1 Introduction

The 3D incompressible Navier–Stokes equations describing Newtonian fluid are given by

$$\begin{aligned} \left\{ \begin{aligned}&u_{t} -\Delta u+ u\cdot \nabla u +\nabla \Pi =0, \\&\mathrm {div} \;u=0,\\&u|_{t=0}=u_0. \end{aligned}\right. \end{aligned}$$
(1.1)

Here, u and \(\Pi \) denote the unknown velocity vector and the pressure, respectively. The initial datum \(u_{0}\) is given and satisfies the divergence-free condition.

The pioneering work involving the global finite energy weak solutions is due to Leray [12] and Hopf [8]. Precisely, they showed that for any given divergence-free datum \(u_0\in L^2(\Omega )\), there exists a global weak solution u of the 3D Navier–Stokes equations such that \(u\in L^{\infty }(0, \infty ; L^{2}(\Omega ))\cap L^{2}(0, \infty ; W^{1,2}(\Omega ))\) where \(\Omega \subseteq \mathbb {R}^{3}\). However, full regularity of Leary–Hopf weak solutions to the 3D Navier–Stokes system is still a fundamental open question. Starting from Serrin’s famous work, regularity criteria of Leray–Hopf weak solutions are extensively studied (see [1,2,3, 5, 7, 9,10,11, 13, 16,17,21, 23,24,26, 28, 29] and references therein). The so-called Serrin type regularity criteria is that a weak solution u is regular on (0, T] if u satisfies

$$\begin{aligned} u\in L^{p} (0,T;L^{q}( \mathbb {R}^{3})) ~~~ \text {with}~~~~2/p+3/q=1, ~~q\ge 3. \end{aligned}$$
(1.2)

Beirao da Veiga found that condition (1.2) can be replaced by the following

$$\begin{aligned} \nabla u\in L^{p} (0,T;L^{q}( \mathbb {R}^{3})) ~~~ \text {with}~~~~2/p+3/q=2, ~~q>3/2. \end{aligned}$$
(1.3)

It is clear that the gradient of the velocity field \(\nabla u\) can be decomposed into a symmetric part \(\mathcal {D}\) and an antisymmetric part \(\Omega \), that is,

$$\begin{aligned} \nabla u=\frac{1}{2}\left( \nabla u+\nabla u^{^{\text {T}}}\right) +\frac{1}{2}\left( \nabla u-\nabla u^{^{\text {T}}}\right) =:\mathcal {D}(u)+\Omega (u). \end{aligned}$$

\(\mathcal {D}(u)\) and \(\Omega (u)\) are usually called the deformation tensor (or rate-of-strain tensor) and the rotation tensor, respectively (See, e.g., [14]). Denote \(\lambda _{1},\lambda _{2},\lambda _{3}\) by the eigenvalue of the deformation tensor \(\mathcal {D}(u)\). In a series of works [17,18,19,20,21], Neustupa–Penel obtained regularity criteria via only the middle eigenvalue \(\lambda _{2}\) of the deformation tensor below

$$\begin{aligned} \lambda _{2}^+=\max \{\lambda _{2},0\}\in L^{p} (0,T;L^{q}( \mathbb {R}^{3})) ~~~ \text {with}~~~~2/p+3/q=2, ~~q>3/2. \end{aligned}$$
(1.4)

Very recently, a new alternative proof (1.4) was presented by Miller in [16]. Subsequently, Wu [28] generalized (1.4) to the anisotropic Lebesgue spaces in the 3D double-diffusive convection equations. Since the Lorentz spaces \(L^{r,s}(\Omega )\) \((s\ge r)\) are larger than the Lebesgues spaces \(L^{r }(\Omega )\), much effort is devoted to the regularity criteria in terms of velocity, gradient, the pressure, velocity directions in Lorentz spaces (see [3, 5, 7, 9,10,11, 25, 26, 29] and references therein). Motivated by the references mentioned above, we extend (1.4) to Lorentz spaces. The main result of this paper reads as follows.

Theorem 1.1

Suppose that u is a weak solution to (1.1) with the divergence-free initial datum \(u_{0}(x)\in W^{1,2}(\mathbb {R}^{3})\). Then, there exists a positive constant \(\varepsilon _{1}\) such that u(x, t) is a regular solution on (0, T] provided holds \( \lambda _{2}^{+} \in L^{p,\infty }(0,T; L ^{q,\infty }(\mathbb {R}^{3}))\)  and 

$$\begin{aligned} \Vert \lambda _{2}^{+}\Vert _{L^{p,\infty }(0,T; L ^{q,\infty }(\mathbb {R}^{3}))} \le \varepsilon _{1}, ~ \text {with} ~~2/p+3/q=2 , ~3/2<q\le \infty ; \end{aligned}$$

Remark 1.1

Theorem 1.1 is an improvement of regularity criteria based on only the middle eigenvalue of the deformation tensor (1.4). It seems that a slightly modified the technique in Theorem 1.1 can be applied to other incompressible fluid equations such as Navier–Stokes equations with fractional dissipation and the double-diffusive convection equations in [28].

As a by-production of Theorem 1.1, by the absolute continuity of norm in Lorentz space \(L^{p,r}(0,T)\) with \(r<\infty \), we have

Corollary 1.2

Suppose that u is a weak solution to (1.1) with the divergence-free initial datum \(u_{0}(x)\in W^{1,2}(\mathbb {R}^{3})\). Then, u(x, t) is a regular solution on (0, T] provided that, for \(p\le r<\infty \),

$$\begin{aligned} \lambda _{2}^{+} \in L^{p,r}(0,T; L ^{q,\infty }(\mathbb {R}^{3}))\quad \text {with} \quad 2/p+3/q=2 , ~q>3/2. \end{aligned}$$

Remark 1.2

Corollary 1.2 is an extension of regularity criteria (1.4).

2 Notations and auxiliary lemmas

In this section, we give some preliminaries on functional settings and some useful lemmas, which will be used in proof of our main results.

For \(p\in [1,\,\infty ]\), the notation \(L^{p}(0,T;X)\) stands for the set of measurable functions f(x, t) on the interval (0, T) with values in X and \(\Vert f(\cdot ,t)\Vert _{X}\) belonging to \(L^{p}(0,T)\). The classical Sobolev space \(W^{k,2}(\mathbb {R}^{3})\) is equipped with the norm \(\Vert f\Vert _{W^{k,2}(\mathbb {R}^{3})}=\sum \limits _{\alpha =0}^{k}\Vert D^{\alpha }f\Vert _{L^{2}(\mathbb {R}^{3})}\). \(|\Omega |\) represents the n-dimensional Lebesgue measure of a set \(\Omega \subset \mathbb {R}^{n}\). We will use the summation convention on repeated indices.

We now present some basic facts on Lorentz spaces. For \(p,q\in [1,\infty ]\), we define

$$\begin{aligned} \Vert f\Vert _{L^{p,q}(\Omega )}=\left\{ \begin{aligned}&\Big (p\int _{0}^{\infty }\alpha ^{q}|\{x\in \Omega :|f(x)|>\alpha \}|^{\frac{q}{p}}\frac{d\alpha }{\alpha }\Big )^{\frac{1}{q}} ,\quad q<\infty ,\\&\sup _{\alpha>0}\alpha |\{x\in \Omega :|f(x)|>\alpha \}|^{\frac{1}{p}} ,~~~q=\infty . \end{aligned}\right. \end{aligned}$$

Moreover,

$$\begin{aligned} L^{p,q}(\Omega )=\big \{f: f~ \text {is a measurable function on}~ \Omega ~\text {and} ~\Vert f\Vert _{L^{p,q}(\Omega )}<\infty \big \}. \end{aligned}$$

Similarly, one can define Lorentz spaces \(L^{p,q}(0,T;X)\) in time for \( p\le q \le \infty \). \(f\in L^{p, q}(0,T;X)\) means that \(\Vert f\Vert _{L^{p,q}(0,T;X)}<\infty \), where

$$\begin{aligned} \Vert f\Vert _{L^{p,q}(0,T;X)}=\left\{ \begin{aligned}&\left( p \int _{0}^{\infty }\alpha ^q|\{t\in (0,T) :\Vert f(t)\Vert _{X}>\alpha \}|^{\frac{q}{p}}\frac{d\alpha }{\alpha }\right) ^{\frac{1}{q}} , q<\infty ,\\&\sup _{\alpha>0}\alpha |\{t\in (0,T) :\Vert f(t)\Vert _{X}>\alpha \}|^{\frac{1}{p}} ,~~~q=\infty . \end{aligned}\right. \end{aligned}$$

Now, we list some properties of Lorentz spaces.

  1. (1)

    Interpolation characteristic of Lorentz spaces [4]:

    $$\begin{aligned} (L^{p_{0},q_{0}}(\mathbb {R}^{n}),L^{p_{1},q_{1}}(\mathbb {R}^{n}))_{\theta ,q}=L^{p,q}(\mathbb {R}^{n}) ~~~~\text {with}~~~ \frac{1}{p}=\frac{1-\theta }{p_{0}}+\frac{\theta }{p_{1}},~0<\theta <1. \end{aligned}$$
    (2.1)
  2. (2)

    Hölder’s inequality in Lorentz spaces [22]:

    $$\begin{aligned}&\Vert fg\Vert _{L^{r,s}(\mathbb {R}^{n})}\le \Vert f\Vert _{L^{r_{1},s_{1}}(\mathbb {R}^{n})}\Vert g\Vert _{L^{r_{2},s_{2}}(\mathbb {R}^{n})},\nonumber \\&\frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}},~~\frac{1}{s}=\frac{1}{s_{1}}+\frac{1}{s_{2}}. \end{aligned}$$
    (2.2)
  3. (3)

    The Lorentz spaces increase as the exponent q increases [6, 15]. For \(1\le p\le \infty \) and \(1\le q_{1}<q_{2}\le \infty ,\)

    $$\begin{aligned} \Vert f\Vert _{L^{p,q_{2}}(\mathbb {R}^{n})}\le \Big (\frac{q_{1}}{p}\Big )^{\frac{1}{q_{1}}-\frac{1}{q_{2}}}\Vert f\Vert _{L^{p,q_{1}}(\mathbb {R}^{n})}. \end{aligned}$$
    (2.3)
  4. (4)

    Sobolev inequality in Lorentz spaces [22, 27]:

    $$\begin{aligned} \Vert f\Vert _{L^{\frac{np}{n-p},p}(\mathbb {R}^{n})}\le \Vert \nabla f\Vert _{L^{p}(\mathbb {R}^{n})}~~\text {with}~~ 1\le p<n.\end{aligned}$$
    (2.4)

We recall the following useful Gronwall lemma first shown by Bosia, Pata and Robinson in [5].

Lemma 2.1

([5]). Let \(\phi \) be a measurable positive function defined on the interval [0, T]. Suppose that there exists \(\kappa _{0}>0\) such that for all \(0<\kappa <\kappa _{0}\) and a.e. \(t\in [0,T]\), \(\phi \) satisfies the inequality

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\phi \le \mu \lambda ^{1-\kappa }\phi ^{1+2\kappa }, \end{aligned}$$

where \(0 <\lambda \in L^{1,\infty }(0,T)\) and \(\mu > 0\) with

$$\begin{aligned} \mu \Vert \lambda \Vert _{L^{1,\infty }(0,T)}<\frac{1}{2}. \end{aligned}$$

Then, \(\phi \) is bounded on [0, T].

To apply the above lemma, we need the following fact obtained in [9].

Lemma 2.2

([9]). Assume that the pair (p, q) satisfies \(\frac{2}{p}+\frac{3}{q}=a\) with \(a,q \ge 1 \) and \(p>0\). Then, for every \(\kappa \in [0,1]\) and given \(b,c_0\ge 1\), there exist \(p_{\kappa } > 0\) and \(\min \{q,b\}\le q_{\kappa }\le \max \{q,b\}\) such that

$$\begin{aligned} \left\{ \begin{aligned}&\frac{2}{p_{\kappa }}+\frac{3}{q_{\kappa }}=a,\\&\frac{p_{\kappa }}{q_{\kappa }}=\frac{p\big (1-\kappa \big )}{q}+\frac{c_0\kappa }{b}. \end{aligned}\right. \end{aligned}$$
(2.5)

To proceed further, we recall the Kronecker symbol \(\delta _{ij}\) and the three-dimensional Levi-Civita symbol \(\varepsilon _{ijk}\), respectively,

$$\begin{aligned} \delta _{ij}= {\left\{ \begin{array}{ll} 0, &{} \text{ for }~~~i\ne j,\\ 1, &{} \text{ for }~~~i=j,\\ \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} \varepsilon _{ijk}= {\left\{ \begin{array}{ll} 1, &{} \text{ if } ~~~(i,j,k)~~\text {is}~~(1,2,3),(2,3,1)~~\text {or}~~(3,1,2),\\ -1, &{} \text{ if } ~~~(i,j,k)~~\text {is}~~(3,2,1),(1,3,2)~~\text {or}~~(2,1,3),\\ 0, &{} \text{ else }. \end{array}\right. } \end{aligned}$$

Then, a useful identity (Levi-Civita and Kronecker Delta identity) holds

$$\begin{aligned} \varepsilon _{ijk}\varepsilon _{imn}=\delta _{jm}\delta _{kn}-\delta _{jn}\delta _{km}. \end{aligned}$$

We denote \(a_{ij}=\frac{1}{2}(\partial _{j}u_{i}-\partial _{i}u_{j})\) and \(d_{ij}=\frac{1}{2}(\partial _{i}u_{j}+\partial _{j}u_{i})\). Next, we recall the following result stated in [17]. Since this result plays an important role in the derivation of regularity criteria via only the middle eigenvalue of the deformation tensor, we present its proof here.

Lemma 2.3

Let \(\omega =\mathrm {curl} u=(\omega _{1},\omega _{2},\omega _{3})\) be the vorticity field. Then,

$$\begin{aligned} d_{kj}a_{ij}a_{ik}=-\frac{1}{4}d_{kj}w_{j}w_{k}. \end{aligned}$$
(2.6)

Proof

Straightforward calculations show that

$$\begin{aligned} w_{i}=\varepsilon _{ijk}\partial _{j}u_{k} =\varepsilon _{ijk}a_{kj}. \end{aligned}$$
(2.7)

Multiplying both side of (2.7) by \(\varepsilon _{imn}\), we obtain

$$\begin{aligned} \varepsilon _{imn}w_{i}=\varepsilon _{imn}\cdot \varepsilon _{ijk}a_{kj}. \end{aligned}$$

This together with the aforementioned Levi-Civita and Kronecker Delta identity yields that

$$\begin{aligned} \varepsilon _{imn}w_{i}=(\delta _{jm}\delta _{kn}-\delta _{jn}\delta _{km})\cdot a_{kj} =\delta _{kn}a_{km}-\delta _{km}a_{kn} =-2a_{mn}, \end{aligned}$$

where the definition of \(a_{ij}\) was used.

As a consequence, we infer that

$$\begin{aligned} a_{mn}=-\frac{1}{2}\varepsilon _{imn}w_{i}. \end{aligned}$$
(2.8)

Using (2.8) twice and (2.7), we see that

$$\begin{aligned} d_{kj}a_{ij}a_{ik}=\frac{1}{4}d_{kj}\varepsilon _{kij}w_{k}\varepsilon _{jik}w_{j} =-\frac{1}{4}d_{jk}w_{k}w_{j}. \end{aligned}$$

This completes the proof of Lemma 2.3. \(\square \)

3 Proof of Theorem 1.1

This section is devoted to proving Theorem 1.1.

Proof of Theorem 1.1

Multiplying both side of the Navier–Stokes equations (1.1) by \(\Delta u\), integrating by parts and using the divergence-free condition \(\mathrm {div} \; u=0\), we obtain

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm{d}x+\int _{\mathbb {R}^{3}}|\Delta u|^{2}\mathrm{d}x=\int _{\mathbb {R}^{3}}(u\cdot \nabla )u\cdot \Delta u \mathrm{d}x. \end{aligned}$$
(3.1)

Using the divergence-free condition \(\mathrm {div}\; u=0\), we write the right-hand side of (3.1) as

$$\begin{aligned} \int _{\mathbb {R}^{3}}(u\cdot \nabla )u\cdot \Delta u \mathrm{d}x&=\int _{\mathbb {R}^{3}}u_{j}\partial _{j}u_{i}\partial _{k}\partial _{k}u_{i}\mathrm{d}x\nonumber \\&=-\int _{\mathbb {R}^{3}}\partial _{k}u_{j}\partial _{j}u_{i}\partial _{k}u_{i}\mathrm{d}x\nonumber \\&=-\int _{\mathbb {R}^{3}}d_{kj}d_{ij}d_{ik}-\int _{\mathbb {R}^{3}}d_{kj}d_{ij}a_{ik}\nonumber \\&\quad -\int _{\mathbb {R}^{n}}d_{kj}a_{ij}d_{ik} -\int _{\mathbb {R}^{3}}d_{kj}a_{ij}a_{ik}\mathrm{d}x. \end{aligned}$$
(3.2)

Interchanging i with k, we have

$$\begin{aligned} d_{kj}d_{ij}a_{ik}=-d_{kj}d_{ij}a_{ki}=-d_{ij}d_{kj}a_{ik}=-d_{kj}d_{ij}a_{ik}. \end{aligned}$$

Hence, we see that

$$\begin{aligned} d_{kj}d_{ij}a_{ik}=0. \end{aligned}$$
(3.3)

Arguing in the same manner as the above, we observe that

$$\begin{aligned} a_{ij}d_{ik}d_{kj}=0. \end{aligned}$$
(3.4)

Substituting (3.3) and (3.4) into (3.2),

$$\begin{aligned} -\int _{\mathbb {R}^{3}}\partial _{k}u_{j}\partial _{j}u_{i}\partial _{k}u_{i}dx=-\int _{\mathbb {R}^{3}}d_{kj}d_{ij}d_{ik}dx-\int _{\mathbb {R}^{3}} d_{kj}a_{ij}a_{ik}\mathrm{d}x. \end{aligned}$$
(3.5)

We derive from Lemma 2.3 that

$$\begin{aligned} -\int _{\mathbb {R}^{3}}d_{kj}a_{ij}a_{ik}\mathrm{d}x=\frac{1}{4}\int _{\mathbb {R}^{3}}d_{kj}w_{j}w_{k}\mathrm{d}x. \end{aligned}$$

Collecting the above estimates, we see that

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm{d}x+\int _{\mathbb {R}^{3}}|\Delta u|^{2}\mathrm{d}x=-\int _{\mathbb {R}^{3}}d_{kj}d_{ij}d_{ik}\mathrm{d}x+\frac{1}{4}\int _{\mathbb {R}^{3}}d_{kj}w_{j}w_{k}\mathrm{d}x. \end{aligned}$$
(3.6)

As in [17], we invoke the vorticity equation to control the first term in the right hand side of (3.6). Recall that the vorticity equation of the Navier–Stokes system is given by

$$\begin{aligned} w_{t}-\Delta w+(u\cdot \nabla )w=(w\cdot \nabla )u. \end{aligned}$$
(3.7)

Multiplying Eq. (3.7) by \(\omega \), we infer that

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Vert \omega \Vert _{L^{2}(\mathbb {R}^{3})}^{2}+\Vert \nabla \omega \Vert _{L^{2}(\mathbb {R}^{3})}^{2}=\int _{\mathbb {R}^{3}}\omega \cdot \nabla u\cdot \omega \mathrm{d}x =\int _{\mathbb {R}^{3}}d_{kj}w_{j}w_{k}\mathrm{d}x. \end{aligned}$$
(3.8)

Notice that,

$$\begin{aligned} \int _{\mathbb {R}^{3}}|\omega |^{2}\mathrm{d}x=\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm{d}x,~~ \int _{\mathbb {R}^{3}}|\nabla \omega |^{2}\mathrm{d}x=\int _{\mathbb {R}^{3}}|\Delta u|^{2}\mathrm{d}x. \end{aligned}$$

Thus, we obtain

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^{3})}^{2}+\Vert \Delta u\Vert _{L^{2}(\mathbb {R}^{3})}^{2}=\int _{\mathbb {R}^{3}}d_{kj}w_{j}w_{k}\mathrm{d}x. \end{aligned}$$
(3.9)

Multiplying Eq. (3.9) by \(-\frac{1}{4}\) and summing with (3.6), it immediately follows that

$$\begin{aligned} \frac{3}{8}\frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^{3})}^{2}+\frac{3}{4}\Vert \Delta u\Vert _{L^{2}(\mathbb {R}^{3})}=-\int _{\mathbb {R}^{3}}d_{kj}d_{ij}d_{ik}\mathrm{d}x. \end{aligned}$$
(3.10)

Without loss of generality, we assume \( \lambda _{1}\le \lambda _{2}\le \lambda _{3}. \) The tensor \((d_{ij})\) is symmetric so that the corresponding eigenvectors are orthogonal. Due to the fact \(\mathrm {div} u=0\), we deduce that

$$\begin{aligned} \lambda _{1}+\lambda _{2}+\lambda _{3}=0, \end{aligned}$$
(3.11)

and

$$\begin{aligned} (d_{kj}d_{ij}d_{ik})(x,t)=\lambda _{1}^{3}+\lambda _{2}^{3}+\lambda _{3}^{3} =\lambda _{1}^{3}+\lambda _{2}^{3}-(\lambda _{1}+\lambda _{2})^{3} =3\lambda _{1}\lambda _{2}\lambda _{3}=3|d_{ij}|.\nonumber \\ \end{aligned}$$
(3.12)

The quantity \(\lambda _{1}\lambda _{2}\lambda _{3}=|d_{ij}|\) is independent of choice of the system of coordinates. Hence, we obtain

$$\begin{aligned} \frac{3}{8}\frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^{3})}^{2}+\Vert \Delta u\Vert _{L^{2}(\mathbb {R}^{3})}^{2}=-3\int _{\mathbb {R}^{3}}\lambda _{1}\lambda _{2}\lambda _{3}\mathrm{d}x. \end{aligned}$$
(3.13)

Using the Hölder inequality (2.2) or interpolation characteristic (2.1), Sobolev embedding (2.4) in Lorentz spaces and the Young inequality, we have

$$\begin{aligned} \int _{\mathbb {R}^{3}}\lambda _{1}\lambda _{2}^{+}\lambda _{3}\mathrm{d}x&\le C\cdot \Vert \lambda _{2}^{+}\Vert _{L^{q,\infty }(\mathbb {R}^{3})} \Vert \lambda _{1}\Vert _{L^{\frac{2q}{q-1},2}(\mathbb {R}^{3})}\Vert \lambda _{3}\Vert _{L^{\frac{2q}{q-1},2}(\mathbb {R}^{3})}\nonumber \\&\le C\cdot \Vert \lambda _{2}^{+}\Vert _{L^{q,\infty }(\mathbb {R}^{3})}\Vert \nabla u\Vert _{L^{\frac{2q}{q-1},2}(\mathbb {R}^{3})}^{2}\nonumber \\&\le C\cdot \Vert \lambda _{2}^{+}\Vert _{L^{q,\infty }(\mathbb {R}^{3})}\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^{3})}^{2-\frac{3}{q}}\Vert \nabla ^{2} u\Vert _{L^{2}(\mathbb {R}^{3})}^{\frac{3}{q}}\nonumber \\&\le C\cdot \Vert \lambda _{2}^{+}\Vert _{L^{q,\infty }(\mathbb {R}^{3})}^{\frac{2q}{q-3}}\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^{3})}^{2}+\frac{1}{8}\Vert \nabla ^{2} u\Vert _{L^{2}(\mathbb {R}^{3})}^{2}. \end{aligned}$$
(3.14)

Inserting (3.14) into (3.13), we get

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^{3})}^{2}\mathrm{d}x\le C\cdot \Vert \lambda _{2}^{+}\Vert _{L^{q,\infty }(\mathbb {R}^{3})}^{\frac{2q}{q-3}}\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^{3})}^{2}. \end{aligned}$$
(3.15)

With the help of interpolation characteristic of Lorentz spaces (2.1) or the Hölder inequality (2.2), applying Lemma 2.2 with \(a=b=2, c_{0}=4\) and (2.3), we see that

$$\begin{aligned} \Vert \lambda _{2}^{+}\Vert _{L^{q_{\kappa },\infty }(\mathbb {R}^{3})}^{p_{\kappa }}&\le \Vert \lambda _{2}^{+}\Vert _{L^{q,\infty }(\mathbb {R}^{3})} ^{p(1-\kappa )}\Vert \lambda _{2}^{+}\Vert _{L^{2,\infty }(\mathbb {R}^{3})}^{4\kappa }\nonumber \\&\le \Vert \lambda _{2}^{+}\Vert _{L^{q,\infty }(\mathbb {R}^{3})}^{p(1-\kappa )}\Vert \nabla u\Vert _{L^{2,\infty }(\mathbb {R}^{3})}^{4\kappa }. \end{aligned}$$
(3.16)

Since the pair \((p_{\kappa }, q_{\kappa })\) also meets \(2/p_{\kappa }+3/q_{\kappa }=2\), we insert (3.16) into (3.15) to obtain

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^{3})}^{2}\mathrm{d}x\le C\cdot \Vert \lambda _{2}^{+}\Vert _{L^{q,\infty }(\mathbb {R}^{3})}^{p(1-\kappa )}\Vert \nabla u\Vert _{L^{2,\infty }(\mathbb {R}^{3})}^{2(1+2\kappa )}. \end{aligned}$$
(3.17)

Applying Lemma 2.1 and (1.2) completes the proof of Theorem 1.1. \(\square \)