniqueness results to the system (1.1), (1.2), and (1.3) can be obtained by the method similar to Chapter 3 in Majda and Bertozzi (2002). More precisely, we state the following local result.

Proposition 6.1

Let initial datum $(u_{0},\,b_{0})\in H^{s}(\mathbb {R}^{n})\times H^{s}(\mathbb {R}^{n})$ with $s>1+\frac{n}{2}$. Assume that $\nabla \cdot u_{0}=0,\,\,\nabla \cdot b_{0}=0$. There exists a positive time $T$ depending on $\Vert u_{0}\Vert _{H^{s}}$ and $\Vert b_{0}\Vert _{H^{s}}$ such that the system (1.4) admits a unique solution $(u, b)$ in $C([0, T]; H^{s}(\mathbb {R}^{n})\times H^{s}(\mathbb {R}^{n}))$.

To prove Proposition 6.1, the main step is to modify the system (1.4) in order to easily produce a family of global smooth solutions. In order to do this, we may, for instance, make use of the Friedrichs method. Now we define the spectral cutoff as follows:

$$\begin{aligned} \widehat{\mathcal {J}_{N}f}(\xi )=\chi _{B(0,N)}(\xi )\widehat{f}(\xi ), \end{aligned}$$

where $N>0, \,B(0,N)=\{\xi \in \mathbb {R}^{n}| \, |\xi |\le N\}$, $\chi _{B(0,N)}$ is the characteristic function on $B(0,N)$. Also we define

$$\begin{aligned} L^{2}_{N}\triangleq \left\{ f\in L^{2}(\mathbb {R}^{n})|\, \text{ supp } \,\widehat{f}\subset B(0,N)\right\} . \end{aligned}$$

It is easy for us to show the following properties (here the proof will be omitted) which will be used frequently later.

Lemma 6.2

Let $N>0$ and for any $f\in H^{s}(\mathbb {R}^{n})$, the followings hold true

$$\begin{aligned}&\Vert \mathcal {J}_{N}f\Vert _{L^{2}(\mathbb {R}^{n})}\le \Vert f\Vert _{L^{2}(\mathbb {R}^{n})},\quad \Vert \nabla ^{k}\mathcal {J}_{N}f\Vert _{H^{s}(\mathbb {R}^{n})}\le C N^{k}\Vert f\Vert _{H^{s}(\mathbb {R}^{n})},\end{aligned}$$

(6.1)

$$\begin{aligned}&\Vert \mathcal {J}_{N}f-f\Vert _{H^{s}(\mathbb {R}^{n})}\rightarrow 0, \,as\,\, N\rightarrow \infty ,\end{aligned}$$

(6.2)

$$\begin{aligned}&\Vert \nabla ^{k}\mathcal {J}_{N}f\Vert _{L^{\infty }(\mathbb {R}^{n})}\le C N^{k+\frac{n}{2}}\Vert f\Vert _{L^{2}(\mathbb {R}^{n})},\end{aligned}$$

(6.3)

$$\begin{aligned}&\int _{\mathbb {R}^{n}}\mathcal {J}_{N}f g \mathrm{d}x=\int _{\mathbb {R}^{n}}f \mathcal {J}_{N} g \mathrm{d}x,\quad \int \mathcal {P}f\cdot g\,\mathrm{d}x=\int f\cdot \mathcal {P}g\,\mathrm{d}x,\end{aligned}$$

(6.4)

$$\begin{aligned}&\Vert \mathcal {J}_{N}f\Vert _{H^{s}(\mathbb {R}^{n})}\le C N^{s}\Vert f\Vert _{L^{2}(\mathbb {R}^{n})},\quad \Vert \mathcal {P} f\Vert _{H^{s}(\mathbb {R}^{n})}\le \Vert f\Vert _{H^{s}(\mathbb {R}^{n})} \end{aligned}$$

(6.5)

where $\mathcal {P}$ denotes the Leray projection onto divergence-free vector fields.

Proof of Proposition 6.1

The first step is to establish a smooth solution $(u^{N},\,b^{N})$ in space $L^{2}_{N}$ satisfying

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}u^{N}+\mathcal {P}\mathcal {J}_{N}((\mathcal {P}\mathcal {J}_{N}u^{N}\cdot \nabla ) \mathcal {P}\mathcal {J}_{N}u^{N})+\kappa \mathcal {P}\mathcal {J}_{N}u^{N}= \mathcal {P}\mathcal {J}_{N}(( \mathcal {J}_{N}b^{N}\cdot \nabla ) \mathcal {J}_{N}b^{N}),\\ \partial _{t}b^{N}+\mathcal {J}_{N}((\mathcal {P}\mathcal {J}_{N}u^{N}\cdot \nabla )\mathcal {J}_{N}b^{N})+\lambda \mathcal {J}_{N} b^{N}=\mathcal {J}_{N}((\mathcal {J}_{N}b^{N}\cdot \nabla )\mathcal {P}\mathcal {J}_{N}u^{N}),\\ \nabla \cdot u^{N}=0,\,\nabla \cdot b^{N}=0,\\ u^{N}(x,0)=\mathcal {J}_{N}u_{0}(x),\,b^{N}(x,0)=\mathcal {J}_{N}b_{0}(x). \end{array} \right. \end{aligned}$$

(6.6)

Claim 1: For any fixed $N>0$, approximate system (6.6) has a unique global (in time) smooth solution $(u^{N},\,b^{N})$ satisfying

$$\begin{aligned} (u^{N},\,b^{N})\in C([0,\,\infty );\,H^{\bar{s}}(\mathbb {R}^{n})),\quad \text{ for } \text{ any }\,\,\bar{s}\ge 0. \end{aligned}$$

Now we give the outline to prove the above claim. First, applying the $L^{2}$ estimate to (6.6) tells us that for any $0\le t\le T$ with any $T\ge 0$

$$\begin{aligned}&\Vert u^{N}(., t)\Vert _{L^{2}}^{2}+\Vert b^{N}(., t)\Vert _{L^{2}}^{2}+2\int _{0}^{T}{(\kappa \Vert \mathcal {J}_{N}u^{N}(., s)\Vert _{L^{2}}^{2}+\lambda \Vert \mathcal {J}_{N}b^{N}(., s)\Vert _{L^{2}}^{2})\,\mathrm{d}t}\nonumber \\&\qquad \le \Vert u_{0}\Vert _{L^{2}}^{2}+\Vert b_{0}\Vert _{L^{2}}^{2}. \end{aligned}$$

(6.7)

We write

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left( \begin{array}{ll} u^{N}\\ b^{N} \end{array} \right) \\&\quad =\left[ \begin{array}{l} -\mathcal {P}\mathcal {J}_{N}((\mathcal {P}\mathcal {J}_{N}u^{N}\cdot \nabla ) \mathcal {P}\mathcal {J}_{N}u^{N})-\kappa \mathcal {P}\mathcal {J}_{N}u^{N}+ \mathcal {P}\mathcal {J}_{N}(( \mathcal {J}_{N}b^{N}\cdot \nabla ) \mathcal {J}_{N}b^{N})\\ -\mathcal {J}_{N}((\mathcal {P}\mathcal {J}_{N}u^{N}\cdot \nabla )\mathcal {J}_{N}b^{N})-\lambda \mathcal {J}_{N} b^{N}+\mathcal {J}_{N}((\mathcal {J}_{N}b^{N}\cdot \nabla )\mathcal {P}\mathcal {J}_{N}u^{N}) \end{array} \right] . \end{aligned}$$

For convenience of notation, we denote the right-hand side of the above differential equations as $F(u^{N},b^{N})$.

It is not difficult to show that $F$ satisfies the local Lipschitz condition for any fixed $N$. That is, the difference $\Vert F(u^{N},b^{N})-F(\widetilde{u}^{N},\widetilde{b}^{N})\Vert _{H^{\bar{s}}}$ satisfies

$$\begin{aligned} \Vert F(u^{N},b^{N})-F(\widetilde{u}^{N},\widetilde{b}^{N})\Vert _{H^{\bar{s}}}\le \widetilde{C}\Vert (u^{N}, b^{N} )-(\widetilde{u}^{N}, \widetilde{b}^{N})\Vert _{H^{\bar{s}}}, \end{aligned}$$

where $\widetilde{C}=C\max (N^{\bar{s}+1+\frac{n}{2}},\kappa , \lambda , \Vert u_{0}\Vert _{L^{2}}+ \Vert b_{0}\Vert _{L^{2}}).$

Taking advantage of the Cauchy-Lipschitz theorem (Picard’s Theorem, see Majda and Bertozzi 2002), we can find that for any fixed $N$, there exists the unique solution $(u^{N},b^{N})$ in in $C([0, T_{N}); H^{\bar{s}}(\mathbb {R}^{n})\times H^{\bar{s}}(\mathbb {R}^{n}))$ with $T_{N}=T(N, u_{0}, b_{0})$. In fact, it is not hard to extend the local solution to the global solution based on the above estimates. In fact, we just set $\widetilde{u}^{N}=\widetilde{b}^{N}=0$, then we can obtain immediately that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}(\Vert u^{N}\Vert _{H^{\bar{s}}}\!+\!\Vert b^{N}\Vert _{H^{\bar{s}}})\!\le C\max (N^{\bar{s}\!+\!1\!+\!\frac{n}{2}},\!\kappa ,\! \lambda , \Vert u_{0}\Vert _{L^{2}}\!+\! \Vert b_{0}\Vert _{L^{2}})(\Vert u^{N}\Vert _{H^{\bar{s}}}\!+\!\Vert b^{N}\Vert _{H^{\bar{s}}}). \end{aligned}$$

Gronwall inequality yields for any $T\ge 0$

$$\begin{aligned} \Vert u^{N}(.,T)\Vert _{H^{\bar{s}}}+\Vert b^{N}(.,T)\Vert _{H^{\bar{s}}}\le e^{C\max (N^{\bar{s}+1+\frac{n}{2}},\kappa , \lambda , \Vert u_{0}\Vert _{L^{2}}+ \Vert b_{0}\Vert _{L^{2}})T}. \end{aligned}$$

Thus, we have proved Claim 1.

Due to $\mathcal {J}_{N}^{2}=\mathcal {J}_{N},\,\mathcal {P}^{2}=\mathcal {P}$, and $\mathcal {P}\mathcal {J}_{N}=\mathcal {J}_{N}\mathcal {P}$, we can discover that $(\mathcal {P}u^{N},\,b^{N})$ and $(\mathcal {J}_{N}u^{N},\,\mathcal {J}_{N}b^{N})$ are also solutions to approximate system (6.6) with the same initial datum. Thanks to the uniqueness, we thus find

$$\begin{aligned} \mathcal {P}u^{N}=u^{N},\,\,\mathcal {J}_{N}u^{N}=u^{N} \,\,and\,\, \mathcal {J}_{N}b^{N}=b^{N}. \end{aligned}$$

Consequently, approximate system (6.6) reduces to

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}u^{N}+\mathcal {P}\mathcal {J}_{N}((u^{N}\cdot \nabla ) u^{N})+\kappa u^{N}=\mathcal {P}\mathcal {J}_{N}(b^{N}\cdot \nabla )b^{N},\\ \partial _{t}b^{N}+\mathcal {J}_{N}((u^{N}\cdot \nabla )b^{N})+\lambda b^{N}=\mathcal {J}_{N}((b^{N}\cdot \nabla )u^{N}),\\ \nabla \cdot u^{N}=0,\,\nabla \cdot b^{N}=0,\\ u^{N}(x,0)=\mathcal {J}_{N}u_{0}(x),\,b^{N}(x,0)=\mathcal {J}_{N}b_{0}(x). \end{array} \right. \end{aligned}$$

(6.8)

The same argument to that used in obtaining (5.8) together with the fact (6.4), we can also get the nonhomogeneous $H^{s}$ bound as follows:

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}(\Vert u^{N}(t)\Vert _{H^{s}}^{2} +\Vert b^{N}(t)\Vert _{H^{s}}^{2})+\kappa \Vert u^{N}\Vert _{H^{s}}^{2}+\lambda \Vert b^{N} \Vert _{H^{s}}^{2}\nonumber \\&\quad \le \,C(\Vert \nabla u^{N}\Vert _{L^{\infty }}+\Vert \nabla b^{N}\Vert _{L^{\infty }}) (\Vert u^{N}(t)\Vert _{H^{s}}^{2}+\Vert b^{N}\Vert _{H^{s}}^{2})\nonumber \\&\quad \le \,C(\Vert u^{N}\Vert _{H^{s}}+\Vert b^{N}\Vert _{H^{s}}) (\Vert u^{N}(t)\Vert _{H^{s}}^{2}+\Vert b^{N}\Vert _{H^{s}}^{2}). \end{aligned}$$

(6.9)

For the convenience of notation, we also denote

$$\begin{aligned} X(t)=\sqrt{\Vert u^{N}(t)\Vert _{H^{s}}^{2}+\Vert b^{N}\Vert _{H^{s}}^{2}}. \end{aligned}$$

Consequently, (6.9) becomes

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}X(t)+\min \{\kappa ,\,\lambda \}X(t)\le CX(t)^{2}. \end{aligned}$$

Standard calculations show that for all $N$

$$\begin{aligned} \sup _{0\le t\le T}(\sqrt{\Vert u^{N}(t)\Vert _{H^{s}}^{2}+\Vert b^{N}\Vert _{H^{s}}^{2}})\le \frac{\sqrt{\Vert u_{0}\Vert _{H^{s}}^{2}+\Vert b_{0}\Vert _{H^{s}}^{2}}}{1-CT\sqrt{\Vert u_{0}\Vert _{H^{s}}^{2}+\Vert b_{0}\Vert _{H^{s}}^{2}}}. \end{aligned}$$

(6.10)

Thus, the family ($u^{N},\,b^{N}$) is uniformly bounded in $C([0, T ]; H^{s})$ with $s>1+\frac{n}{2}$, provided that $T<\Big (C(\Vert u_{0}\Vert _{{H}^{s}}^{2}+\Vert b_{0}\Vert _{H^{s}}^{2}) \Big )^{-\frac{1}{2}}$.

Therefore, one can conclude from (6.7) and (6.10) that

• $(u^{N},\,b^{N})_{N\in \mathbb {N}}$ is bounded in $L^{\infty }([0, T]; L^{2}(\mathbb {R}^{n}))$,

• $(u^{N},\,b^{N})_{N\in \mathbb {N}}$ is bounded in $L^{\infty }([0, T]; H^{s}(\mathbb {R}^{n}))$ for some $s>1+\frac{n}{2}$.

This is enough to pass to the limit (up to extraction) in (6.8). In fact, we have

$$\begin{aligned}&\Vert \mathcal {P}\mathcal {J}_{N}((u^{N}u^{N})\Vert _{L^{2}}\le C \Vert u^{N}\Vert _{L^{4}}^{2}\le C,\quad \Vert \mathcal {P}\mathcal {J}_{N}((b^{N}b^{N})\Vert _{L^{2}}\le C \Vert b^{N}\Vert _{L^{4}}^{2}\le C,\\&\Vert \mathcal {P}\mathcal {J}_{N}((u^{N}b^{N})\Vert _{L^{2}}\le C \Vert u^{N}\Vert _{L^{4}}\Vert b^{N}\Vert _{L^{4}}\le C, \end{aligned}$$

where we have used the following interpolation:

$$\begin{aligned} \Vert f\Vert _{L^{4}}\le C\Vert f\Vert _{L^{2}}^{\frac{4s-n}{4s}}\Vert f\Vert _{H^{s}} ^{\frac{n}{4s}}, \quad s\ge \frac{n}{4}. \end{aligned}$$

Note that

$$\begin{aligned}&\partial _{t}u^{N}=-\mathcal {P}\mathcal {J}_{N}((u^{N}\cdot \nabla ) u^{N})-\kappa u^{N}+\mathcal {P}\mathcal {J}_{N}(b^{N}\cdot \nabla )b^{N},\\&\partial _{t}b^{N}=-\mathcal {J}_{N}((u^{N}\cdot \nabla )b^{N})-\lambda b^{N}+\mathcal {J}_{N}((b^{N}\cdot \nabla )u^{N}). \end{aligned}$$

Thus, it is not hard to see that

$$\begin{aligned} \partial _{t}u^{N},\,\,\partial _{t}b^{N}\in L_{t}^{\infty } ([0, T]);\,H_{x}^{-\sigma }(\mathbb {R}^{n})\quad \text{ for } \text{ any } \,\, \sigma \ge 1. \end{aligned}$$

Consequently, we assume that

$$\begin{aligned} \partial _{t}u^{N},\,\,\partial _{t}b^{N}\in L_{Loc}^{4}([0, T]);\,H_{x}^{-2}(\mathbb {R}^{n}). \end{aligned}$$

Since the embedding $L^{2}\hookrightarrow H^{-2}$ is locally compact, the well-known Aubin-Lions argument (see e.g., Constantin and Foias 1988; Temam 2002) allows us to conclude that, up to extraction, subsequence $(u^{N},\,b^{N})_{N\in \mathbb {N}}$ satisfies

$$\begin{aligned} \Vert u^{N}-u^{N'}\Vert _{L^{2}},\quad \Vert b^{N}-b^{N'}\Vert _{L^{2}}\rightarrow 0,\quad as\quad N,\,\,N'\rightarrow \infty . \end{aligned}$$

By the interpolation ($\Vert u\Vert _{H^{s'}}\le C \Vert u\Vert _{L^{2}}^{1-\frac{s'}{s}}\Vert u\Vert _{H^{s}}^{\frac{s'}{s}}$ for any $s'<s$), we can show that

$$\begin{aligned} \Vert u^{N}-u^{N'}\Vert _{H^{s'}},\quad \Vert b^{N}-b^{N'}\Vert _{H^{s'}}\rightarrow 0,\quad as\quad N,\,\,N'\rightarrow \infty ,\,\, \text{ for } \text{ any }\,\,s'<s, \end{aligned}$$

which imply that we have strong convergence limit $(u, b)$ in $C([0, T]; H^{s'})$ (see Claim 2 below for detailed proof) with any $s'<s$. Therefore, it is enough for us to show that up to extraction, sequence $(u^{N},\,b^{N})_{N\in \mathbb {N}}$ has a limit $(u,\,b)$ satisfying

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}u+\mathcal {P}((u\cdot \nabla ) u)+\kappa u=\mathcal {P}((b\cdot \nabla )b),\\ \partial _{t}b+(u\cdot \nabla )b+\lambda b=(b\cdot \nabla )u,\\ \nabla \cdot u=0,\,\nabla \cdot b=0,\\ u(x,0)=u_{0}(x),\,b(x,0)=b_{0}(x). \end{array} \right. \end{aligned}$$

(6.11)

Claim 2: $(u, b)\in L^{\infty }([0, T]; H^{s}(\mathbb {R}^{n}) \times H^{s}(\mathbb {R}^{n}))$ and $(u, b)\in$Lip$([0, T]; H^{s-1}(\mathbb {R}^{n}) \times H^{s-1}(\mathbb {R}^{n}))$. Moreover, $(u, b)\in C([0, T]; H^{s}(\mathbb {R}^{n}) \times H^{s}(\mathbb {R}^{n}))$.

From the above argument, it is easy to show that

$$\begin{aligned}&\sup _{0\le t\le T}\Vert (u^{N},b^{N})\Vert _{H^{s}} \le M_{1}<\infty ,\end{aligned}$$

(6.12)

$$\begin{aligned}&\sup _{0\le t\le T} \Vert (\partial _{t}u^{N},\partial _{t}b^{N})\Vert _{H^{s-1}} \le M_{2}<\infty . \end{aligned}$$

(6.13)

Therefore, $(u^{N},b^{N})$ is uniformly bounded in the Hilbert space $L^{2}([0, T]; H^{s}(\mathbb {R}^{n}) \times H^{s}(\mathbb {R}^{n}))$ such that there exists a subsequence that converges weakly to

$$\begin{aligned} (u,b)\in L^{2}([0, T]; H^{s}(\mathbb {R}^{n}) \times H^{s}(\mathbb {R}^{n})). \end{aligned}$$

(6.14)

For the fixed $t\in [0, T]$, the sequence $(u^{N}(., t),b^{N}(., t))$ is uniformly bounded in $H^{s}(\mathbb {R}^{n}) \times H^{s}(\mathbb {R}^{n})$, so that it also has a subsequence that converges weakly to $(u(t), b(t))\in H^{s}(\mathbb {R}^{n}) \times H^{s}(\mathbb {R}^{n})$. Consequently, $\Vert (u, b)\Vert _{H^{s}}$ is bounded for any $t\in [0, T]$ which together with (6.14) implies that $(u, b)\in L^{\infty }([0, T]; H^{s}(\mathbb {R}^{n}) \times H^{s}(\mathbb {R}^{n}))$. Applying the same arguments and (6.13), we can show that $(u, b)\in \text{ Lip } ([0, T]; H^{s-1}(\mathbb {R}^{n}) \times H^{s-1}(\mathbb {R}^{n}))$.

In fact, we can get from the local existence theorem that

$$\begin{aligned} (u^{N}, b^{N})\in C^{1}([0, T]; H^{s}(\mathbb {R}^{n})\times H^{s}(\mathbb {R}^{n})), \end{aligned}$$

(6.15)

and

$$\begin{aligned} (u^{N}, b^{N})\rightarrow (u, b) \in L^{\infty } ([0, T]; H^{s'}(\mathbb {R}^{n})\times H^{s'}(\mathbb {R}^{n}))\quad \text{ for } \text{ any }\,\, s'\le s. \end{aligned}$$

(6.16)

Now we will show that $(u,b)$ is strongly continuous in $H^{s}(\mathbb {R}^{n})\times H^{s}(\mathbb {R}^{n})$ in time. It suffices to consider $u\in C([0, T]; H^{s}(\mathbb {R}^{n})$ as the same fashion can be applied to $b$ to obtain the desired result.

By the equivalent norm, it yields

$$\begin{aligned} \Vert u(t_{1})-u(t_{2})\Vert _{H^{s}}=\Big \{\Big (\sum _{j<N}+ \sum _{j\ge N}\Big ) (2^{js}\Vert \Delta _{j}u(t_{1})-\Delta _{j}u(t_{2})\Vert _{L^{2}})^{2} \Big \}^{\frac{1}{2}}. \end{aligned}$$

(6.17)

Let $\varepsilon >0$ be arbitrarily small. Due to $u\in L^{\infty }([0, T]; H^{s}(\mathbb {R}^{n}))$, there exists a integer $N>0$ such that

$$\begin{aligned} \Big \{\sum _{j\ge N} (2^{js}\Vert \Delta _{j}u(t_{1})-\Delta _{j}u(t_{2})\Vert _{L^{2}})^{2} \Big \}^{\frac{1}{2}}<\frac{\varepsilon }{2}. \end{aligned}$$

(6.18)

Recalling the system (6.11)$_{1}$, we obtain

$$\begin{aligned} \Delta _{j}u(t_{1})-\Delta _{j}u(t_{2})&= \int _{t_{1}}^{t_{2}}{\frac{\mathrm{d}}{\mathrm{d}\tau } \Delta _{j}u(\tau )\,\mathrm{d}\tau }\nonumber \\&= \int _{t_{1}}^{t_{2}}{ \Delta _{j}\mathcal {P}[(b\cdot \nabla ) b-(u\cdot \nabla ) u-\kappa u](\tau )\,\mathrm{d}\tau }. \end{aligned}$$

(6.19)

Therefore, we can get

$$\begin{aligned}&\sum _{j<N} 2^{2js}\Vert \Delta _{j}u(t_{1})-\Delta _{j}u(t_{2})\Vert _{L^{2}}^{2}\nonumber \\&\quad =\sum _{j<N} 2^{2js}\Big (\Big \Vert \int _{t_{1}}^{t_{2}}{ \Delta _{j}\mathcal {P}[(b\cdot \nabla ) b-(u\cdot \nabla ) u-\kappa u](\tau )\,\mathrm{d}\tau }\Big \Vert _{L^{2}}\Big )^{2}\nonumber \\&\quad \le \sum _{j<N} 2^{2js}\Big (\int _{t_{1}}^{t_{2}}{ \Vert \Delta _{j}[(b\cdot \nabla ) b-(u\cdot \nabla ) u-\kappa u]\Vert _{L^{2}}(\tau )\,\mathrm{d}\tau }\Big )^{2} \nonumber \\&\quad \le \sum _{j<N} 2^{2js}\Big (\int _{t_{1}}^{t_{2}}{ [\Vert \Delta _{j}(b\cdot \nabla ) b\Vert _{L^{2}}+ \Vert (u\cdot \nabla ) u\Vert _{L^{2}}+\kappa \Vert u\Vert _{L^{2}}](\tau )\,\mathrm{d}\tau }\Big )^{2} \nonumber \\&\quad =\sum _{j<N} 2^{2j}\!\Big (\!\int _{t_{1}}^{t_{2}}{ [2^{j(s\!-\!1)}\Vert \!\Delta _{j}(b\!\cdot \nabla ) b\Vert _{L^{2}}\!+\! 2^{j(s\!-\!1)}\Vert (u\!\cdot \!\nabla ) u\Vert _{L^{2}}\!+\!\kappa 2^{j(s\!-\!1)}\Vert u\Vert _{L^{2}}]\!(\tau )\,\mathrm{d}\!\tau }\!\Big )^{2}\nonumber \\&\quad \le C\sum _{j<N} 2^{2j}\!\Big (\Vert (b\cdot \nabla ) b\Vert _{H^{s\!-\!1}}^{2}|t_{1}\!-\!t_{2}|\!+\!\Vert (u\cdot \nabla ) u\Vert _{H^{s-1}}^{2}|t_{1}-t_{2}|\!+\!\kappa \Vert u\Vert _{H^{s-1}}^{2}|t_{1}\!-\!t_{2}|\Big )\nonumber \\&\quad \le C\sum _{j<N} 2^{2j}|t_{1}-t_{2}|\Big (\Vert b\Vert _{L^{\infty }}^{2}\Vert \nabla b\Vert _{H^{s-1}}^{2}+\Vert \nabla b\Vert _{L^{\infty }}^{2}\Vert b\Vert _{H^{s-1}}^{2}+ \Vert u\Vert _{L^{\infty }}^{2}\Vert \nabla u\Vert _{H^{s-1}}^{2}\nonumber \\&\qquad +\;\Vert \nabla u\Vert _{L^{\infty }}^{2}\Vert u\Vert _{H^{s-1}}^{2} +\kappa \Vert u\Vert _{H^{s-1}}^{2}\Big )\nonumber \\&\quad \le C 2^{2N}|t_{1}-t_{2}|\Big (\Vert b\Vert _{H^{s}}^{2}\Vert b\Vert _{H^{s}}^{2}+\Vert u\Vert _{H^{s}}^{2}\Vert u\Vert _{H^{s}}^{2} +\kappa \Vert u\Vert _{H^{s}}^{2}\Big ), \end{aligned}$$

(6.20)

where the Sobolev imbeddings $H^{s}(\mathbb {R}^{n})\hookrightarrow H^{s-1}(\mathbb {R}^{n})$ and $H^{s-1}(\mathbb {R}^{n})\hookrightarrow L^{\infty }(\mathbb {R}^{n})$ with $s>1+\frac{n}{2}$ are used several times in the last inequality.

Thus, the following holds true

$$\begin{aligned} \Big \{\sum _{j< N} (2^{js}\Vert \Delta _{j}u(t_{1})-\Delta _{j}u(t_{2})\Vert _{L^{2}})^{2} \Big \}^{\frac{1}{2}}<\frac{\varepsilon }{2} \end{aligned}$$

(6.21)

provided $|t_{1}-t_{2}|$ small enough.

Combining (6.18) with (6.21) implies $u\in C([0, T]; H^{s}(\mathbb {R}^{n})$. Consequently, the Claim 2 holds true. The uniqueness can be easily obtained as the velocity field and magnetic field are both in Lipschitz space. Therefore, the proof of Proposition 6.1 is completed. $\square$

Now we will state the following fundamental commutator estimates which have been used repeatedly in the proofs of Lemmas 2.6 and 2.7.

Lemma 6.3

(See Bahour et al. 2011) Let $\theta$ be a $C^{1}$ function on $\mathbb {R}^{n}$ such that $|x|\check{\theta }(x)\in L^{1}$. There exists a constant C such that for any Lipschitz function $a$ with gradient in $L^{p}$ and any function $b$ in $L^{q}$, we have, for any positive $\lambda$,

$$\begin{aligned} \Vert [\theta (\lambda ^{-1}D), a]b\Vert _{L^{r}}\le C\lambda ^{-1}\Vert \nabla a\Vert _{L^{p}}\Vert b\Vert _{L^{q}} \, \, with \, \, \frac{1}{p}+\frac{1}{q}=\frac{1}{r},\,\,(p,q)\in [1,\infty ]^{2}. \end{aligned}$$

With the aid of above Lemma, we will prove the commutator (2.3) and Lemma 2.7.

Proof of Commutator 2.3

The proof of (2.2) follows the general line of presentation in Bahour et al. (2011), Miao et al. (2012) and is standard; thus, we omit it. Now our efforts focus on the commutator (2.3). Using the notion of para-products, we write

$$\begin{aligned}{}[{\Delta }_{j}, u\cdot \nabla ]\omega =K_{1}+K_{2}+K_{3} \end{aligned}$$

where

$$\begin{aligned} K_{1}=&\sum _{|k-j|\le 2}[{\Delta }_{j}, {S}_{k-1}u\cdot \nabla ]{\Delta }_{k}\omega ,\quad K_{2}=\sum _{|k-j|\le 2}[{\Delta }_{j}, {\Delta }_{k}u\cdot \nabla ]{S}_{k-1}\omega ,\\ K_{3}=&\sum _{k+1\ge j}[{\Delta }_{j}, {\Delta }_{k}u\cdot \nabla ]\widetilde{{\Delta }_{k}}\omega \quad \text{ and } \quad \widetilde{{\Delta }_{k}}={\Delta }_{k-1}+{\Delta }_{k}+{\Delta }_{k+1}. \end{aligned}$$

By using Bernstein inequality and Lemma 6.3 above, we arrive at

$$\begin{aligned} \Vert K_{1}\Vert _{L^{p}}&\le C \sum _{i=1}^{n}\sum _{|k-j|\le 2}2^{-j}\Vert \nabla {S}_{k-1}{u_{i}}\Vert _{L^{\infty }} \Vert \partial _{i} {\Delta }_{k}\omega \Vert _{L^{p}}\\&\le C\sum _{|k-j|\le 2}2^{k-j}\Vert \nabla {S}_{k-1}{u}\Vert _{L^{\infty }} \Vert {\Delta }_{k}\omega \Vert _{L^{p}}\\&\le C\Vert \nabla {u}\Vert _{L^{\infty }}\sum _{|k-j|\le 2}2^{k-j} \Vert {\Delta }_{k}\omega \Vert _{L^{p}}. \end{aligned}$$

We can bound $K_{2}$ ($k\ge 1$ otherwise $K_{2}\equiv 0$) similar to $K_{1}$ as follows:

$$\begin{aligned} \Vert K_{2}\Vert _{L^{p}}&\le C \sum _{i=1}^{n}\sum _{|k-j|\le 2}2^{-j}\Vert \nabla { {\Delta }_{k}u_{i}}\Vert _{L^{p}} \Vert \partial _{i} {S}_{k-1}\omega \Vert _{L^{\infty }}\\&\le C\sum _{|k-j|\le 2}2^{-j}\Vert \nabla { {\Delta }_{k}u}\Vert _{L^{p}} 2^{k}\Vert \ {S}_{k-1}\omega \Vert _{L^{\infty }}\\&\le C\sum _{|k-j|\le 2}2^{-j}\Vert {{\Delta }_{k}\nabla u}\Vert _{L^{p}} 2^{k}\Vert {S}_{k-1}\omega \Vert _{L^{\infty }}\\&\le C\sum _{|k-j|\le 2}2^{k-j}\Vert { {\Delta }_{k}\nabla u}\Vert _{L^{p}} \Vert \omega \Vert _{L^{\infty }}\\&\le C\Vert \nabla u\Vert _{L^{\infty }}\sum _{|k-j|\le 2}2^{k-j}\Vert { {\Delta }_{k}\omega }\Vert _{L^{p}}, \end{aligned}$$

where $\Vert \omega \Vert _{L^{\infty }}\le C\Vert \nabla u\Vert _{L^{\infty }}$ and $\Vert { {\Delta }_{k}\nabla u}\Vert _{L^{p}}\le C \Vert { {\Delta }_{k}\omega }\Vert _{L^{p}}$ for any $k\ge 0$ are applied (in fact, $k\ge 0$ is only needed when $p=1$ or $p=\infty$).

We decompose $K_{3}$ into the following two parts:

$$\begin{aligned} K_{3}=\sum _{ j-1\le k\le j}[{\Delta }_{j}, {\Delta }_{k}u\cdot \nabla ]\widetilde{{\Delta }_{k}}\omega +\sum _{k> j}[{\Delta }_{j}, {\Delta }_{k}u\cdot \nabla ]\widetilde{{\Delta }_{k}}\omega \triangleq K_{3}^{1}+K_{3}^{2}. \end{aligned}$$

For $K_{3}^{1}$, making use of Lemma 6.3 above, we have

$$\begin{aligned} \Vert K_{3}^{1}\Vert _{L^{p}}&\le C \sum _{ j-1\le k\le j}\Vert [{\Delta }_{j}, {\Delta }_{k}u\cdot \nabla ]\widetilde{{\Delta }_{k}}\omega \Vert _{L^{p}}\\&\le C \sum _{ j-1\le k\le j}2^{-j}\Vert \nabla {\Delta }_{k}u\Vert _{L^{\infty }} \Vert \nabla \widetilde{{\Delta }_{k}}\omega \Vert _{L^{p}}\\&\le C\Vert \nabla u\Vert _{L^{\infty }}\sum _{ j-1\le k\le j}2^{k-j} \Vert \widetilde{{\Delta }_{k}}\omega \Vert _{L^{p}}\\&\le C\Vert \nabla u\Vert _{L^{\infty }}\sum _{ j-1\le k\le j}2^{k-j} \Vert {{\Delta }_{k}}\omega \Vert _{L^{p}}. \end{aligned}$$

For the term $K_{3}^{2}$, we do not need to use the structure of the commutator. Thanks to divergence-free condition, we can rewrite $K_{3}^{2}$ as follows:

$$\begin{aligned} K_{3}^{2}=\sum _{k> j}{\Delta }_{j}\partial _{l} ({\Delta }_{k}u_{l}\widetilde{{\Delta }_{k}}\omega )+\sum _{|k-j|\le 3, k>j} {\Delta }_{k}u_{l}{\Delta }_{j}\partial _{l}\widetilde{{\Delta }_{k}}\omega . \end{aligned}$$

By using Bernstein inequality, we can obtain (bearing in mind $k>j\Rightarrow k\ge 0$)

$$\begin{aligned} \Vert K_{3}^{2}\Vert _{L^{p}}\le C&\sum _{k> j}\Vert {\Delta }_{j}\partial _{l} ({\Delta }_{k}u_{l}\widetilde{{\Delta }_{k}}\omega ) \Vert _{L^{p}}+C\sum _{|k-j|\le 3, k>j}\Vert {\Delta }_{k}u_{l}{\Delta }_{j}\partial _{l}\widetilde{{\Delta }_{k}}\omega \Vert _{L^{p}}\\ \!\le \! C&\!\sum _{k> j}2^{j-k}\Vert \!\nabla {\Delta }_{k}u\Vert _{L^{\infty }} \Vert \widetilde{{\Delta }_{k}}\omega \Vert _{L^{p}}\!+\! C\sum _{|k-j|\le 3, k>j}2^{j\!-\!k}\Vert \nabla {\Delta }_{k}u\Vert _{L^{\infty }} \Vert \widetilde{{\Delta }_{k}}\omega \Vert _{L^{p}}\\ \le C&\Vert \nabla u\Vert _{L^{\infty }}\sum _{k> j}2^{j-k} \Vert {{\Delta }_{k}}\omega \Vert _{L^{p}}. \end{aligned}$$

Plugging all the obtained estimates together, we have

$$\begin{aligned} \Vert [{\Delta }_{j}, u\!\cdot \!\nabla ]\omega \Vert _{L^{p}}&\le C\Vert \!\nabla {u}\Vert _{L^{\infty }}\!\!\sum _{|k\!-j|\!\le 2}2^{k\!-j} \Vert {\!\Delta }_{k}\!\omega \Vert _{L^{p}} +\;C\Vert \nabla u\Vert _{L^{\infty }}\!\!\sum _{ j-1\!\le k\le j}2^{k\!-j}\Vert {{\Delta }_{k}}\omega \Vert _{L^{p}}\\&+\,C \Vert \nabla u\Vert _{L^{\infty }}\sum _{k> j}2^{j-k} \Vert {{\Delta }_{k}}\omega \Vert _{L^{p}}. \end{aligned}$$

Multiplying above inequality by $2^{js}$, taking $l_{j}^{r}$ then applying the discrete Young inequality, we can show that

$$\begin{aligned} \big \Vert 2^{js}\Vert [{\Delta }_{j}, u\cdot \nabla ]w\Vert _{L^{p}}\big \Vert _{l_{j}^{r}} \le&\; C \Vert \nabla u\Vert _{L^{\infty }}\big \Vert \sum _{|k-j|\le 2}2^{(j-k)(s-1)}2^{ks}\Vert {\Delta }_{k}\omega \Vert _{L^{p}}\big \Vert _{l_{j}^{r}}\\&+C \Vert \nabla u\Vert _{L^{\infty }}\big \Vert \sum _{ j-1\le k\le j}2^{(j-k)(s-1)}2^{ks}\Vert {\Delta }_{k}\omega \Vert _{L^{p}}\big \Vert _{l_{j}^{r}}\\&+\,C \Vert \nabla u\Vert _{L^{\infty }}\big \Vert \sum _{k> j}2^{(j-k)(s+1)}2^{ks} \Vert {{\Delta }_{k}}\omega \Vert _{L^{p}}\big \Vert _{l_{j}^{r}}\\ \le&\; C \Vert \nabla u\Vert _{L^{\infty }}\Vert \omega \Vert _{{B}_{p,r}^{s}}, \end{aligned}$$

where $s+1>0$ can guarantee that

$$\begin{aligned} \big \Vert \sum _{k> j}2^{(j-k)(s+1)}2^{ks} \Vert {{\Delta }_{k}}\omega \Vert _{L^{p}}\big \Vert _{l_{j}^{r}}\le C\Vert \omega \Vert _{{B}_{p,r}^{s}}. \end{aligned}$$

Therefore, the commutator (2.3) is then proved. $\square$

Proof of Lemma 2.7

Using the notion of para-products, we write

$$\begin{aligned}{}[\dot{\Delta }_{j}, u\cdot \nabla ]v=J_{1}+J_{2}+J_{3} \end{aligned}$$

where

$$\begin{aligned} J_{1}=&\sum _{|k-j|\le 2}[\dot{\Delta }_{j}, \dot{S}_{k-1}u\cdot \nabla ]\dot{\Delta }_{k}v,\quad J_{2}=\sum _{|k-j|\le 2}[\dot{\Delta }_{j}, \dot{\Delta }_{k}u\cdot \nabla ]\dot{S}_{k-1}v ,\\ J_{3}=&\sum _{k+1\ge j}[\dot{\Delta }_{j}, \dot{\Delta }_{k}u\cdot \nabla ]\widetilde{\dot{\Delta }_{k}}v\quad \text{ and } \quad \widetilde{\dot{\Delta }_{k}}=\dot{\Delta }_{k-1}+\dot{\Delta }_{k}+\dot{\Delta }_{k+1}. \end{aligned}$$

Since the summation above is for $k$ satisfying $|j-k|\le 2$ and can be replaced by a constant multiple of the representative term with $k=j$, then it follows from Bernstein inequality and Lemma 6.3 above that

$$\begin{aligned} \Vert J_{1}\Vert _{L^{p}}\le&\,C \sum _{i=1}^{n}\sum _{|k-j|\le 2}2^{-j}\Vert \nabla \dot{S}_{k-1}{u_{i}}\Vert _{L^{\infty }} \Vert \nabla \dot{\Delta }_{k}v\Vert _{L^{p}}\\ \le&\, C\sum _{|k-j|\le 2}2^{k-j}\Vert \nabla \dot{S}_{k-1}{u}\Vert _{L^{\infty }} \Vert \dot{\Delta }_{k}v\Vert _{L^{p}}\\ \le&\, C\Vert \nabla u\Vert _{L^{\infty }} \Vert \dot{\Delta }_{j}v\Vert _{L^{p}}. \end{aligned}$$

Similarly, we can deal with $J_{2}$ as follows

$$\begin{aligned} \Vert J_{2}\Vert _{L^{p}}\le&\, C \sum _{i=1}^{n}\sum _{|k-j|\le 2}2^{-j}\Vert \nabla {\dot{\Delta }_{k}u_{i}}\Vert _{L^{p}} \Vert \nabla \dot{S}_{k-1}v\Vert _{L^{\infty }}\\ \le&\,C\sum _{|k-j|\le 2}2^{-j}\Vert \nabla {\dot{\Delta }_{k}u}\Vert _{L^{p}} \Vert \nabla \dot{S}_{k-1}v\Vert _{L^{\infty }}\\ \le&\,C\sum _{|k-j|\le 2}2^{-j}2^{k}\Vert {\dot{\Delta }_{k}u}\Vert _{L^{p}} 2^{k}\Vert \dot{S}_{k-1}v\Vert _{L^{\infty }}\\ \le&\, C\sum _{|k-j|\le 2}2^{2k-2j}\Vert {\dot{\Delta }_{k}u}\Vert _{L^{p}} 2^{j}\Vert v\Vert _{L^{\infty }}\\ \le&\, C 2^{j}\Vert v\Vert _{L^{\infty }}\Vert {\dot{\Delta }_{j}u}\Vert _{L^{p}}. \end{aligned}$$

It is much more involved to handle the remainder term $J_{3}$. We split it into two terms: high frequencies and low frequencies:

$$\begin{aligned} J_{3}=\sum _{k\ge j-1}\dot{\Delta }_{j}[\dot{\Delta }_{k}u\cdot \nabla \widetilde{\dot{\Delta }_{k}}v] +\sum _{|k-j|\le 3}\dot{\Delta }_{k}u\cdot \nabla \dot{\Delta }_{j}\widetilde{\dot{\Delta }_{k}}v\triangleq J_{3}^{1}+J_{3}^{2}. \end{aligned}$$

For the first term we do not need to use the structure of the commutator. We estimate separately each term of the commutator by using Bernstein inequalities: Thanks to divergence-free condition, we can rewrite $J_{3}^{1}$ as follows:

$$\begin{aligned} J_{3}^{1}=\sum _{k\ge j-1}\sum _{l=1}^{n} \dot{\Delta }_{j}\partial _{l}[\dot{\Delta }_{k}u_{l}\widetilde{\dot{\Delta }_{k}}v]. \end{aligned}$$

Hence,

$$\begin{aligned} \Vert J_{3}^{1}\Vert _{L^{p}}\le C&\sum _{k\ge j-1}2^{j}\Vert \dot{\Delta }_{k}u\widetilde{\dot{\Delta }_{k}}v \Vert _{L^{p}}\\ \le C&\sum _{k\ge j-1}2^{j-k}\Vert \dot{\Delta }_{k}\nabla u\Vert _{L^{\infty }}\Vert \widetilde{\dot{\Delta }_{k}}v \Vert _{L^{p}}\\ \le C&\sum _{k\ge j-1}2^{j-k}\Vert \nabla u\Vert _{L^{\infty }}\Vert {\dot{\Delta }_{k}}v \Vert _{L^{p}}\\ \le C&\Vert \nabla u\Vert _{L^{\infty }} \sum _{k\ge j-1}2^{j-k}\Vert {\dot{\Delta }_{k}}v \Vert _{L^{p}}. \end{aligned}$$

For the second term, we use Bernstein inequalities to obtain

$$\begin{aligned} \Vert J_{3}^{2}\Vert _{L^{p}}\le C&\sum _{|k-j|\le 3}\Vert \dot{\Delta }_{k}u\cdot \nabla \dot{\Delta }_{j}\widetilde{\dot{\Delta }_{k}}v \Vert _{L^{p}}\\ \le C&2^{j} \Vert v\Vert _{L^{\infty }}\Vert \dot{\Delta }_{j}u\Vert _{L^{p}}. \end{aligned}$$

Putting all the above estimates together, we have

$$\begin{aligned} \Vert [\dot{\Delta }_{j}, u\cdot \nabla ]v\Vert _{L^{p}}\le&\, C\Vert \nabla u\Vert _{L^{\infty }} \Vert \dot{\Delta }_{j}v\Vert _{L^{p}}+ C 2^{j}\Vert v\Vert _{L^{\infty }}\Vert {\dot{\Delta }_{j}u}\Vert _{L^{p}}\\&+\,C \Vert \nabla u\Vert _{L^{\infty }} \sum _{k\ge j-1}2^{j-k}\Vert {\dot{\Delta }_{k}}v \Vert _{L^{p}}. \end{aligned}$$

Multiplying above inequality by $2^{js}$ then taking $l_{j}^{r}$ yields

$$\begin{aligned} \big \Vert 2^{js}\Vert [\dot{\Delta }_{j}, u\cdot \nabla ]v\Vert _{L^{p}}\big \Vert _{l_{j}^{r}} \le&\,C \Vert \nabla u\Vert _{L^{\infty }}\Vert v\Vert _{\dot{B}_{p,r}^{s}}+C \Vert v\Vert _{L^{\infty }}\Vert u\Vert _{\dot{B}_{p,r}^{s+1}}\\&+\,C \Vert \nabla u\Vert _{L^{\infty }} \big \Vert \sum _{k\ge j-1}2^{(j-k)(s+1)}2^{ks}\Vert {\dot{\Delta }_{k}}v \Vert _{L^{p}}\big \Vert _{l_{j}^{r}}\\ \le&\, C \Vert \nabla u\Vert _{L^{\infty }}\Vert v\Vert _{\dot{B}_{p,r}^{s}}+C \Vert v\Vert _{L^{\infty }}\Vert u\Vert _{\dot{B}_{p,r}^{s+1}}+\Vert C_{k}\star D_{m}(j)\Vert _{l_{j}^{r}} \\ \le&\, C \Vert \nabla u\Vert _{L^{\infty }}\Vert v\Vert _{\dot{B}_{p,r}^{s}}+C \Vert v\Vert _{L^{\infty }}\Vert u\Vert _{\dot{B}_{p,r}^{s+1}}, \end{aligned}$$

where $C_{k}=\chi _{[k\le 1]}2^{(s+1)k}$ and $D_{m}=2^{ms}\Vert {\dot{\Delta }_{m}}v \Vert _{L^{p}}$. Here, the discrete Young inequality has been applied. Note the following fact

$$\begin{aligned} \Vert u\Vert _{\dot{B}_{p,r}^{s+1}}\thickapprox \Vert \nabla u\Vert _{\dot{B}_{p,r}^{s}}\thickapprox \Vert \omega \Vert _{\dot{B}_{p,r}^{s}}; \end{aligned}$$

thus, the desired inequalities can be obtained immediately. Therefore, we have completed the proof Lemma 2.7. $\square$