1 Introduction

In this paper, we study the Cauchy problem of three-dimensional (3D) Navier–Stokes/Poisson–Nernst–Planck (NSPNP) system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t} u+(u\cdot \nabla ) u-\Delta u+\nabla \Pi =\Delta \varPsi \nabla \varPsi ,\ \ &{} x\in \mathbb {R}^{3},\ t>0,\\ \nabla \cdot u=0,\ \ &{} x\in \mathbb {R}^{3},\ t>0,\\ \partial _{t} v+(u\cdot \nabla ) v=\nabla \cdot (\nabla v-v\nabla \varPsi ),\ \ &{} x\in \mathbb {R}^{3},\ t>0,\\ \partial _{t} w+(u\cdot \nabla ) w=\nabla \cdot (\nabla w+w\nabla \varPsi ),\ \ &{} x\in \mathbb {R}^{3},\ t>0,\\ \Delta \varPsi =v-w,\ \ &{} x\in \mathbb {R}^{3},\ t>0,\\ (u, v, w)|_{t=0}=(u_0, v_0, w_0), \ \ &{} x\in \mathbb {R}^{3}, \end{array}\right. } \end{aligned}$$
(1.1)

where the unknown functions u, \(\Pi \), \(\varPsi \), v and w denote the velocity vector field, the scalar pressure, the electrostatic potential, the densities of binary diffuse negative and positive charges, respectively. For the sake of simplicity of presentation, in this paper, we have assumed that the fluid density, viscosity, charge mobility and dielectric constant are unity.

When formally setting \(u=0\), Eqs. (1.1)\(_{3}\)–(1.1)\(_{5}\) are known as the Poisson–Nernst–Planck equations, which was formulated by W. Nernst and M. Planck at the end of the nineteenth century as a basic model for the diffusion of ions in an electrolytes [3, 6]. It is also referred as the van Roosbroeck system in semiconductor devices [14, 25], as the drift-diffusion Poisson system in plasma physics [2, 15] and as the chemotaxis model in biology [5]. On the other hand, if the flow is charge-free (i.e., \(v=w=\varPsi =0\)), then Eqs. (1.1)\(_{1}\)–(1.1)\(_{2}\) are known as the conventional Navier–Stokes equations of incompressible flow.

In a nanoscopic fluid-dynamical view of electro-hydrodynamics, the NSPNP system (1.1) was first proposed by Rubinstein [22] to model electro-kinetic fluids by describing the dynamic coupling between incompressible flows and electric charges. To best of our knowledge, mathematical analysis of the NSPNP system (1.1) was initiated by Jerome [16], where the local smooth theory has been established under the Kato’s semigroup framework, we refer to [8, 10, 17, 29,30,31,32] for more results concerning about the global existence, uniqueness, regularity of strong solutions and asymptotic stability of self-similar solutions and other related topics in various scaling invariant spaces.

The global existence of weak solution to various initial/boundary-value problems of the 3D NSPNP system (1.1) has already been established, e.g., we refer the readers to see mixed Dirichlet boundary condition [18], no-flux boundary condition [23] and Neumann boundary condition [24]. However, similar to the 3D incompressible Navier–Stokes equations, the global regularity of weak solution is still open, which in this paper we aim to study. In [12], the authors proved that if the velocity field u satisfies

$$\begin{aligned} \int _{0}^{T}\Vert u(\cdot ,t)\Vert _{L^{p}}^{q}\mathrm{d}t<\infty \ \ \ \text {with}\ \ \ \frac{2}{q}+\frac{3}{p}=1\ \ \ \text {and} \ \ \ 3<p\le \infty , \end{aligned}$$
(1.2)

or

$$\begin{aligned} \int _{0}^{T}\Vert \nabla u(\cdot ,t)\Vert _{L^{p}}^{q}\mathrm{d}t<\infty \ \text { with }\ \frac{2}{q}+\frac{3}{p}= 2\ \ \text { and }\ \ \frac{3}{2}<p\le \infty , \end{aligned}$$
(1.3)

then the weak solution (u, v, w) of (1.1) is regular on (0, T]. Recently, in [13], the authors established the following Beale–Kato–Majda type regularity criterion:

$$\begin{aligned} \int _{0}^{T}\Vert \omega (\cdot ,t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}\mathrm{d}t<\infty , \end{aligned}$$
(1.4)

where \(\omega :=\nabla \times u\) is the vorticity field, and \(\dot{B}^{0}_{\infty ,\infty }(\mathbb {R}^3)\) is the homogeneous Besov space.

The main results in this paper now read:

Theorem 1.1

Let (u, v, w) be a local smooth solution to the NSPNP system (1.1) with initial data \(u_{0}\in H^{3}(\mathbb {R}^{3})\) and \(\nabla \cdot u_{0}=0\), \((v_{0}, w_{0})\in L^{1}(\mathbb {R}^{3})\cap H^{2}(\mathbb {R}^{3})\) and \(v_{0}, w_{0}\ge 0\). Suppose that the vorticity field \(\omega \) satisfies

$$\begin{aligned} \int _{0}^{T}\frac{\Vert \omega (\cdot ,t)\Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}^{\frac{2}{2-\alpha }}}{1+\ln \left( e+\Vert \omega (\cdot ,t)\Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}\right) }\mathrm{d}t<\infty \ \ \ \mathrm{for}\ \ \ 0<\alpha <2. \end{aligned}$$
(1.5)

Then the solution (u, v, w) is smooth up to time T. As a consequence, if we denote by \(T_{*}<\infty \) the maximal existence time of the solution (u, v, w), then for any \(0<\alpha <2\) we have

$$\begin{aligned} \int _{0}^{T_{*}}\frac{\Vert \omega (\cdot ,t)\Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}^{\frac{2}{2-\alpha }}}{1+\ln \left( e+\Vert \omega (\cdot ,t)\Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}\right) }\mathrm{d}t =\infty . \end{aligned}$$

In the marginal case \(\alpha =0\), we have the follow regularity result.

Theorem 1.2

Let (u, v, w) be a local smooth solution to the NSPNP system (1.1) with initial data \(u_{0}\in H^{3}(\mathbb {R}^{3})\) and \(\nabla \cdot u_{0}=0\), \((v_{0}, w_{0})\in L^{1}(\mathbb {R}^{3})\cap H^{2}(\mathbb {R}^{3})\) and \(v_{0}, w_{0}\ge 0\). Suppose that the vorticity field \(\omega \) satisfies

$$\begin{aligned} \int _{0}^{T}\frac{\Vert \omega (\cdot ,t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}}{\sqrt{1+\ln \left( e+\Vert \omega (\cdot ,t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}\right) }}\mathrm{d}t<\infty . \end{aligned}$$
(1.6)

Then the local smooth solution (u, v, w) is smooth up to time T. As a consequence, if we denote by \(T_{*}<\infty \) the maximal existence time of the solution (u, v, w), then we have

$$\begin{aligned} \int _{0}^{T_{*}}\frac{\Vert \omega (\cdot ,t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}}{\sqrt{1+\ln \left( e+\Vert \omega (\cdot ,t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}\right) }}\mathrm{d}t=\infty . \end{aligned}$$

In the marginal case \(\alpha =2\), owing to the fact that

$$\begin{aligned} \Vert \omega \Vert _{\dot{B}^{-2}_{\infty ,\infty }}\approx \Vert \nabla u\Vert _{\dot{B}^{-2}_{\infty ,\infty }}\approx \Vert u\Vert _{\dot{B}^{-1}_{\infty ,\infty }}, \end{aligned}$$

we have the following regularity result.

Theorem 1.3

Let (u, v, w) be a local smooth solution to the NSPNP system (1.1) with initial data \(u_{0}\in H^{3}(\mathbb {R}^{3})\) and \(\nabla \cdot u_{0}=0\), \((v_{0}, w_{0})\in L^{1}(\mathbb {R}^{3})\cap H^{2}(\mathbb {R}^{3})\) and \(v_{0}, w_{0}\ge 0\). There exists a small positive constant \(\varepsilon \) such that if

$$\begin{aligned} \sup _{0\le t\le T}\Vert u(\cdot ,t)\Vert _{\dot{B}^{-1}_{\infty ,\infty }}\le \varepsilon , \end{aligned}$$
(1.7)

then the local smooth solution (u, v, w) is smooth up to time T.

Remark 1.1

It should be noticed that, for the 3D Navier–Stokes equations, Yuan and Zhang [26] first derived the regularity criterion

$$\begin{aligned} \omega \in L^{\frac{2}{2-\alpha }}\left( 0,T; \dot{B}^{-\alpha }_{\infty ,\infty }\left( \mathbb {R}^{3}\right) \right) \cap L^{\frac{2}{1-\alpha }}\left( 0,T; \dot{B}^{-1-\alpha }_{\infty ,\infty }\left( \mathbb {R}^{3}\right) \right) \end{aligned}$$

for all \(0<\alpha <1\). Zhang and Yang [28] recently improved the above regularity criterion to the final form (1.5). Moreover, the regularity criterion (1.6) was first established in [11], and (1.7) in [4]. In this paper we intend to generalize these regularity criteria in Theorems 1.1–1.3 to the more complicated coupled NSPNP system (1.1). We refer to [9, 27, 33, 34] for further studies on regularity criterion issues for the Navier–Stokes equations and other related equations.

Remark 1.2

In [7], the authors proved that, for the 3D Navier–Stokes equations, if the weak solution u belongs to \( L^{\infty }(0,T;L^{3}(\mathbb {R}^{3}))\); then, it is actually smooth up to time T. Compared this result with (1.7), the smallness condition was additionally imposed on u due to the functions in \(\dot{B}^{-1}_{\infty ,\infty }(\mathbb {R}^{3})\) have no decay at infinity; thus, the backward uniqueness theorem cannot be applied.

Remark 1.3

It is clear that (1.6) is a logarithmically improved regularity criterion of (1.4).

Remark 1.4

For simplicity, we just consider initial data \(u_{0}\in H^{3}(\mathbb {R}^{3})\), \((v_{0},w_{0}) \in H^{2}(\mathbb {R}^{3})\). As a matter of fact, one can also prove the same results for initial data \(u_{0}\in H^{s}(\mathbb {R}^{3})\), \((v_{0},w_{0}) \in H^{s-1}(\mathbb {R}^{3})\) with \(s>\frac{5}{2}\).

At the end of this section, we introduce the structure of this paper. In Sect. 2, we first present the Littlewood–Paley decomposition theory and basic functional setting; then, we give some analytical tools frequently used in the proofs of main results. In Sect. 3, we examine a priori estimates of local smooth solutions, and prove Theorems 1.1–1.3 in Sect. 4. Throughout the paper, we denote by C the harmless positive constants, which may depend on initial data and its value may change from line to line, the special dependence will be pointed out explicitly in the text if necessary.

2 Preliminaries

We first recall some basic notions and preliminary results used in the proofs of Theorems 1.1–1.3. Let \(\mathcal {S}(\mathbb {R}^{3})\) be the Schwartz class of rapidly decreasing function, and \(\mathcal {S}'(\mathbb {R}^{3})\) the space of all tempered distributions on \(\mathbb {R}^{3}\), given \(f\in \mathcal {S}(\mathbb {R}^{3})\), its Fourier transformation \(\mathcal {F}(f)\) or \(\widehat{f}\) is defined by

$$\begin{aligned} \mathcal {F}(f)(\xi )=\widehat{f}(\xi ):=\frac{1}{(2\pi )^{\frac{3}{2}}}\int _{\mathbb {R}^{3}}f(x)e^{-ix\cdot \xi }\mathrm{d}x. \end{aligned}$$

More generally, the Fourier transform of a tempered distribution \(f\in \mathcal {S}'(\mathbb {R}^{3})\) is defined by the dual argument in the standard way.

Let \(\mathcal {D}_{1}:=\{\xi \in \mathbb {R}^{3},\ |\xi |\le \frac{4}{3}\}\) and \(\mathcal {D}_{2}:=\{\xi \in \mathbb {R}^{3},\ \frac{3}{4}\le |\xi |\le \frac{8}{3}\}\). Choose two nonnegative radial functions \(\phi , \psi \in \mathcal {S}(\mathbb {R}^{3})\) supported, respectively, in \(\mathcal {D}_{1}\) and \(\mathcal {D}_{2}\) such that

$$\begin{aligned}&\psi (\xi )+\sum _{j\ge 0}\phi (2^{-j}\xi )=1, \quad \xi \in \mathbb {R}^{3},\\&\sum _{j\in \mathbb {Z}}\phi (2^{-j}\xi )=1, \quad \xi \in \mathbb {R}^{3}\backslash \{0\}. \end{aligned}$$

Let \(h:=\mathcal {F}^{-1}\phi \) and \(\tilde{h}:=\mathcal {F}^{-1}\psi \), where \(\mathcal {F}^{-1}\) is the inverse Fourier transform. Then we define the dyadic blocks \(\Delta _{j}\) and \(S_{j}\) as follows:

$$\begin{aligned} \Delta _{j}f:=\phi (2^{-j}D)f=2^{3j}\int _{\mathbb {R}^{3}}h(2^{j}y)f(x-y)\mathrm{d}y,\\ S_{j}f:=\psi (2^{-j}D)f=2^{3j}\int _{\mathbb {R}^{3}}\tilde{h}(2^{j}y)f(x-y)\mathrm{d}y. \end{aligned}$$

Informally, \(\Delta _{j}\) is a frequency projection to the annulus \(\{|\xi |\sim 2^{j}\}\), while \(S_{j}\) is a frequency projection to the ball \(\{|\xi |\le 2^{j}\}\).

Let \(\mathcal {P}(\mathbb {R}^{3})\) be the class of all polynomials on \(\mathbb {R}^{3}\) and denote by \(\mathcal {S}'_{h}(\mathbb {R}^{3}):=\mathcal {S}'(\mathbb {R}^{3})/\mathcal {P}(\mathbb {R}^{3})\) the tempered distributions modulo polynomials. By telescoping the series, for any \(f\in \mathcal {S}'_{h}(\mathbb {R}^{3})\), one has the following Littlewood–Paley decomposition:

$$\begin{aligned} f=\sum _{j\in \mathbb {Z}}\Delta _{j}f. \end{aligned}$$

Moreover, from the Young inequality, we have the following classical Bernstein inequality.

Lemma 2.1

[1] For any nonnegative integer k and any couple of real numbers (p, q) with \(1\le p\le q\le \infty \), we have

$$\begin{aligned} \sup _{|\alpha |=k}\Vert \partial ^{\alpha }\Delta _{j}f\Vert _{L^{q}}\le C2^{jk+3j(\frac{1}{p}-\frac{1}{q})}\Vert \Delta _{j}f\Vert _{L^{p}}, \end{aligned}$$
(2.1)

where C being a positive constant independent of f and j.

Next we give the definition of the homogeneous Besov space.

Definition 2.2

For \(s\in \mathbb {R}\), \(1\le p,r\le \infty \), the homogeneous Besov space is defined by

$$\begin{aligned} \dot{B}^{s}_{p,r}(\mathbb {R}^{3}):=\Big \{f\in \mathcal {S}'_{h}(\mathbb {R}^{3}):\ \ \Vert f\Vert _{\dot{B}^{s}_{p,r}}<\infty \Big \}, \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{\dot{B}^{s}_{p,r}}:= {\left\{ \begin{array}{ll} \left( \sum _{j\in \mathbb {Z}}2^{jsr}\Vert \Delta _{j}f\Vert _{L^{p}}^{r}\right) ^{\frac{1}{r}} \ \ &{}\text {for}\ \ 1\le r<\infty ,\\ \sup _{j\in \mathbb {Z}}2^{js}\Vert \Delta _{j}f\Vert _{L^{p}}\ \ &{}\text {for}\ \ r=\infty . \end{array}\right. } \end{aligned}$$

Notice that if we denote \(D^s f=\mathcal {F}^{-1}(|\xi |^{s}\mathcal {F}(f))\), then for any function f defined on \(\mathbb {R}^{3}\backslash \{0\}\) which is smooth and homogeneous of degree k, the corresponding pseudo-differential operator f(D) is a bounded linear map from \(\dot{B}^{s}_{p,r}(\mathbb {R}^{3})\) to \(\dot{B}^{s-k}_{p,r}(\mathbb {R}^{3})\). Besides, the classical homogeneous Sobolev space \(\dot{H}^{s}(\mathbb {R}^{3})\) can be characterized by the homogeneous Besov space \(\dot{B}^{s}_{2,2}(\mathbb {R}^{3})\) equipped with an equivalent norms.

Now we present some analytical lemmas which play an important roles in Sect. 3.

Lemma 2.3

[1] Let \(\alpha >0\) and \(1\le q<p<\infty \). Then there exists a constant C depending only on \(\alpha \), \(\beta \), p and q such that

$$\begin{aligned} \Vert f\Vert _{L^{p}}\le C\Vert f\Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}^{1-\theta }\Vert f\Vert _{\dot{B}^{\beta }_{q,q}}^{\theta } \ \ \ \text {with}\ \ \ \beta =\alpha \left( \frac{p}{q}-1 \right) \ \ \ \text {and}\ \ \ \theta =\frac{q}{p} \end{aligned}$$
(2.2)

holds for all \(f\in \dot{B}^{-\alpha }_{\infty ,\infty }(\mathbb {R}^{3})\cap \dot{B}^{\beta }_{q,q}(\mathbb {R}^{3})\).

The proof (2.2) can be found in [1] (see Theorem 2.42). Especially, in Sect. 3, we shall use the following specific case of (2.2) (by taking \(p=3\), \(q=2\) and using the fact \(\dot{H}^{\beta }(\mathbb {R}^{3})= \dot{B}^{\beta }_{2,2}(\mathbb {R}^{3})\)):

$$\begin{aligned} \Vert f\Vert _{L^{3}}\le C\Vert f\Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}^{\frac{1}{3}}\Vert f\Vert _{\dot{H}^{\frac{\alpha }{2}}}^{\frac{2}{3}}\ \ \text {with}\ \ \alpha >0. \end{aligned}$$
(2.3)

Lemma 2.4

[19] Let \(s>1\). Then we have

$$\begin{aligned} \Vert \nabla ^{s}(fg)-f\nabla ^{s}g\Vert _{L^{p}}\le C\big (\Vert \nabla f\Vert _{L^{p_{1}}}\Vert \nabla ^{s-1}g\Vert _{L^{q_{1}}}+\Vert \nabla ^{s}f\Vert _{L^{p_{2}}}\Vert g\Vert _{L^{q_{2}}}\big ), \end{aligned}$$
(2.4)

where \(1<p, q_{1}\), \(p_{2}<\infty \) and \(\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{q_{1}}=\frac{1}{p_{2}}+\frac{1}{q_{2}}\).

Lemma 2.5

[21] Let \(1<p<\infty \). Then there exists a constant C depending only on p such that

$$\begin{aligned} \Vert fg\Vert _{L^{p}} \le C\big (\Vert f\Vert _{L^{p}}\Vert g\Vert _{BMO}+\Vert f\Vert _{BMO}\Vert g\Vert _{L^{p}}\big ) \end{aligned}$$
(2.5)

holds for all \(f,g \in BMO\cap L^{p}(\mathbb {R}^{3})\).

We shall use the particular form of (2.5) in Sect. 3 by taking \(f=g\) and \(p=2\):

$$\begin{aligned} \Vert f\Vert _{L^{4}}=\Vert f^{2}\Vert _{L^{2}}^{\frac{1}{2}} \le C\Vert f\Vert _{L^{2}}^{\frac{1}{2}}\Vert f\Vert _{BMO}^{\frac{1}{2}}. \end{aligned}$$
(2.6)

Lemma 2.6

[20] For all \(f\in H^{s-1}(\mathbb {R}^{3})\) with \(s>\frac{5}{2}\), we have

$$\begin{aligned} \Vert f\Vert _{BMO}\le C\big (1+\Vert f\Vert _{\dot{B}^{0}_{\infty ,\infty }}\ln ^{\frac{1}{2}}(e+\Vert f\Vert _{H^{s-1}})\big ). \end{aligned}$$
(2.7)

Actually, the original form of (2.7) in [20] (see Theorem 2.1) is

$$\begin{aligned} \Vert f\Vert _{\dot{B}^{0}_{\infty ,2}}\le C\big (1+\Vert f\Vert _{\dot{B}^{0}_{\infty ,\infty }}\ln ^{\frac{1}{2}}(e+\Vert f\Vert _{H^{s-1}})\big ), \end{aligned}$$

which combining the Sobolev embedding relation \(\dot{B}^{0}_{\infty ,2}(\mathbb {R}^{3})\hookrightarrow BMO\) implies (2.7) immediately.

Lemma 2.7

Let u be the velocity field and \(\omega =\nabla \times u\) the vorticity. Then for any \(0\le \alpha <2\), we have

$$\begin{aligned} \Vert \omega \Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}\le C(\Vert u\Vert _{L^{2}}+\Vert \nabla ^{3} u\Vert _{L^{2}}). \end{aligned}$$
(2.8)

Proof

Thanks to the classical Bernstein inequality (2.1) in Lemma 2.1, we have for all \(0\le \alpha <2\),

$$\begin{aligned} \Vert \omega \Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}&\le \sup _{j\le -1}2^{-\alpha j}\Vert \Delta _{j}\nabla \times u\Vert _{L^{\infty }}+\sup _{j\ge 0}2^{-\alpha j}\Vert \Delta _{j}\nabla \times u\Vert _{L^{\infty }}\\&\le C\sup _{j\le -1}2^{-\alpha j}2^{\frac{5}{2}j}\Vert \Delta _{j}u\Vert _{L^{2}}+C\sup _{j\ge 0}2^{-\alpha j}2^{\frac{5}{2}j}2^{-3j}\Vert \Delta _{j}\nabla ^{3} u\Vert _{L^{2}}\\&\le C\sup _{j\le -1}2^{\left( \frac{5}{2}-\alpha \right) j}\Vert u\Vert _{L^{2}}+C\sup _{j\ge 0}2^{-(\frac{1}{2}+\alpha ) j}\Vert \nabla ^{3} u\Vert _{L^{2}}\\&\le C\left( \Vert u\Vert _{L^{2}}+\Vert \nabla ^{3} u\Vert _{L^{2}}\right) . \end{aligned}$$

The proof of Lemma 2.7 is achieved. \(\square \)

3 A Priori Estimates

The proofs of Theorems 1.1–1.3 are based on the a priori estimates for local smooth solutions of the system (1.1) described in the following.

Lemma 3.1

Let \(u_{0}\in H^{3}(\mathbb {R}^{3})\) with \(\nabla \cdot u_{0}=0\), \(v_{0}, w_{0}\in L^{1}(\mathbb {R}^{3})\cap H^{2}(\mathbb {R}^{3})\) with \(v_{0}, w_{0}\ge 0\). Assume that (u, v, w) is the corresponding local smooth solution of (1.1) on \(\mathbb {R}^{3}\times [0,T)\) and satisfies the condition (1.5). Then

$$\begin{aligned} \sup _{0\le t<T}\left( \Vert u(\cdot ,t)\Vert _{H^{3}}+\Vert \left( v(\cdot ,t),w(\cdot ,t)\right) \Vert _{H^{2}}\right) \le C, \end{aligned}$$
(3.1)

where C is a constant depending on the bound of the left hand side of (1.5), \(\Vert u_{0}\Vert _{H^{3}}\), \(\Vert (v_{0}, w_{0})\Vert _{L^{1}\cap H^{2}}\) and T.

Proof

By the maximum principle, we deduce that if \(v_0\) and \(w_0\) are nonnegative, then v and w are also nonnegative, see [24] for more details. Moreover, the fundamental energy inequalities have already been established, e.g., see [24] and [30], thus we have for all \(0\le t<T\),

$$\begin{aligned}&\Vert v(t)\Vert _{L^{2}}^{2}+\Vert w(t)\Vert _{L^{2}}^{2}+2\int _{0}^{t}\big (\Vert \nabla v(\tau )\Vert _{L^{2}}^{2}+\Vert \nabla w(\tau )\Vert _{L^{2}}^{2}\big )d\tau \le \Vert v_{0}\Vert _{L^{2}}^{2}+\Vert w_{0}\Vert _{L^{2}}^{2},\nonumber \\ \end{aligned}$$
(3.2)
$$\begin{aligned}&\Vert u(t)\Vert _{L^{2}}^{2}+\Vert \nabla \varPsi (t)\Vert _{L^{2}}^{2}+2\int _{0}^{t}\big (\Vert \nabla u(\tau )\Vert _{L^{2}}^{2}+\Vert \Delta \varPsi (\tau )\Vert _{L^{2}}^{2}\big )d\tau \le C, \end{aligned}$$
(3.3)

where C is a constant depending only on \(\Vert u_{0}\Vert _{L^{2}}\) and \(\Vert (v_{0}, w_{0})\Vert _{L^{1}\cap L^{2}}\).

Next we derive the desired estimate for the vorticity \(\omega \). Taking the curl \(\nabla \times \) on (1.1)\(_{1}\), and taking the inner product of the resulting equations with \(\omega \), after integrating by parts, one shows that

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Vert \omega \Vert _{L^{2}}^{2}+\Vert \nabla \omega \Vert _{L^{2}}^{2}&=\int _{\mathbb {R}^{3}}(\omega \cdot \nabla )u\cdot \omega \mathrm{d}x-\int _{\mathbb {R}^{3}}(\Delta \varPsi \nabla \varPsi )\cdot (\nabla \times \omega )\mathrm{d}x. \end{aligned}$$
(3.4)

Notice that, via the Biot–Savart law, we have for any \(1<p<\infty \),

$$\begin{aligned} \Vert \nabla u\Vert _{L^{p}}\le C\Vert \omega \Vert _{L^{p}}\ \ \text {and}\ \ \Vert \Delta u\Vert _{L^{2}}\le C\Vert \nabla \omega \Vert _{L^{2}}. \end{aligned}$$

Thus applying (2.3), (3.2), (3.3) and (1.1)\(_5\), the right-hand side of (3.4) can be majorized by

$$\begin{aligned}&\int _{\mathbb {R}^{3}}(\omega \cdot \nabla )u\cdot \omega \mathrm{d}x \le C\Vert \omega \Vert _{L^{3}}^{3}\nonumber \\&\quad \le C\Vert \omega \Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}\Vert \omega \Vert _{\dot{H}^{\frac{\alpha }{2}}}^{2}\nonumber \\&\quad \le C\Vert \omega \Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}\Vert \omega \Vert _{L^{2}}^{2-\alpha }\Vert \nabla \omega \Vert _{L^{2}}^{\alpha }\nonumber \\&\quad \le \frac{1}{4}\Vert \nabla \omega \Vert _{L^{2}}^{2}+C\Vert \omega \Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}^{\frac{2}{2-\alpha }}\Vert \omega \Vert _{L^{2}}^{2}, \end{aligned}$$
(3.5)
$$\begin{aligned}&-\int _{\mathbb {R}^{3}}(\Delta \varPsi \nabla \varPsi )\cdot (\nabla \times \omega )\mathrm{d}x \le C \Vert \nabla \varPsi \Vert _{L^{4}}\Vert \Delta \varPsi \Vert _{L^{4}}\Vert \nabla \omega \Vert _{L^{2}}\nonumber \\&\quad \le \frac{1}{4}\Vert \nabla \omega \Vert _{L^{2}}^{2}+C\Vert (v,w)\Vert _{L^{4}}^{2}\Vert \nabla \varPsi \Vert _{L^{4}}^{2}\nonumber \\&\quad \le \frac{1}{4}\Vert \nabla \omega \Vert _{L^{2}}^{2}+C\Vert (v,w)\Vert _{L^{2}}^{\frac{1}{2}}\Vert (\nabla v,\nabla w)\Vert _{L^{2}}^{\frac{3}{2}}\Vert \nabla \varPsi \Vert _{L^{2}}^{\frac{1}{2}}\Vert \Delta \varPsi \Vert _{L^{2}}^{\frac{3}{2}}\nonumber \\&\quad \le \frac{1}{4}\Vert \nabla \omega \Vert _{L^{2}}^{2}+C\Vert (v,w)\Vert _{L^{2}}^{2}\Vert (\nabla v,\nabla w)\Vert _{L^{2}}^{\frac{3}{2}}\Vert \nabla \varPsi \Vert _{L^{2}}^{\frac{1}{2}}\nonumber \\&\quad \le \frac{1}{4}\Vert \nabla \omega \Vert _{L^{2}}^{2}+C\Vert (v,w)\Vert _{L^{2}}^{2}\Vert (\nabla v,\nabla w)\Vert _{L^{2}}^{2}+C\Vert (v,w)\Vert _{L^{2}}^{2}\Vert \nabla \varPsi \Vert _{L^{2}}^{2}\nonumber \\&\quad \le \frac{1}{4}\Vert \nabla \omega \Vert _{L^{2}}^{2}+C(\Vert (\nabla v,\nabla w)\Vert _{L^{2}}^{2}+1), \end{aligned}$$
(3.6)

where we have employed the following interpolation inequalities (\(0<\alpha <2\)):

$$\begin{aligned} \Vert f\Vert _{\dot{H}^{\frac{\alpha }{2}}}\le \Vert f\Vert _{L^{2}}^{1-\frac{\alpha }{2}}\Vert \nabla f\Vert _{L^{2}}^{\frac{\alpha }{2}} \ \ \ \text {and}\ \ \ \Vert f\Vert _{L^{4}}\le \Vert f\Vert _{L^{2}}^{\frac{1}{4}}\Vert \nabla f\Vert _{L^{2}}^{\frac{3}{4}}. \end{aligned}$$

Plugging (3.5) and (3.6) into (3.4), and using (2.8) in Lemma 2.7, we deduce that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Vert \omega \Vert _{L^{2}}^{2}&+\Vert \nabla \omega \Vert _{L^{2}}^{2} \le C\big (1+\Vert (\nabla v,\nabla w)\Vert _{L^{2}}^{2}+\Vert \omega \Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}^{\frac{2}{2-\alpha }}\big )\big (\Vert \omega \Vert _{L^{2}}^{2}+1\big )\nonumber \\&\le C\Big (1+\Vert (\nabla v,\nabla w)\Vert _{L^{2}}^{2}+\frac{\Vert \omega \Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}^{\frac{2}{2-\alpha }}}{1+\ln (e+\Vert \omega \Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }})} \Big )\nonumber \\&\quad \times \Big (1+\ln (e+\Vert \omega \Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }})\Big )\Big (\Vert \omega \Vert _{L^{2}}^{2}+1\Big )\nonumber \\&\le C\Big (1+\Vert (\nabla v,\nabla w)\Vert _{L^{2}}^{2}+\frac{\Vert \omega \Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}^{\frac{2}{2-\alpha }}}{1+\ln (e+\Vert \omega \Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }})} \Big )\nonumber \\&\quad \times \ln \Big (e+\Vert \nabla ^{3} u\Vert _{L^{2}}^{2}+\Vert (\nabla ^{2} v, \nabla ^{2} w)\Vert _{L^{2}}^{2}\Big )\Big (\Vert \omega \Vert _{L^{2}}^{2}+1\Big ). \end{aligned}$$
(3.7)

By the fact (3.2) and the growth condition (1.5), one deduces that for any small constant \(\sigma >0\), there exists \(T_{0}<T\) such that

$$\begin{aligned} \int _{T_{0}}^{T}\big (1+\Vert (\nabla v,\nabla w)\Vert _{L^{2}}^{2}\big )\mathrm{d}t<\sigma \end{aligned}$$

and

$$\begin{aligned} \int _{T_{0}}^{T}\frac{\Vert \omega \Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}^{\frac{2}{2-\alpha }}}{1+\ln (e+\Vert \omega \Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }})}\mathrm{d}t<\sigma . \end{aligned}$$

Therefore, by setting

$$\begin{aligned} Y(t):=\sup _{T_{0}\le \tau \le t}\big (\Vert \nabla ^{3} u(\cdot ,\tau )\Vert _{L^{2}}^{2}+\Vert (\nabla ^{2} v(\cdot ,\tau ), \nabla ^{2} w(\cdot ,\tau ))\Vert _{L^{2}}^{2}\big ), \end{aligned}$$

we integrate (3.7) from \(T_{0}\) to t for any \(T_{0}\le t<T\) to obtain that

$$\begin{aligned} \Vert \omega (t)\Vert _{L^{2}}^{2}&\le C\exp \left( \int _{T_{0}}^{t}C\left( \phantom {\Vert \omega \Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}^{\frac{2}{2-\alpha }}}1+\Vert (\nabla v,\nabla w)\Vert _{L^{2}}^{2} \right. \right. \nonumber \\&\quad \left. \left. +\frac{\Vert \omega \Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}^{\frac{2}{2-\alpha }}}{1+\ln (e+\Vert \omega \Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }})}\right) d\tau \ln (e+Y(t)) \right) \nonumber \\&\le C\exp \big (2C\sigma \ln (e+Y(t))\big )\nonumber \\&\le C(e+Y(t))^{2C\sigma }, \end{aligned}$$
(3.8)

where C is a constant depending on \(\Vert \omega (\cdot ,T_{0})\Vert _{L^{2}}^{2}\).

Finally, it remains to derive the \(H^{3}\times H^{2}\times H^{2}\) estimates of the solution (u, v, w) under the above inequality (3.8). Applying \(\nabla ^{3}\) on (1.1)\(_{1}\), multiplying the resulting equality by \(\nabla ^{3} u\) and integrating over \(\mathbb {R}^{3}\), observing that the pressure \(\Pi \) can be eliminated by the incompressible condition \(\nabla \cdot u=0\), one obtains that

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla ^{3} u\Vert _{L^{2}}^{2}+\Vert \nabla ^{4}u\Vert _{L^{2}}^{2}=&-\int _{\mathbb {R}^{3}}\nabla ^{3}((u\cdot \nabla ) u)\cdot \nabla ^{3} u \mathrm{d}x\nonumber \\&+\int _{\mathbb {R}^{3}}\nabla ^{3}(\Delta \varPsi \nabla \varPsi )\cdot \nabla ^{3} u \mathrm{d}x. \end{aligned}$$
(3.9)

Based on the commutator estimate (2.4) in Lemma 2.4, and the interpolation inequality:

$$\begin{aligned} \Vert \nabla ^{3} u\Vert _{L^{4}}\le C\Vert \nabla u\Vert _{L^{2}}^{\frac{1}{12}}\Vert \nabla ^{4} u\Vert _{L^{2}}^{\frac{11}{12}}, \end{aligned}$$

we employ the Leibniz’s rule to bound the right-hand side of (3.9) as

$$\begin{aligned}&-\int _{\mathbb {R}^{3}}\nabla ^{3}((u\cdot \nabla ) u)\cdot \nabla ^{3} u \mathrm{d}x=-\int _{\mathbb {R}^{3}}\big [\nabla ^{3}((u\cdot \nabla ) u)-(u\cdot \nabla )\nabla ^{3} u\big ]\cdot \nabla ^{3} u \mathrm{d}x\nonumber \\&\quad \le C\Vert \nabla ^{3}((u\cdot \nabla ) u)-(u\cdot \nabla )\nabla ^{3} u\Vert _{L^{\frac{4}{3}}}\Vert \nabla ^{3} u\Vert _{L^{4}}\nonumber \\&\quad \le C\Vert \nabla u\Vert _{L^{2}}\Vert \nabla ^{3} u \Vert _{L^{4}}^{2}\nonumber \\&\quad \le C\Vert \nabla u\Vert _{L^{2}}^{\frac{7}{6}}\Vert \nabla ^{4} u\Vert _{L^{2}}^{\frac{11}{6}}\nonumber \\&\quad \le \frac{1}{4}\Vert \nabla ^{4} u\Vert _{L^{2}}^{2} +C\Vert \nabla u\Vert _{L^{2}}^{14},\\&\int _{\mathbb {R}^{3}}\nabla ^{3}(\Delta \varPsi \nabla \varPsi )\cdot \nabla ^{3} u \mathrm{d}x=-\int _{\mathbb {R}^{3}}\nabla ^{2}((v-w)\nabla \varPsi )\cdot \nabla ^{4}u \mathrm{d}x\nonumber \\&\quad \le \frac{1}{4}\Vert \nabla ^{4}u\Vert _{L^{2}}^{2}+C\Vert \nabla ^{2}((v-w)\nabla \varPsi )\Vert _{L^{2}}^{2}\nonumber \\&\quad \le \frac{1}{4}\Vert \nabla ^{4}u\Vert _{L^{2}}^{2}+C\big (\Vert \nabla ^{2} (v-w)\nabla \varPsi \Vert _{L^{2}}^{2}+2\Vert (\nabla v-\nabla w)\nabla ^{2}\varPsi \Vert _{L^{2}}^{2}\nonumber \\&\qquad +\Vert (v-w)\nabla ^{3}\varPsi \Vert _{L^{2}}^{2}\big )\nonumber \\&\quad \le \frac{1}{4}\Vert \nabla ^{4}u\Vert _{L^{2}}^{2}+C\big (\Vert (\nabla ^{2} v,\nabla ^{2} w)\Vert _{L^{3}}^{2}\Vert \nabla \varPsi \Vert _{L^{6}}^{2}+\Vert (v,w)\Vert _{L^{3}}^{2}\Vert (\nabla v,\nabla w)\Vert _{L^{6}}^{2}\big )\nonumber \\&\quad \le \frac{1}{4}\Vert \nabla ^{4}u\Vert _{L^{2}}^{2}+C\big (\Vert (\nabla ^{2} v, \nabla ^{2} w)\Vert _{L^{2}}\Vert (\nabla ^{3} v, \nabla ^{3} w)\Vert _{L^{2}}\nonumber \\&\qquad +\Vert (\nabla v,\nabla w)\Vert _{L^{2}}\Vert (\nabla ^{2} v, \nabla ^{2} w)\Vert _{L^{2}}^{2}\big )\nonumber \\&\quad \le \frac{1}{4}\Vert \nabla ^{4}u\Vert _{L^{2}}^{2}+\frac{1}{6}\Vert (\nabla ^{3} v, \nabla ^{3} w)\Vert _{L^{2}}^{2}+C\big (1+\Vert (\nabla v, \nabla w)\Vert _{L^{2}}^{2}\big )\Vert (\nabla ^{2} v, \nabla ^{2} w)\Vert _{L^{2}}^{2}. \end{aligned}$$

Therefore we obtain

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla ^{3} u\Vert _{L^{2}}^{2}&+\Vert \nabla ^{4}u\Vert _{L^{2}}^{2}\le \frac{1}{3}\Vert (\nabla ^{3} v, \nabla ^{3} w)\Vert _{L^{2}}^{2}\nonumber \\&+C\Vert \nabla u\Vert _{L^{2}}^{14}+C\big (1+\Vert (\nabla v, \nabla w)\Vert _{L^{2}}^{2}\big )\Vert (\nabla ^{2} v, \nabla ^{2} w)\Vert _{L^{2}}^{2}. \end{aligned}$$
(3.10)

Applying \(\nabla ^{2}\) to (1.1)\(_{3}\), then taking the inner product of the resulting equality by \(\nabla ^{2} v\), after integration by parts, one has

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla ^{2} v\Vert _{L^{2}}^{2}+\Vert \nabla ^{3} v\Vert _{L^{2}}^{2}=&-\int _{\mathbb {R}^{3}}\nabla ^{2}((u\cdot \nabla ) v)\nabla ^{2} v \mathrm{d}x\nonumber \\&-\int _{\mathbb {R}^{3}}\nabla ^{2}\nabla \cdot (v\nabla \varPsi )\nabla ^{2} v \mathrm{d}x. \end{aligned}$$
(3.11)

Based on the basic energy inequalities (3.2) and (3.3), it follows from \(H^{2}(\mathbb {R}^{3})\hookrightarrow L^{ \infty }(\mathbb {R}^{3})\) that \(\Vert \nabla u\Vert _{L^{\infty }}^{2}\le C(\Vert u\Vert _{L^{2}}^{2}+\Vert \nabla ^{3} u\Vert _{L^{2}}^{2})\le C(1+\Vert \nabla ^{3} u\Vert _{L^{2}}^{2})\), moreover, by using the interpolation inequality:

$$\begin{aligned} \Vert \nabla ^{2} v\Vert _{L^{4}}\le C\Vert v\Vert _{L^{2}}^{\frac{1}{12}}\Vert \nabla ^{3} v\Vert _{L^{2}}^{\frac{11}{12}}, \end{aligned}$$

the right-hand side of (3.11) can be majorized by

$$\begin{aligned}&-\int _{\mathbb {R}^{3}}\nabla ^{2}((u\cdot \nabla ) v)\nabla ^{2} v \mathrm{d}x=\int _{\mathbb {R}^3}\nabla ((u\cdot \nabla ) v)\cdot \nabla ^{3} vdx\nonumber \\&\quad \le \frac{1}{8}\Vert \nabla ^{3} v\Vert _{L^{2}}^{2}+C\big (\Vert (\nabla u\cdot \nabla )v\Vert _{L^{2}}^{2}+\Vert (u\cdot \nabla )\nabla v\Vert _{L^{2}}^{2}\big )\nonumber \\&\quad \le \frac{1}{8}\Vert \nabla ^{3} v\Vert _{L^{2}}^{2}+C\big (\Vert \nabla u\Vert _{L^{\infty }}^{2}\Vert \nabla v\Vert _{L^{2}}^{2}+\Vert u\Vert _{L^{4}}^{2}\Vert \nabla ^{2} v\Vert _{L^{4}}^{2}\big )\nonumber \\&\quad \le \frac{1}{8}\Vert \nabla ^{3} v\Vert _{L^{2}}^{2}+C\left( \Vert \nabla v\Vert _{L^{2}}^{2}(1+\Vert \nabla ^{3} u\Vert _{L^{2}}^{2})+\Vert u\Vert _{L^{2}}^{\frac{1}{2}}\Vert \nabla u\Vert _{L^{2}}^{\frac{3}{2}}\Vert v\Vert _{L^{2}}^{\frac{1}{6}}\Vert \nabla ^{3} v\Vert _{L^{2}}^{\frac{11}{6}}\right) \nonumber \\&\quad \le \frac{1}{6}\Vert \nabla ^{3} v\Vert _{L^{2}}^{2}+C\Vert \nabla v\Vert _{L^{2}}^{2}\big (1+\Vert \nabla ^{3} u\Vert _{L^{2}}^{2}\big )+C\Vert \nabla u\Vert _{L^{2}}^{18},\\&-\int _{\mathbb {R}^{3}}\nabla ^{2}\nabla \cdot (v\nabla \varPsi )\nabla ^{2} v \mathrm{d}x=\int _{\mathbb {R}^3}\nabla ^{2}(v\nabla \varPsi )\cdot \nabla ^{3} vdx\nonumber \\&\quad \le \frac{1}{8}\Vert \nabla ^{3} v\Vert _{L^{2}}^{2}+C\big (\Vert \nabla ^{2} v\nabla \varPsi \Vert _{L^{2}}^{2}+\Vert \nabla v\nabla ^{2}\varPsi \Vert _{L^{2}}^{2}+\Vert v\nabla ^{3}\varPsi \Vert _{L^{2}}^{2}\big )\nonumber \\&\quad \le \frac{1}{8}\Vert \nabla ^{3} v\Vert _{L^{2}}^{2}+C\big (\Vert \nabla ^{2} v\Vert _{L^{3}}^{2}\Vert \nabla \varPsi \Vert _{L^{6}}^{2}+\Vert \nabla v\Vert _{L^{6}}^{2}\Vert \nabla ^{2} \varPsi \Vert _{L^{3}}^{2}\nonumber \\&\qquad +\Vert v\Vert _{L^{3}}^{2}\Vert \nabla (v-w)\Vert _{L^{6}}^{2}\big )\nonumber \\&\quad \le \frac{1}{8}\Vert \nabla ^{3} v\Vert _{L^{2}}^{2}+C\big (\Vert \nabla ^{2} v\Vert _{L^{2}}\Vert \nabla ^{3} v\Vert _{L^{2}}+\Vert v\Vert _{L^{2}}\Vert \nabla v\Vert _{L^{2}}\Vert (\nabla ^{2} v, \nabla ^{2} w)\Vert _{L^{2}}^{2}\big )\nonumber \\&\quad \le \frac{1}{6}\Vert \nabla ^{3} v\Vert _{L^{2}}^{2}+C\big (1+\Vert (\nabla v, \nabla w)\Vert _{L^{2}}^{2}\big )\big (1+\Vert (\nabla ^{2} v, \nabla ^{2} w)\Vert _{L^{2}}^{2}\big ). \end{aligned}$$

Taking the above two inequalities into (3.11) yields that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla ^{2} v\Vert _{L^{2}}^{2}+\frac{4}{3}\Vert \nabla ^{3} v\Vert _{L^{2}}^{2}\le C\Vert \nabla u\Vert _{L^{2}}^{18}\nonumber \\&\quad +C\big (1+\Vert (\nabla v, \nabla w)\Vert _{L^{2}}^{2}\big )\big (1+\Vert \nabla ^{3} u\Vert _{L^{2}}^{2}+\Vert (\nabla ^{2} v, \nabla ^{2} w)\Vert _{L^{2}}^{2}\big ). \end{aligned}$$
(3.12)

Similarly, we can derive the bound of w; thus, we have

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla ^{2} w\Vert _{L^{2}}^{2}+\frac{4}{3}\Vert \nabla ^{3} w\Vert _{L^{2}}^{2}\le C\Vert \nabla u\Vert _{L^{2}}^{18}\nonumber \\&\quad +C\big (1+\Vert (\nabla v, \nabla w)\Vert _{L^{2}}^{2}\big )\big (1+\Vert \nabla ^{3} u\Vert _{L^{2}}^{2}+\Vert (\nabla ^{2} v, \nabla ^{2} w)\Vert _{L^{2}}^{2}\big ). \end{aligned}$$
(3.13)

Now we are in a position to complete the proof of Lemma 2.1. We deduce from (3.8), (3.10), (3.12) and (3.13) that

$$\begin{aligned} \frac{d(e+Y(t))}{\mathrm{d}t}&\le C\big (\Vert \nabla u\Vert _{L^{2}}^{14}+\Vert \nabla u\Vert _{L^{2}}^{18}\big )+C\big (1+\Vert (\nabla v, \nabla w)\Vert _{L^{2}}^{2}\big )(e+Y(t))\nonumber \\&\le C(1+(e+Y(t))^{18C\sigma })+C\big (1+\Vert (\nabla v, \nabla w)\Vert _{L^{2}}^{2}\big )(e+Y(t)). \end{aligned}$$
(3.14)

Choosing \(\sigma \) small enough such that \(18C\sigma \le 1\), the above inequality (3.14) implies that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}(e+Y(t))\le C\big (1+\Vert (\nabla v, \nabla w)\Vert _{L^{2}}^{2}\big )(e+Y(t)). \end{aligned}$$
(3.15)

For any \(T_{0}\le t<T\), applying the Gronwall’s inequality to (3.15), we get

$$\begin{aligned} Y(t)\le (e+Y(T_{0}))\exp \Big (C\int _{T_{0}}^{t}\big (1+\Vert (\nabla v, \nabla w)\Vert _{L^{2}}^{2}\big )d\tau \Big ), \end{aligned}$$

where \(Y(T_{0})=e+\Vert \nabla ^{3} u(\cdot ,T_{0})\Vert _{L^{2}}^{2}+\Vert (\nabla ^{2} v(\cdot ,T_{0}),\nabla ^{2} w(\cdot ,T_{0}))\Vert _{L^{2}}^{2}\). This together with the basic energy inequalities (3.2) and (3.3) lead to

$$\begin{aligned} \sup _{0\le t<T}\big (\Vert u(\cdot ,t)\Vert _{H^{3}}+\Vert (v(\cdot ,t),w(\cdot ,t))\Vert _{H^{2}}\big )\le C. \end{aligned}$$

The proof of Lemma 2.1 is achieved. \(\square \)

Similarly, under the condition (1.6), we have the following a priori estimate of local smooth solutions.

Lemma 3.2

Under the assumptions of Lemma 3.1 if we replace the condition (1.5) by (1.6), we still have the desired bound (3.1).

Proof

Based on the condition (1.6), we employ (2.6) to derive the bound of \(\Vert \omega \Vert _{L^{2}}\) that

$$\begin{aligned} \int _{\mathbb {R}^{3}}(\omega \cdot \nabla )u\cdot \omega \mathrm{d}x&\le \Vert \omega \Vert _{L^{4}}^{2}\Vert \nabla u\Vert _{L^{2}}\nonumber \\&\le C\Vert \omega \Vert _{BMO}\Vert \omega \Vert _{L^{2}}^{2}. \end{aligned}$$
(3.16)

Taking (3.16) and (3.6) into (3.4), and employing (2.7) in Lemma 2.6 and (2.8) with \(\alpha =0\) in Lemma 2.7, we find that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\Vert \omega \Vert _{L^{2}}^{2}+\Vert \nabla \omega \Vert _{L^{2}}^{2} \le C\big (1+\Vert (\nabla v,\nabla w)\Vert _{L^{2}}^{2}+\Vert \omega \Vert _{BMO}\big )\big (\Vert \omega \Vert _{L^{2}}^{2}+1\big )\nonumber \\&\quad \le C\big (1+\Vert (\nabla v,\nabla w)\Vert _{L^{2}}^{2}+\Vert \omega \Vert _{\dot{B}^{0}_{\infty ,\infty }}\sqrt{1+\ln (1+\Vert \omega \Vert _{H^{2}})}\big )\big (\Vert \omega \Vert _{L^{2}}^{2}+1\big )\nonumber \\&\quad \le C\Big (1+\Vert (\nabla v,\nabla w)\Vert _{L^{2}}^{2}+\frac{\Vert \omega \Vert _{\dot{B}^{0}_{\infty ,\infty }}}{\sqrt{1+\ln (e+\Vert \omega \Vert _{\dot{B}^{0}_{\infty ,\infty }})}} \Big )\nonumber \\&\qquad \times \Big (1+\ln (e+\Vert \omega \Vert _{H^{2}})\Big )\Big (\Vert \omega \Vert _{L^{2}}^{2}+1\Big )\nonumber \\&\quad \le C\Big (1+\Vert (\nabla v,\nabla w)\Vert _{L^{2}}^{2}+\frac{\Vert \omega \Vert _{\dot{B}^{0}_{\infty ,\infty }}}{\sqrt{1+\ln (e+\Vert \omega \Vert _{\dot{B}^{0}_{\infty ,\infty }})}} \Big )\nonumber \\&\qquad \times \ln \Big (e+\Vert \nabla ^{3} u\Vert _{L^{2}}^{2}+\Vert (\nabla ^{2} v, \nabla ^{2} w)\Vert _{L^{2}}^{2}\Big )\Big (\Vert \omega \Vert _{L^{2}}^{2}+1\Big ). \end{aligned}$$
(3.17)

Under the growth condition (1.6), one deduces that for any small constant \(\sigma >0\), there exists \(T_{0}<T\) such that

$$\begin{aligned} \int _{T_{0}}^{T}\frac{\Vert \omega \Vert _{\dot{B}^{0}_{\infty ,\infty }}}{\sqrt{1+\ln (e+\Vert \omega \Vert _{\dot{B}^{0}_{\infty ,\infty }})}}\mathrm{d}t<\sigma . \end{aligned}$$

Therefore, integrating (3.17) from \(T_{0}\) to t, \(T_{0}\le t<T\), one obtains that

$$\begin{aligned} \Vert \omega (t)\Vert _{L^{2}}^{2}&\le C\exp \Big (\int _{T_{0}}^{t}C\big (1+\Vert (\nabla v,\nabla w)\Vert _{L^{2}}^{2} \nonumber \\&\quad +\frac{\Vert \omega \Vert _{\dot{B}^{0}_{\infty ,\infty }}}{\sqrt{1+\ln (e+\Vert \omega \Vert _{\dot{B}^{0}_{\infty ,\infty }})}}\big )d\tau \ln (e+Y(t)) \Big )\nonumber \\&\le C\exp \big (2C\sigma \ln (e+Y(t))\big )\nonumber \\&\le C(e+Y(t))^{2C\sigma }. \end{aligned}$$
(3.18)

As soon as (3.18) is established, one can exactly follow the same procedures as that in the proof of Lemma 3.1 to get the desired assertion (3.1); thus, we safely omit the proof here. The proof of Lemma 3.2 is achieved. \(\square \)

Furthermore, under the condition (1.7), we have the following a priori estimate of local smooth solutions.

Lemma 3.3

Under the assumptions of Lemma 3.1 if we replace the condition (1.5) by (1.7), we still have the desired bound (3.1).

Proof

We claim that, under the growth condition (1.7), we can directly derive the \(L^{2}\)-bound of \(\nabla u\). In order to do so, taking inner product of (1.1)\(_{1}\) with \(-\Delta u\), and performing integration by parts, we deduce that

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla u\Vert _{L^{2}}^{2}+\Vert \Delta u\Vert _{L^{2}}^{2}=-\int _{\mathbb {R}^{3}}(\nabla u\cdot \nabla )u\cdot \nabla udx -\int _{\mathbb {R}^{3}}\Delta \varPsi \nabla \varPsi \cdot \Delta udx. \end{aligned}$$
(3.19)

Applying (2.3) with \(p=3\), \(q=2\) and \(\alpha =2\) in Lemma 2.3, the first term in the right-hand side of (3.19) can be bounded as

$$\begin{aligned} -\int _{\mathbb {R}^{3}}(\nabla u\cdot \nabla ) u\cdot \nabla udx&\le \Vert \nabla u\Vert _{L^{3}}^{3}\nonumber \\&\le C\Vert \nabla u\Vert _{\dot{B}^{-2}_{\infty ,\infty }}\Vert \nabla u\Vert _{\dot{H}^{1}}^{2}\nonumber \\&\le C\Vert u\Vert _{\dot{B}^{-1}_{\infty ,\infty }}\Vert \Delta u\Vert _{L^{2}}^{2}. \end{aligned}$$
(3.20)

Since \(\nabla \times \omega =-\Delta u\), the second term in the right-hand side of (3.19) can be exactly tackled the same as (3.6), thus it follows that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla u\Vert _{L^{2}}^{2}+\frac{3}{2}\Vert \Delta u\Vert _{L^{2}}^{2}\le C\Vert u\Vert _{\dot{B}^{-1}_{\infty ,\infty }}\Vert \Delta u\Vert _{L^{2}}^{2}+C\big (\Vert (\nabla v,\nabla w)\Vert _{L^{2}}^{2}+1\Big ). \end{aligned}$$
(3.21)

By taking \(\varepsilon \) small enough such that \( C\Vert u\Vert _{L^{\infty }(0,T; \dot{B}^{-1}_{\infty ,\infty })}\le C\varepsilon \le \frac{1}{2}\), then integrating (3.21) from 0 to t for any \(0<t<T\), we get the \(L^{2}\)-bound of \(\nabla u\):

$$\begin{aligned} \Vert \nabla u(t)\Vert _{L^{2}}^{2}+\int _{0}^{t}\Vert \Delta u(\tau )\Vert _{L^{2}}^{2}d\tau \le \Vert \nabla u_{0}\Vert _{L^{2}}^{2}+C\int _{0}^{t}\big (1+\Vert (\nabla v(\tau ),\nabla w(\tau ))\Vert _{L^{2}}^{2}\big )d\tau . \end{aligned}$$
(3.22)

Based on the above estimate (3.22), the derivation of \(H^{3}\times H^{2}\times H^{2}\) estimates of (u, v, w) is simpler than that in Lemma 3.1. Actually, we have derived the desired bound for u as (3.11). It suffices to deduce the estimation for v (w can be done analogously). By (3.22), the first term in the right-hand side of (3.12) can be majorized as

$$\begin{aligned} -\int _{\mathbb {R}^{3}}&\nabla ^{2}((u\cdot \nabla ) v)\nabla ^{2} v \mathrm{d}x=\int _{\mathbb {R}^3}\nabla ((u\cdot \nabla ) v)\cdot \nabla ^{3} vdx\nonumber \\&\le \frac{1}{8}\Vert \nabla ^{3} v\Vert _{L^{2}}^{2}+C\big (\Vert (\nabla u\cdot \nabla )v\Vert _{L^{2}}^{2}+\Vert (u\cdot \nabla )\nabla v\Vert _{L^{2}}^{2}\big )\nonumber \\&\le \frac{1}{8}\Vert \nabla ^{3} v\Vert _{L^{2}}^{2}+C\big (\Vert \nabla u\Vert _{L^{\infty }}^{2}\Vert \nabla v\Vert _{L^{2}}^{2}+\Vert u\Vert _{L^{6}}^{2}\Vert \nabla ^{2} v\Vert _{L^{3}}^{2}\big )\nonumber \\&\le \frac{1}{8}\Vert \nabla ^{3} v\Vert _{L^{2}}^{2}+C\big (\Vert \nabla v\Vert _{L^{2}}^{2}(1+\Vert \nabla ^{3} u\Vert _{L^{2}}^{2})+\Vert \nabla u\Vert _{L^{2}}^{2}\Vert \nabla ^{2} v\Vert _{L^{2}}\Vert \nabla ^{3} v\Vert _{L^{2}}\big )\nonumber \\&\le \frac{1}{6}\Vert \nabla ^{3} v\Vert _{L^{2}}^{2}+C\big (1+\Vert \nabla v\Vert _{L^{2}}^{2}\big )\big (1+\Vert \nabla ^{3} u\Vert _{L^{2}}^{2}+\Vert \nabla ^{2} v\Vert _{L^{2}}^{2}\big ). \end{aligned}$$

Notice that the second term in the right-hand side of (3.12) can be proceeded the same as that in Lemma 3.1, thus we get

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla ^{2}v\Vert _{L^{2}}^{2}+\frac{4}{3}\Vert \nabla ^{3} v\Vert _{L^{2}}^{2}&\le C\big (1+\Vert (\nabla v, \nabla w)\Vert _{L^{2}}^{2}\big )\nonumber \\&\quad \times \big (1+\Vert \nabla ^{3} u\Vert _{L^{2}}^{2}+\Vert (\nabla ^{2} v, \nabla ^{2} w)\Vert _{L^{2}}^{2}\big ). \end{aligned}$$
(3.23)

Therefore, by (3.22), we obtain

$$\begin{aligned} \frac{d(e+Y(t))}{\mathrm{d}t}&\le C\Vert \nabla u\Vert _{L^{2}}^{14}+C\big (1+\Vert (\nabla v, \nabla w)\Vert _{L^{2}}^{2}\big )(e+Y(t))\nonumber \\&\le C\big (1+\Vert (\nabla v, \nabla w)\Vert _{L^{2}}^{2}\big )(e+Y(t)). \end{aligned}$$
(3.24)

Applying Gronwall’s inequality to (3.24) yields (3.1). The proof of Lemma 3.3 is achieved. \(\square \)

4 Proofs of Theorems 1.1–1.3

We are now in a position to prove Theorems 1.1–1.3. Since \(u_{0}\in H^{3}(\mathbb {R}^{3})\) with \(\nabla \cdot u_{0}=0\), \(v_{0}, w_{0}\in L^{1}(\mathbb {R}^{3})\cap H^{2}(\mathbb {R}^{3})\), it follows from the local existence theorem in [16] that there exists a time \(T_{*}\) and a unique solution \((\widetilde{u},\widetilde{v},\widetilde{w})\) on \([0,T_{*})\) with initial data \((u_{0},v_{0},w_{0})\) satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} \widetilde{u}\in C([0,T_{*}), H^{3}(\mathbb {R}^{3}))\cap L^{\infty }(0,T_{*}; H^{3}(\mathbb {R}^{3}))\cap L^{2}(0, T_{*}; H^{4}(\mathbb {R}^{3})),\\ \widetilde{v},\widetilde{w}\in C([0,T_{*}), H^{2}(\mathbb {R}^{3}))\cap L^{\infty }(0,T_{*}; H^{2}(\mathbb {R}^{3}))\cap L^{2}(0, T_{*}; H^{3}(\mathbb {R}^{3})). \end{array}\right. } \end{aligned}$$

Notice that it follows from [24] that the weak solution coincides with the above local smooth solution

$$\begin{aligned} \widetilde{u}\equiv u, \ \ \widetilde{v}\equiv v, \ \ \widetilde{w} \equiv w\ \ \ \ \text {on}\ \ \ \ [0,T_{*}). \end{aligned}$$

Thus it is sufficient to show that \(T=T_{*}\). If \(T<T_{*}\), then we have already the conclusion that (u, v, w) is smooth up to time T. Suppose that \(T_{*}<T\), without loss of generality, we may assume that \(T_{*}\) is the maximal existence time for \((\widetilde{u},\widetilde{v},\widetilde{w})\). Since \(\widetilde{u}\equiv u\), \(\widetilde{v}\equiv v\) and \(\widetilde{w}\equiv w\) on \([0,T_{*})\), by the assumptions (1.5)–(1.7), we have

$$\begin{aligned} \int _{0}^{T}\frac{\Vert \widetilde{\omega }(\cdot ,t)\Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}^{\frac{2}{2-\alpha }}}{1+\ln (e+\Vert \widetilde{\omega }(\cdot ,t)\Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }})}\mathrm{d}t<\infty \ \ \ \text {for}\ \ \ 0<\alpha <2 \end{aligned}$$

or

$$\begin{aligned} \int _{0}^{T}\frac{\Vert \widetilde{\omega }(\cdot ,t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}}{\sqrt{1+\ln (e+\Vert \widetilde{\omega }(\cdot ,t)\Vert _{\dot{B}^{0}_{\infty ,\infty }})}}\mathrm{d}t<\infty \end{aligned}$$

or

$$\begin{aligned} \sup _{0\le t\le T}\Vert \widetilde{u}(\cdot ,t)\Vert _{\dot{B}^{-1}_{\infty ,\infty }}<\varepsilon , \end{aligned}$$

where \(\widetilde{\omega }=\nabla \times \widetilde{u}\). Therefore, it follows from Lemmas 3.1–3.3 that we can extend the existence time of \((\widetilde{u},\widetilde{v},\widetilde{w})\) beyond the time \(T_{*}\), which contradicts with the maximality of \(T_{*}\). We complete the proofs of Theorems 1.1–1.3.