Global Regularity of the Three-Dimensional Fractional Micropolar Equations

Abstract

The global well-posedness of the smooth solution to the three-dimensional (3D) incompressible micropolar equations is a difficult open problem. This paper focuses on the 3D incompressible micropolar equations with fractional dissipations $(-\Delta )^{\alpha }u$ and $(-\Delta )^{\beta }w$. Our objective is to establish the global regularity of the fractional micropolar equations with the minimal amount of dissipations. We prove that, if $\alpha \ge \frac{5}{4}$, $\beta \ge 0$ and $\alpha +\beta \ge \frac{7}{4}$, the fractional 3D micropolar equations always possess a unique global classical solution for any sufficiently smooth data. In addition, we also obtain the global regularity of the 3D micropolar equations with the dissipations given by Fourier multipliers that are logarithmically weaker than the fractional Laplacian.

Appendices

Appendix A. Besov Spaces

This appendix provides the definition of the Besov spaces and related facts that have been used in the previous sections. Some of the materials are taken from [2].

We start with the partition of unity. Let B(0, r) and ${{\mathcal {C}}}(0, r_1, r_2)$ denote the standard ball and the annulus, respectively,

$$\begin{aligned} B(0, r) = \left\{ \xi \in {\mathbb {R}}^n: \, |\xi | \le r \right\} , \quad {\mathcal {C}} (0, r_1, r_2) = \left\{ \xi \in {\mathbb {R}}^n:\, r_1 \le |\xi |\le r_2 \right\} . \end{aligned}$$

There are two compactly supported smooth radial functions $\phi$ and $\psi$ satisfying

$$\begin{aligned}&\text{ supp } \,\phi \subset B(0, 4/3), \quad \text{ supp } \,\psi \subset {\mathcal {C}}(0, 3/4, 8/3), \nonumber \\&\phi (\xi ) + \sum _{j\ge 0} \psi (2^{-j} \xi ) = 1 \quad \text{ for } \text{ all }\,\, \xi \in {\mathbb {R}}^n. \end{aligned}$$

(A.1)

We use ${{\widetilde{h}}}$ and h to denote the inverse Fourier transforms of $\phi$ and $\psi$ respectively,

$$\begin{aligned} {{\widetilde{h}}} = {\mathcal {F}}^{-1} \phi , \quad h = {\mathcal {F}}^{-1} \psi . \end{aligned}$$

In addition, for notational convenience, we write $\psi _j(\xi ) = \psi (2^{-j} \xi )$. By a simple property of the Fourier transform,

$$\begin{aligned} h_j(x) :={\mathcal {F}}^{-1} (\psi _j)(x) = 2^{n j} \, h(2^j x). \end{aligned}$$

The inhomogeneous dyadic block operator $\Delta _j$ are defined as follows

$$\begin{aligned}&\Delta _j f=0 \quad \text{ for } j\le -2,\\&\Delta _{-1} f = {{\widetilde{h}}} *f = \int _{{\mathbb {R}}^n} f(x-y) \, {{\widetilde{h}}}(y)\,dy,\\&\Delta _j f = h_j *f = 2^{n j} \int _{{\mathbb {R}}^n} f(x-y) \, h(2^j y)\,dy \quad \text{ for } j\ge 0. \end{aligned}$$

The corresponding inhomogeneous low frequency cut-off operator $S_j$ is defined by

$$\begin{aligned} S_j f = \sum _{k\le j-1} \Delta _k f. \end{aligned}$$

For any function f in the usual Schwarz class ${\mathcal {S}}$, (A.1) implies

$$\begin{aligned} {{\widehat{f}}} (\xi ) = \phi (\xi )\, {{\widehat{f}}} (\xi ) + \sum _{j\ge 0} \psi (2^{-j} \xi )\,{{\widehat{f}}} (\xi ) \end{aligned}$$

(A.2)

or, in terms of the inhomogeneous dyadic block operators,

$$\begin{aligned} f= \sum _{j\ge -1} \Delta _j f \quad \text{ or }\quad \text{ Id } = \sum _{j\ge -1} \Delta _j, \end{aligned}$$

where Id denotes the identity operator. More generally, for any F in the space of tempered distributions, denoted ${{\mathcal {S}}}'$, (A.2) still holds but in the distributional sense. That is, for $F \in {{\mathcal {S}}}'$,

$$\begin{aligned} F = \sum _{j\ge -1} \Delta _j F \quad \text{ or }\quad \text{ Id } = \sum _{j\ge -1} \Delta _j \quad \text{ in } \quad {{\mathcal {S}}}'. \end{aligned}$$

(A.3)

In fact, one can verify that

$$\begin{aligned} S_j F := \sum _{k\le j-1} \Delta _k F \quad \rightarrow \quad F \quad \text{ in } \quad {{\mathcal {S}}}'. \end{aligned}$$

(A.3) is referred to as the Littlewood-Paley decomposition for tempered distributions.

The inhomogeneous Besov space can be defined in terms of $\Delta _j$ specified above.

Definition A.1

For $1\le p,q \le \infty$ and $s\in {{\mathbb {R}}}$, the inhomogeneous Besov space $B^s_{p,q}$ consists of the functions $f\in {\mathcal S}'$ satisfying $\Vert f\Vert _{B^s_{p,q}} \equiv \Vert 2^{js} \Vert \Delta _j f\Vert _{L^p} \Vert _{l^q} <\infty .$

Bernstein’s inequality is a useful tool on Fourier localized functions and these inequalities trade derivatives for integrability. The following proposition provides Bernstein type inequalities for fractional derivatives.

Lemma A.1

For $\alpha \ge 0$, $1\le p\le q\le \infty$, and $f\in L^p({\mathbb {R}}^n)$,

• (1) if there exist some integer j and a constant $K>0$, such that, $\text{ supp }\, {\widehat{f}} \subset \{\xi \in {\mathbb {R}}^n: \,\, |\xi | \le K 2^j \}$, then

  $$\begin{aligned} \Vert (-\Delta )^\alpha f\Vert _{L^q({\mathbb {R}}^n)} \le C_1\, 2^{2\alpha j + j n(\frac{1}{p}-\frac{1}{q})} \Vert f\Vert _{L^p({\mathbb {R}}^n)}; \end{aligned}$$

• (2) if there exist some integer j and constants $0<K_1\le K_2$, such that, $\text{ supp }\, {\widehat{f}} \subset \{\xi \in {\mathbb {R}}^n: \,\, K_12^j\le |\xi | \le K_2 2^j \}$, then

  $$\begin{aligned} C_1\, 2^{2\alpha j} \Vert f\Vert _{L^q({\mathbb {R}}^n)} \le \Vert (-\Delta )^\alpha f\Vert _{L^q({\mathbb {R}}^n)} \le C_2\, 2^{2\alpha j + j n(\frac{1}{p}-\frac{1}{q})} \Vert f\Vert _{L^p({\mathbb {R}}^n)}, \end{aligned}$$

  where $C_1$ and $C_2$ are constants depending only on $\alpha ,p$ and q.

Appendix B. A Global Regularity Result When $\nabla \nabla \cdot w$ is Eliminated

As we mentioned in the introduction, the term $\nabla \nabla \cdot w$ in the equation of w in the micropolar system is a “bad” term in the sense that it prevents us from deriving the estimate $\Vert w\Vert _{L^{q}}$ with $q>2$ directly by the standard $L^{q}$-estimate. This appendix provides a global regularity result for the micropolar system without this term, namely (1.2) with $\mu =0$. As we shall see in Theorem B.1, the requirement on the fractional powers can be reduced to $\alpha \ge \frac{5}{4}$ and $\beta =0$, which is the best one at this moment.

Theorem B.1

Consider the following 3D incompressible micropolar equations, namely,

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t u+(u\cdot \nabla ) u +(-\Delta )^{\alpha } u +\nabla p = \nabla \times w , \quad x\in {\mathbb {R}}^{3},\,t>0,\\ \partial _tw + (u\cdot \nabla ) w +2 w = \nabla \times u,\\ \nabla \cdot u=0,\\ u(x,0)=u_{0}(x),\quad w(x,0)=w_{0}(x). \end{array}\right. \end{aligned}$$

(B.1)

Let $(u_{0},\,w_{0})\in H^{s}({\mathbb {R}}^{3})$ with $s>\frac{5}{2}$ and $\nabla \cdot u_{0}=0$. If $\alpha \ge \frac{5}{4}$, then the system (B.1) admits a unique global solution $(u,\,w)$ such that for any given $T>0$,

$$\begin{aligned} (u,\,w)\in L^{\infty }([0, T]; H^{s}({\mathbb {R}}^{3})),\quad \Lambda ^{\alpha }u\in L^{2}([0, T]; H^{s}({\mathbb {R}}^{3})). \end{aligned}$$

It suffices to consider the endpoint case $\alpha =\frac{5}{4}$ since $\alpha >\frac{5}{4}$ is even simpler. Combining Lemma 2.1 and Lemma 2.2, we still have

$$\begin{aligned} \Vert u(t)\Vert _{H^{\frac{1}{4}}}^{2} +\int _{0}^{t}{\Vert u(\tau )\Vert _{H^{\frac{3}{2}}}^{2}\,d\tau }\le C(t,\,u_{0},\,w_{0}). \end{aligned}$$

(B.2)

With (B.2) at our disposal, we are in the position to establish the following key estimates.

Lemma B.1

Assume $(u_{0},w_{0})$ satisfies the assumptions stated in Theorem B.1, then the smooth solution (u, w) of (B.1) admits the following bounds

$$\begin{aligned}&\int _{0}^{t}{ \Vert \nabla u(\tau )\Vert _{L^{\infty }}\,d\tau }\le C(t,\,u_{0},\,w_{0}), \end{aligned}$$

(B.3)

$$\begin{aligned}&\Vert w(t)\Vert _{L^{\infty }} \le C(t,\,u_{0},\,w_{0}). \end{aligned}$$

(B.4)

Proof

By $\nabla \cdot u=0$, we rewrite (B.1)${}_{1}$ as follows

$$\begin{aligned} \partial _{t}u+\Lambda ^{\frac{5}{2}} u=-\left( {\mathbb {I}}_{3}+(-\Delta )^{-1}\nabla \nabla \cdot \right) \Big [\nabla \cdot (u\otimes u)-\nabla \times w\Big ], \end{aligned}$$

where we have eliminated the pressure term by $\nabla \cdot u=0$. Applying $\Lambda ^{-1}$ yields

$$\begin{aligned} \partial _{t}\Lambda ^{-1}u+\Lambda ^{\frac{5}{2}} \Lambda ^{-1} u=-\Lambda ^{-1}\left( {\mathbb {I}}_{3}+(-\Delta )^{-1}\nabla \nabla \cdot \right) \Big [\nabla \cdot (u\otimes u)-\nabla \times w\Big ]. \end{aligned}$$

(B.5)

Applying Lemma 3.1 of [53] to (B.5) leads to

$$\begin{aligned} \Vert \Lambda ^{\frac{3}{2}-\epsilon }u\Vert _{L_{t}^{1}L^{8}}&= \Vert \Lambda ^{\frac{5}{2}-\epsilon }\Lambda ^{-1}u\Vert _{L_{t}^{1}L^{8}}\nonumber \\&\le C(t,u_{0})+ C(t)\left\| \Lambda ^{-1}\left( {\mathbb {I}}+(-\Delta )^{-1}\nabla \nabla \cdot \right) \Big [\nabla \cdot (u\otimes u)-\nabla \times w\Big ]\right\| _{L_{t}^{1}L^{8}}\nonumber \\&\le C(t,u_{0}) + C(t)\left\| \Lambda ^{-1} \Big [\nabla \cdot (u\otimes u)-\nabla \times w\Big ]\right\| _{L_{t}^{1}L^{8}}\nonumber \\&\le C(t,u_{0}) + C(t)\Vert uu\Vert _{L_{t}^{1}L^{8}} + C(t)\Vert w\Vert _{L_{t}^{1}L^{8}}\nonumber \\&\le C(t,u_{0}) + C(t)\Vert u\Vert _{L_{t}^{2}L^{16}}^{2} + C(t)\Vert w\Vert _{L_{t}^{1}L^{8}} \nonumber \\&\le C(t,u_{0}) + C(t)\Vert u\Vert _{L_{t}^{2}H^{\frac{3}{2}}}^{2} + C(t)\Vert w\Vert _{L_{t}^{1}L^{8}} \nonumber \\&\le C(t,\,u_{0},\,w_{0}) + C(t)\Vert w\Vert _{L_{t}^{1}L^{8}}, \end{aligned}$$

(B.6)

where in the last line we have used (B.2). By the equation of w in (B.1),

$$\begin{aligned} \frac{d}{dt}\Vert w(t)\Vert _{L^{8}}\le \Vert \nabla u\Vert _{L^{8}}. \end{aligned}$$

By an interpolation inequality, one derives

$$\begin{aligned} \Vert w(t)\Vert _{L^{8}}&\le \Vert w_{0}\Vert _{L^{8}}+\int _{0}^{t}\Vert \nabla u(\tau )\Vert _{L^{8}}\,d\tau \\&\le \Vert w_{0}\Vert _{L^{8}}+C\int _{0}^{t}\Vert u(\tau )\Vert _{L^{2}}^{1-\frac{17}{21-8\epsilon }} \Vert \Lambda ^{\frac{3}{2}-\epsilon }u(\tau )\Vert _{L^{8}}^{\frac{17}{21-8\epsilon }}\,d\tau \\&\le \Vert w_{0}\Vert _{L^{8}} +C\left( \int _{0}^{t}\Vert u(\tau )\Vert _{L^{2}}\,d\tau \right) ^{1-\frac{17}{21-8\epsilon }} \left( \int _{0}^{t} \Vert \Lambda ^{\frac{3}{2}-\epsilon }u(\tau )\Vert _{L^{8}} \,d\tau \right) ^{\frac{17}{21-8\epsilon }}, \end{aligned}$$

where $0<\epsilon <\frac{1}{2}$. Therefore, we conclude

$$\begin{aligned} \Vert w\Vert _{L_{t}^{1}L^{8}}&\le t\Vert w_{0}\Vert _{L^{8}} +Ct\left( \int _{0}^{t}\Vert u(\tau )\Vert _{L^{2}}\,d\tau \right) ^{1-\frac{17}{21-8\epsilon }} \left( \int _{0}^{t} \Vert \Lambda ^{\frac{3}{2}-\epsilon }u(\tau )\Vert _{L^{8}} \,d\tau \right) ^{\frac{17}{21-8\epsilon }} \nonumber \\&\le t\Vert w_{0}\Vert _{L^{8}}+C(t,\,u_{0},\,w_{0}) \Vert \Lambda ^{\frac{3}{2}-\epsilon }u\Vert _{L_{t}^{1}L^{8}}^{\frac{17}{21-8\epsilon }}. \end{aligned}$$

(B.7)

Combining (B.6) and (B.7), we have

$$\begin{aligned} \Vert \Lambda ^{\frac{3}{2}-\epsilon }u\Vert _{L_{t}^{1}L^{8}}&\le C(t,\,u_{0},\,w_{0})+C(t,\,u_{0},\,w_{0}) \Vert \Lambda ^{\frac{3}{2}-\epsilon }u\Vert _{L_{t}^{1}L^{8}}^{\frac{17}{21-8\epsilon }}\\&\le C(t,\,u_{0},\,w_{0})+\frac{1}{2}\Vert \Lambda ^{\frac{3}{2}-\epsilon }u\Vert _{L_{t}^{1}L^{8}}, \end{aligned}$$

which yields

$$\begin{aligned} \Vert \Lambda ^{\frac{3}{2}-\epsilon }u\Vert _{L_{t}^{1}L^{8}} \le C(t,\,u_{0},\,w_{0}). \end{aligned}$$

(B.8)

By further taking $0<\epsilon <\frac{1}{8}$, we obtain from (B.8) that

$$\begin{aligned} \int _{0}^{t}\Vert \nabla u(\tau )\Vert _{L^{\infty }}\,d\tau&\le C\int _{0}^{t}\Vert u(\tau )\Vert _{L^{2}}\,d\tau +C\int _{0}^{t} \Vert \Lambda ^{\frac{3}{2}-\epsilon }u(\tau )\Vert _{L^{8}}\,d\tau \nonumber \\&\le C(t,\,u_{0},\,w_{0}). \end{aligned}$$

(B.9)

By the equation of w in (B.1), we again have, for any $2\le q<\infty$,

$$\begin{aligned} \frac{d}{dt}\Vert w(t)\Vert _{L^{q}}\le \Vert \nabla u\Vert _{L^{q}} \quad \text{ or }\quad \Vert w(t)\Vert _{L^{q}} \le \Vert w_0\Vert _{L^{q}} + \int _0^t \Vert \nabla u\Vert _{L^{q}}\, d\tau . \end{aligned}$$

Letting $q\rightarrow \infty$ and invoking (B.9), we find

$$\begin{aligned} \Vert w(t)\Vert _{L^{\infty }} \le C(t,\,u_{0},\,w_{0}). \end{aligned}$$

Thus, we complete the proof of Lemma B.1. $\square$

By (B.3) and (B.4), we can obtain our ultimate global $H^s$-estimate for u and w.

Proof of Theorem B.1

Similar to (3.16), we have

$$\begin{aligned}&\frac{d}{dt}(\Vert \Lambda ^{s} u\Vert _{L^{2}}^{2}+\Vert \Lambda ^{s} w\Vert _{L^{2}}^{2})+\Vert \Lambda ^{s+\frac{5}{4}} u\Vert _{L^{2}}^{2}\\&\quad \le C(1+\Vert w\Vert _{L^{\infty }}^{2}+\Vert \nabla u\Vert _{L^{\infty }}) (\Vert \Lambda ^{s}u\Vert _{L^{2}}^{2}+\Vert \Lambda ^{s}w\Vert _{L^{2}}^{2}), \end{aligned}$$

which along with the Gronwall inequality, (B.3) and (B.4) yield

$$\begin{aligned} \Vert \Lambda ^{s} u(t)\Vert _{L^{2}}^{2}+\Vert \Lambda ^{s} w(t)\Vert _{L^{2}}^{2}+\int _{0}^{t}{ \Vert \Lambda ^{s+\frac{5}{4}} u(\tau )\Vert _{L^{2}}^{2} \,d\tau }\le C(t,\,u_{0},\,w_{0}). \end{aligned}$$

This finish the proof of Theorem B.1. $\square$

Appendix C. Local Well-Posedness Result on (1.2)

For the sake of completeness, this appendix presents the local well-posedness result of (1.2) with initial data $(u_{0}, w_{0}) \in H^{s}({\mathbb {R}}^{3})$ with $s>\frac{5}{2}$. More precisely, we prove the following local well-posedness result.

Proposition C.1

Let $(u_{0}, w_{0}) \in H^{s}({\mathbb {R}}^{3})$ with $s>\frac{5}{2}$ and $\nabla \cdot u_{0}=0$. If $\alpha +\beta >1$, then there exists a positive time T depending on $\Vert u_{0}\Vert _{H^{s}}$ and $\Vert w_{0}\Vert _{H^{s}}$ such that (1.2) admits a unique solution $(u, w)\in C([0, T]; H^{s}({\mathbb {R}}^{3}))$.

We remark that the same local well-posedness result also holds true for (1.8). Similarly to [10, 34] (also see [56]), the main ingredient of the proof of the Proposition C.1 is to approximate (1.2) by the Friedrichs method to obtain a family of global smooth solutions.

For $N>0$, set $B(0,N)=\{\xi \in {\mathbb {R}}^{3}|\,|\xi |\le N\}$ and denote by $\chi _{B(0,N)}$ the characteristic function on B(0, N). Define the functional space

$$\begin{aligned} L^{2}_{N}:=\{f\in L^{2}({\mathbb {R}}^{3})|\, \text{ supp } \,{\widehat{f}}\subset B(0,N)\}, \end{aligned}$$

and the spectral cut-off

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

Proof of Proposition C.1

We first consider the following approximate system of (1.2),

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}u^{N}+{\mathcal {P}}{\mathcal {J}}_{N}(({\mathcal {J}}_{N}u^{N}\cdot \nabla ) {\mathcal {J}}_{N}u^{N})+ \Lambda ^{2\alpha }{\mathcal {J}}_{N}u^{N} ={\mathcal {P}}\mathcal \nabla \times {J}_{N}w^{N}, \\&\partial _{t}w^{N}+{\mathcal {J}}_{N} (({\mathcal {J}}_{N}u^{N} \cdot \nabla ){\mathcal {J}}_{N}w^{N})+2{\mathcal {J}}_{N}w^{N}+\Lambda ^{2\beta }{\mathcal {J}}_{N}w^{N}=\nabla \times {J}_{N}u^{N}+ \nabla \nabla \cdot {J}_{N}w^{N},\\&\nabla \cdot u^{N}=0,\\&u^{N}(x, 0)={\mathcal {J}}_{N}u_{0}(x), \quad w^{N}(x,0)={\mathcal {J}}_{N}w_{0}(x) , \end{aligned}\right. \end{aligned}$$

(C.1)

where ${\mathcal {P}}$ denotes the standard projection onto divergence-free vector fields. Thanks to the Cauchy-Lipschitz theorem (Picard’s Theorem, see [34]), we can find that for any fixed N, there exists a unique local solution $(u^{N},w^{N})$ on $[0,\,T_{N})$ in the functional setting $L^{2}_{N}$ with $T_{N}=T(N, u_{0}, w_{0})$. By ${\mathcal {J}}_{N}^{2}={\mathcal {J}}_{N},\,{\mathcal {P}}^{2}={\mathcal {P}}$ and ${\mathcal {P}}{\mathcal {J}}_{N}={\mathcal {J}}_{N}{\mathcal {P}}$, we can check that $({\mathcal {J}}_{N}u^{N},\,{\mathcal {J}}_{N}w^{N})$ is also a solution to (C.1) with the same initial datum. Based on the uniqueness, it yields

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

Consequently, the approximate system (C.1) reduces to

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}u^{N}+{\mathcal {P}}{\mathcal {J}}_{N}((u^{N}\cdot \nabla )u^{N})+ \Lambda ^{2\alpha } u^{N} ={\mathcal {P}}\mathcal \nabla \times w^{N}, \\&\partial _{t}w^{N}+{\mathcal {J}}_{N} ((u^{N} \cdot \nabla )w^{N})+2w^{N}+\Lambda ^{2\beta }w^{N}=\nabla \times u^{N}+ \nabla \nabla \cdot w^{N},\\&\nabla \cdot u^{N}=0,\\&u^{N}(x, 0)={\mathcal {J}}_{N}u_{0}(x), \quad w^{N}(x,0)={\mathcal {J}}_{N}w_{0}(x). \end{aligned}\right. \end{aligned}$$

(C.2)

A basic energy estimate implies $(u^{N},w^{N})$ of (C.2) satisfies

$$\begin{aligned}&\Vert u^{N}(t)\Vert _{L^{2}}^{2}+\Vert w^{N}(t)\Vert _{L^{2}}^{2}+ \int _{0}^{t}{ (\Vert \Lambda ^{\alpha } u^{N}\Vert _{L^{2}}^{2}+\Vert w^{N}\Vert _{H^{\beta }}^{2})(\tau )\,d\tau } \le C(\Vert u_{0}\Vert _{L^{2}}^{2}+\Vert w_{0}\Vert _{L^{2}}^{2},t). \end{aligned}$$

Therefore, the local solution can be extended into a global one, via the classical Picard Extension Theorem (see, e.g., [34]). By the direct $H^s$-estimates (see for example (3.16)), we deduce from (C.2) that

$$\begin{aligned}&\frac{d}{dt}(\Vert u^{N}(t)\Vert _{H^{s}}^{2} +\Vert w^{N}(t)\Vert _{H^{s}}^{2})+\Vert \Lambda ^{\alpha }u^{N}\Vert _{H^{s}}^{2} +\Vert \Lambda ^{\beta }w^{N}\Vert _{H^{s}}^{2}\nonumber \\&\quad \le C(1+\Vert \nabla u^{N}\Vert _{L^{\infty }}+\Vert \nabla w^{N}\Vert _{L^{\infty }}) (\Vert u^{N}\Vert _{H^{s}}^{2}+\Vert w^{N}\Vert _{H^{s}}^{2})\nonumber \\&\quad \le C(1+\Vert u^{N}\Vert _{H^{s}}+\Vert w^{N}\Vert _{H^{s}}) (\Vert u^{N}\Vert _{H^{s}}^{2}+\Vert w^{N}\Vert _{H^{s}}^{2}), \end{aligned}$$

(C.3)

where we use the fact that

$$\begin{aligned} \Vert \nabla f\Vert _{L^{\infty }({\mathbb {R}}^{3})}\le C\Vert f\Vert _{H^{s}({\mathbb {R}}^{3})},\quad s>\frac{5}{2}. \end{aligned}$$

We assume in (C.3) that $\Vert u^{N}\Vert _{H^{s}}+\Vert w^{N}\Vert _{H^{s}}\ge 1$ since, otherwise, we replace $\Vert u^{N}\Vert _{H^{s}}+\Vert w^{N}\Vert _{H^{s}}$ by $1+\Vert u^{N}\Vert _{H^{s}}+\Vert w^{N}\Vert _{H^{s}}$. Denoting

$$\begin{aligned} X(t):={\Vert u^{N}(t)\Vert _{H^{s}}^{2}+\Vert w^{N}(t)\Vert _{H^{s}}^{2}}, \end{aligned}$$

we get from (C.3) that

$$\begin{aligned} \frac{d}{dt}X(t) \le \kappa X(t)^{\frac{3}{2}}, \end{aligned}$$

where $\kappa >0$ is an absolute constant. By direct calculations, we show that for all N

$$\begin{aligned} \sup _{0\le t\le T}( {\Vert u^{N}(t)\Vert _{H^{s}}^{2}+\Vert w^{N}(t)\Vert _{H^{s}}^{2}}) \le \frac{ {4\Vert u_{0}\Vert _{H^{s}}^{2}+4\Vert w_{0}\Vert _{H^{s}}^{2}}}{\big (2-\kappa T \sqrt{{\Vert u_{0}\Vert _{H^{s}}^{2} +\Vert w_{0}\Vert _{H^{s}}^{2} }}\big )^{2}}, \end{aligned}$$

where $T>0$ satisfies

$$\begin{aligned} T<\frac{2}{\kappa \sqrt{\Vert u_{0}\Vert _{{H}^{s}}^{2}+\Vert w_{0}\Vert _{H^{s}}^{2}}}. \end{aligned}$$

As a result, the family $(u^{N},w^{N})$ is uniformly bounded in $C([0, T]; H^{s})$ with $s>\frac{5}{2}$. We can also show that

$$\begin{aligned} \partial _{t}u^{N},\,\,\partial _{t}w^{N}\in L_{t}^{\infty } ([0, T]);\,H_{x}^{-\vartheta }({\mathbb {R}}^{3})\quad \text{ for } \text{ some } \,\, \vartheta \ge 2. \end{aligned}$$

As the embedding $L^{2}\hookrightarrow H^{-\vartheta }$ is locally compact, the well-known Aubin-Lions argument allows us to conclude that a subsequence $(u^{N},w^{N})_{N\in {\mathbb {N}}}$ satisfies, on any compact subset of ${\mathbb {R}}^3$,

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

Noticing that $\Vert f\Vert _{H^{s'}}\le C \Vert f\Vert _{L^{2}}^{1-\frac{s'}{s}}\Vert f\Vert _{H^{s}}^{\frac{s'}{s}}$ for $s>s'$, we have

$$\begin{aligned} \Vert u^{N}-u^{N'}\Vert _{H^{s'}}\rightarrow 0,\quad \Vert w^{N}-w^{N'}\Vert _{H^{s'}} \rightarrow 0,\quad as\quad N,\,\,N'\rightarrow \infty , \end{aligned}$$

which imply that we have strong convergence limit $(u, w)\in C([0, T]; H^{s'}({\mathbb {R}}^{3}))$ for any $s'<s$. Therefore, this is enough for us to show that up to extraction, sequence $(u^{N},w^{N})_{N\in {\mathbb {N}}}$ has a limit $(u,\,w)$ satisfying

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t u+{\mathcal {P}}(u\cdot \nabla ) u +\Lambda ^{2\alpha } u = {\mathcal {P}}\nabla \times w,\\ \partial _tw + (u\cdot \nabla ) w +2 w+\Lambda ^{2\beta }w = \nabla \times u+ \nabla \nabla \cdot w,\\ \nabla \cdot u=0,\\ u(x,0)=u_{0}(x),\quad w(x,0)=w_{0}(x). \end{array}\right. \end{aligned}$$

(C.4)

Furthermore, it is not hard to check that $(u, w)\in L^{\infty }([0, T]; H^{s}({\mathbb {R}}^{3}))$. Finally, we claim that $(u, w)\in C([0, T]; H^{s}({\mathbb {R}}^{3}))$. It suffices to consider $u\in C([0, T]; H^{s}({\mathbb {R}}^{3})$ as the same fashion can be applied to w to obtain the desired result. First, one has

$$\begin{aligned} \sup _{0\le t\le T}(\Vert u\Vert _{H^{s}}+\Vert w\Vert _{H^{s}})\le C(T)<\infty . \end{aligned}$$

By the equivalent norm, we get

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

(C.5)

Let $\varepsilon >0$ be arbitrarily small. Thanks to $u\in L^{\infty }([0, T]; H^{s}({\mathbb {R}}^{3}))$, there exists an integer $N=N(\varepsilon )>0$ such that

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

(C.6)

Appealing to (C.4)${}_{1}$ implies

$$\begin{aligned} \Delta _{k}u(t_{1})-\Delta _{k}u(t_{2})&= \int _{t_{1}}^{t_{2}}{\frac{d}{d\tau } \Delta _{k}u(\tau )\,d\tau }\\&= -\int _{t_{1}}^{t_{2}}{ \Delta _{k}{\mathcal {P}}[\nabla \times w+(u\cdot \nabla ) u+ \Lambda ^{2\alpha } u](\tau )\,d\tau }. \end{aligned}$$

This allows us to derive

$$\begin{aligned}&\sum _{k<N} 2^{2ks}\Vert \Delta _{k}u(t_{1})-\Delta _{k}u(t_{2})\Vert _{L^{2}}^{2}\\&\quad = \sum _{k<N} 2^{2ks}\Big (\Big \Vert \int _{t_{1}}^{t_{2}}{ \Delta _{k}{\mathcal {P}}[\nabla \times w+(u\cdot \nabla ) u+\Lambda ^{2\alpha } u](\tau )\,d\tau }\Big \Vert _{L^{2}}\Big )^{2}\\&\quad \le \sum _{k<N} 2^{2ks}\Big (\int _{t_{1}}^{t_{2}}{ \Vert \Delta _{k}[\nabla \times w+(u\cdot \nabla ) u+\Lambda ^{2\alpha } u]\Vert _{L^{2}}(\tau )\,d\tau }\Big )^{2}\\&\quad \le \sum _{k<N} 2^{2ks}\Big (\int _{t_{1}}^{t_{2}}{ [\Vert \Delta _{k}\nabla \times w \Vert _{L^{2}}+ \Vert \Vert \Delta _{k}(u\cdot \nabla u)\Vert _{L^{2}}+ \Vert \Vert \Delta _{k}\Lambda ^{2\alpha }u\Vert _{L^{2}}](\tau )\,d\tau }\Big )^{2} \\&\quad = \sum _{k<N} 2^{2k}\Big (\int _{t_{1}}^{t_{2}}{ 2^{k(s-1)}\Vert \Delta _{k}\nabla \times w (\tau )\Vert _{L^{2}}\,d\tau }\Big )^{2}\\&\quad \quad + \sum _{k<N} 2^{2k}\Big (\int _{t_{1}}^{t_{2}}{ 2^{k(s-1)}\Vert \Delta _{k}\nabla \cdot (u\otimes u)(\tau )\Vert _{L^{2}} \,d\tau }\Big )^{2}\\&\quad \quad + \sum _{k<N} 2^{4\alpha k}\Big (\int _{t_{1}}^{t_{2}}{ 2^{ks}\Vert \Delta _{k}u(\tau )\Vert _{L^{2}}\,d\tau }\Big )^{2}\\&\quad \le C\sum _{k<N} 2^{2k}\Big (\Vert w\Vert _{L_{t}^{\infty }H^{s}}^{2}|t_{1}-t_{2}|^{2} +\Vert uu\Vert _{L_{t}^{\infty }H^{s}}^{2}|t_{1}-t_{2}|^{2} \Big )\\&\quad \quad +C\sum _{k<N} 2^{4\alpha k} |t_{1}-t_{2}|^{2} \Vert u\Vert _{L_{t}^{\infty }H^{s}}^{2} \\&\quad \le C\sum _{k<N} 2^{2k}|t_{1}-t_{2}|^{2}\Big (\Vert w\Vert _{L_{t}^{\infty }H^{s}}^{2}+ \Vert u\Vert _{L_{t}^{\infty }L^{\infty }}^{2}\Vert u\Vert _{L_{t}^{\infty }H^{s}}^{2}\Big )\\&\quad \quad +C\sum _{k<N} 2^{4\alpha k} |t_{1}-t_{2}|^{2} \Vert u\Vert _{L_{t}^{\infty }H^{s}}^{2} \\&\quad \le C 2^{2N}|t_{1}-t_{2}|^{2}\Big (\Vert w\Vert _{L_{t}^{\infty }H^{s}}^{2}+\Vert u\Vert _{L_{t}^{\infty } H^{s}}^{4} \Big )+C2^{4\alpha N} |t_{1}-t_{2}|^{2} \Vert u\Vert _{L_{t}^{\infty }H^{s}}^{2}, \end{aligned}$$

which implies

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

(C.7)

provided that $|t_{1}-t_{2}|$ is small enough. The desired $u\in C([0, T]; H^{s}({\mathbb {R}}^{3})$ follows from (C.5), (C.6) and (C.7). Since (u, w) are all in Lipschitz space, the uniqueness follows directly (see the end of Sect. 2). This completes the proof of Proposition C.1. $\square$