Abstract
We study the second spatial derivatives of suitable weak solutions to the incompressible Navier–Stokes equations in dimension three. We show that it is locally \(L ^{\frac{4}{3}, q}\) for any \(q > \frac{4}{3}\), which improves from the current result of \(L ^{\frac{4}{3}, \infty }\). Similar improvements in Lorentz space are also obtained for higher derivatives of the vorticity for smooth solutions. We use a blow-up technique to obtain nonlinear bounds compatible with the scaling. The local study works on the vorticity equation and uses De Giorgi iteration. In this local study, we can obtain any regularity of the vorticity without any a priori knowledge of the pressure. The local-to-global step uses a recently constructed maximal function for transport equations.
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1 Introduction
We study the three dimensional incompressible Navier–Stokes equations
Here \(u: (0, T) \times {\mathbb {R}}^3 \rightarrow {\mathbb {R}}^3\) and \(P: (0, T) \times {\mathbb {R}}^3 \rightarrow {\mathbb {R}}\) represent the velocity field and the pressure field of a fluid in \({\mathbb {R}}^3\), within a finite or infinite timespan of length T. Initial condition
is given by a divergence-free velocity profile \(u _0\) of finite energy.
Leray [11] and Hopf [8] proved the existence of weak solutions for all time. They constructed solutions \(u \in C _w (0, \infty ; L ^2 ({\mathbb {R}}^3)) \cap L ^2 (0, \infty ; \dot{H} ^1 ({\mathbb {R}}^3))\) corresponding to each aforementioned initial value, and satisfying (1) in the sense of distribution. A weak solution is called a Leray-Hopf solution if it satisfies energy inequality
for every \(t > 0\). Since Leray and Hopf much work has been developed in regard to the uniqueness and regularity of weak solutions. Nonuniqueness of weak solutions was proven very recently by Buckmaster and Vicol using convex integration scheme [1]. However, the question of the uniqueness of Leray-Hopf solutions still remains open. The uniqueness is related with the regularity of solutions by the Ladyženskaya-Prodi-Serrin criteria [7, 10, 14, 20, 21]: if the velocity belongs to any space interpolating \(L ^2 _t L ^\infty _x\) and \(L ^\infty _t L ^3 _x\) then it is actually smooth, and hence unique. The endpoint case \(L ^\infty _t L ^3 _x\) came much later by Iskauriaza, Serëgin and Shverak [9]. These spaces require \(\frac{1}{6}\) higher spatial integrability than the energy space provides, which is \({\mathcal {E}} = L ^\infty _t L ^2 _x \cap L ^2 _t \dot{H} ^1 _x\).
At the level of energy space, Scheffer began to study the partial regularity for a class of Leray-Hopf solutions, called suitable weak solutions [16,17,18,19]. These solutions exist globally and satisfy the following local energy inequality:
Scheffer showed the singular set, at which the solution is unbounded nearby, has time-space Hausdorff dimension at most \(\frac{5}{3}\). This result was later improved by Caffarelli, Kohn and Nirenberg in [2] (see also [12, 23]), where they showed the 1-dimensional Hausdorff measure of the singular set is zero. We will investigate the regularity of suitable weak solutions. In the periodic setting, Constantin constructed suitable weak solutions whose second derivatives have space-time integrability \(L ^{\frac{4}{3} - \varepsilon }\) for any \(\varepsilon > 0\), provided the initial vorticities are bounded measures [6]. This was improved by Lions to a slightly better space \(L ^{\frac{4}{3}, \infty }\), a Lorentz space which corresponds to weak \(L ^\frac{4}{3}\) space [13]. These estimates are extended to higher derivatives of smooth solutions by one of the authors and Choi using blow-up arguments: \(L ^{p, \infty } _{\mathrm {loc}}\) space-time boundedness for \((-\varDelta ) ^\frac{\alpha }{2} \nabla ^n u\), where \(p = \frac{4}{n + \alpha + 1}\), \(n \ge 1\), \(0 \le \alpha < 2\) [5, 22]. They also constructed suitable weak solutions satisfying these bounds for \(n + \alpha < 3\).
The aim of this paper is to improve these regularity results in Lorentz space. The main result is the following. Note that the estimate does not rely on the size of the pressure.
Theorem 1
Suppose we have a smooth solution u to the Navier–Stokes equations in \((0, T) \times {\mathbb {R}}^3\) for some \(0 < T \le \infty \) with smooth divergence free initial data \(u _0 \in L ^2\). Then for any integer \(n \ge 0\), for any real number \(q > 1\), the vorticity \(\omega = {\text {curl}}u\) satisfies
for some constant \(C _n\) depending on n and \(C _{q, n}\) depending only on q and n, uniform in T. The above estimate (2) also holds for suitable weak solutions with only \(L ^2\) divergence free initial data in the case \(n = 1\).
This theorem gives the following improvement on the second derivatives:
Corollary 1
Let u be a suitable weak solution in \((0, \infty ) \times {\mathbb {R}}^3\) with initial data \(u _0 \in L ^2\). Then for any \(q > \frac{4}{3}\), \(K \subset \subset (0, \infty ) \times {\mathbb {R}}^3\), there exists a constant \(C _{q, K}\) depending on q and K such that the following holds
Let us explain the main ideas of the proof. Similar as previous work on higher derivatives, the proof is also based on blow-up techniques. In particular, we blow up the equation along a trajectory, using the scaling symmetry and the Galilean invariance of the Navier–Stokes equations. That is, if we fix an initial time \(t _0\) and move the frame of reference along some X(t), and zoom in into \(\varepsilon \) scale, then it is easy to verify that \({{\tilde{u}}} (s, y)\) and \({{\tilde{P}}} (s, y)\), defined by
also satisfy the Navier–Stokes equation
We develop the following local theorem for \({{\tilde{u}}}\) and \({{\tilde{P}}}\). Note that it needs nothing from the pressure. Denote \(B _r \subset {\mathbb {R}}^3\) to be a ball centered at the origin with radius r, and \(Q _r = (-r ^2, 0) \times B _r \subset {\mathbb {R}}^4\) to be a space-time cylinder.
Theorem 2
(Local Theorem) There exists a universal constant \(\eta _1 > 0\), such that for any suitable weak solution u to the Navier–Stokes equations in \((-4, 0) \times {\mathbb {R}}^3\) satisfying
where \(\phi \in C _c ^\infty (B _1)\) is a non-negative function with \(\int \phi = 1\), \(\omega = {\text {curl}}u\) is the vorticity, \(\frac{4}{3} \le p _1 \le \infty \), \(1 \le p _2 \le \infty \), \(1 \le q _1, q _2 < 3\) satisfying
then for any integer \(n \ge 0\), we have
for some constant \(C _n\) depending only on n.
Let us illustrate the ideas of how to go from this local theorem towards the main result. We want to choose a “pivot quantity”, blow up near a point, and use this quantity to control \(\nabla ^n \omega \). When we patch the local results together, we will obtain a nonlinear bound with the same scaling as the pivot quantity, so we want the pivot quantity to have the best possible scaling. The ideal pivot quantities would be \(\int |\nabla u| ^2 {\mathrm {d}}x {\mathrm {d}}t\) and \(\int |\nabla ^2 P| {\mathrm {d}}x {\mathrm {d}}t\). \(\int |u| ^\frac{10}{3} {\mathrm {d}}x {\mathrm {d}}t\) has a worse scaling and should not be used. However, we still need to control the flux in the local theorem, so we want to take out the mean velocity and control u by \(\nabla u\) using Poincaré’s inequality.
In order to take out the mean velocity, we choose X(t) to be the trajectory of the mollified flow so that (4) can be realized. Notice that a cylinder \(Q _r\) in the local (s, y) coordinate will be transformed into a “skewed cylinder” growing along X(t) in the global (t, x) coordinate. One of the authors recently constructed a maximal function \({\mathcal {M}}_{\mathcal {Q}}\) associated with these cylinders [24], which serves as a bridge between the local theorem and the global result, and is one of the main reasons for the improvement in this paper. The idea is, if locally the vorticity gradient can be controlled in \(L ^\infty \) by the integral of something in the skewed cylinder, and the integral in a skewed cylinder can be controlled by the maximal function \({\mathcal {M}}_{\mathcal {Q}}\), then vorticity gradient is pointwise bounded by the maximal function.
If one uses \(\int |\nabla u| ^2 {\mathrm {d}}x {\mathrm {d}}t\) and \(\int |\nabla ^2 P| {\mathrm {d}}x {\mathrm {d}}t\) as the pivot quantity, then unfortunately the best possible outcome would just be an \(L ^{1, \infty }\) bound, as obtained in [24]. The reason is, the maximal function is bounded on \(L ^p\) for \(p > 1\), but for \(p = 1\) it is only bounded from \(L ^1\) to \(L ^{1, \infty }\). Unfortunately \(|\nabla u| ^2\) and \(|\nabla ^2 P|\) are both \(L ^1\) quantities, so \({\mathcal {M}}_{\mathcal {Q}}\left( |\nabla u| ^2 + |\nabla ^2 P|\right) \) is only \(L ^{1, \infty }\). We need two things to improve from \(L ^{1, \infty }\): replace \(\int |\nabla u| ^2\) by \(\int |\nabla u| ^p\), and drop the pressure \(\nabla ^2 P\).
Suppose we could use \((\int |\nabla u| ^p {\mathrm {d}}x {\mathrm {d}}t) ^\frac{2}{p}\) as the pivot quantity for some \(p < 2\), then we can majorize it by \({\mathcal {M}}_{\mathcal {Q}}\left( |\nabla u| ^p\right) ^\frac{2}{p} \in L ^1\), since \(\frac{2}{p}>1\) and \({\mathcal {M}}_{\mathcal {Q}}\) is bounded in \(L ^\frac{2}{p}\). However, this poses significant difficulties in the local theorem. The nonlinear term \(u \cdot \nabla u\) is quadratic, and if we only have a subquadratic integrability to begin with, we cannot treat this quadratic transport term as a source term because it is not integrable. Observe that what we lack is the temporal integrability rather than the spatial one: if p is slightly smaller than two, then \(u \cdot \nabla u\) is still \(L ^{\frac{3}{2}-}\) in space, but \(L ^{1-}\) in time. To overcome this difficulty, we write \(u \cdot \nabla u\) as \(\omega \times u\) up to a gradient term, and put \(L ^{2-} _t L^{6-} _x\) on u and \(L ^{2+} _t L ^{2-} _x\) on \(\omega \). We compensate the lower integrability term by pairing with a higher integrability term to make \(\omega \times u\) integrable. \(L ^{2+} _t L ^{2-} _x\) of \(\omega \) can be interpolated between \(L ^{2-} _t L^{2-} _x\) and \(L ^\infty _t L ^1 _x\), while the latter is controlled by \(L ^2 _{t, x}\) of \(\nabla u\). Since \(L ^{2+} _t L ^{2-} _x\) is closer to \(L ^{2-} _t L^{2-} _x\) than to \(L ^\infty _t L ^1 _x\), the pivot quantity that we use is actually \(\delta ^{-\nu } \Vert \nabla u \Vert ^2 _{L ^p} + \delta \Vert \nabla u \Vert ^2 _{L ^2}\) for \(\nu \) close to 0. By using more subquadratic integrability and a tiny bit of the quadratic one, we can complete the task by interpolation. That is why we obtain \(L ^{1, q}\) in the end: it interpolates \(L ^1\) bound from \(\Vert \nabla u\Vert _{L ^p}\) and \(L ^{1, \infty }\) bound from \(\Vert \nabla u \Vert _{L ^2}\). Unfortunately we still miss the endpoint \(L ^1\).
The second task is more subtle and technical. Without any information on the pressure, we don’t have any control on the nonlocal effect. However, the role of the pressure is not important at the vorticity level; if we take the curl of the Navier–Stokes equation, the pressure will disappear and we are left with the vorticity equation involving only local quantities:
Inspired by Chamorro, Lemarié-Rieusset and Mayoufi [4], we introduce a new velocity variable \(v = -{\text {curl}}\varphi ^\sharp \varDelta ^{-1}\varphi \omega \) using only local information of vorticity (\(\varphi \) and \(\varphi ^\sharp \) are spatial cut-off functions), and this helps us to prove the local theorem. This is another main reason for the improvement in this paper. Consequently, the bounds we obtain in the end are on the vorticity \(\omega \) rather than on the velocity u.
This paper is organized as follow: in the preliminary Section 2 we introduce the analysis tools to the reader. We show how to rigorously derive the main results from the local theorem in Section 3, and then deal with technicalities of the local theorem in the later sections. The proof of the local theorem consists of three parts. Section 4 introduces the new variables v, and shows the smallness of v in the energy space. Then we use De Giorgi iteration argument in Section 5 to prove boundedness of v. Finally, we inductively bound \(\omega \) and all its higher derivatives in Section 6.
2 Preliminary
In this section, we introduce a few tools that we are going to use in the paper, including the maximal function, Lorentz space, and Helmholtz decomposition.
2.1 Maximal Function Associated with Skewed Cylinders
This is recently developed for incompressible flows in [24]. We quote useful results here without proof.
Suppose \(u \in L ^p (0, T; \dot{W} ^{1, p} ({\mathbb {R}}^3; {\mathbb {R}}^3))\) is a vector field in \({\mathbb {R}}^3\). Fix \(\phi \in C _c ^\infty (B _1)\) to be a nonnegative function with \(\int \phi = 1\) through out the paper. For \(\varepsilon > 0\) define \(\phi _\varepsilon (x) = \varepsilon ^{-3} \phi (-x/\varepsilon )\), and let \(u _\varepsilon (t, \cdot ) = u (t, \cdot ) * \phi _\varepsilon \) be the mollified velocity. For a fixed (t, x) we let X(s) solve the following initial value problem:
The skewed parabolic cylinder \(Q _\varepsilon (t, x)\) is then defined to be
We use \({\mathcal {M}}\) to denote the spatial Hardy-Littlewood maximal function, which is defined by
Then we construct the space-time maximal function adapted to the flow.
Theorem 3
(\({\mathcal {Q}}\)-Maximal Function) There exists a universal constant \(\eta _0\) such that the following is true. We say \(Q _\varepsilon (t, x)\) is admissible if \(Q _\varepsilon (t, x) \subset (0, T) \times {\mathbb {R}}^3\) and
Define the maximal function
If u is divergence free and \({\mathcal {M}}(|\nabla u|) \in L ^q\) for some \(1 \le q \le \infty \), then \({\mathcal {M}}_{\mathcal {Q}}\) is bounded from \(L ^1 ((0, T) \times {\mathbb {R}}^3)\) to \(L ^{1, \infty } ((0, T) \times {\mathbb {R}}^3)\) and from \(L ^p ((0, T) \times {\mathbb {R}}^3)\) to itself for any \(p > 1\) with norm depending on p.
An important consequence of the weak type (1, 1) bound of the Hardy-Littlewood maximal function is the Lebesgue differentiation theorem in \({\mathbb {R}}^n\). Similarly, we can use the \({\mathcal {Q}}\)-maximal function to prove the \({\mathcal {Q}}\)-Lebesgue differentiation theorem.
Theorem 4
(\({\mathcal {Q}}\)-Lebesgue Differentiation Theorem) Let \(f \in L ^1 _{\mathrm {loc}} ((0, T) \times {\mathbb {R}}^3)\). Then for almost every \((t, x) \in (0, T) \times {\mathbb {R}}^3\),
In this case we say (t, x) is a \({\mathcal {Q}}\)-Lebesgue point of f.
2.2 Lorentz Space
Let \((X, \mu )\) be a measure space. Recall that for a measurable function f, its decreasing rearrangement is defined as
For \(0< p < \infty \), \(0 < q \le \infty \), Lorentz space \(L ^{p,q} (X)\) is defined as the set of functions f for which
Now we introduct the interpolation lemma for Lorentz spaces.
Lemma 1
(Interpolation of Lorentz Spaces) Let \(\nu > 0\) be a fixed positive number. Assume \(f _0 \in L ^{p _0, q _0}\), \(f _1 \in L ^{p _1, q _1}\), where \(0< p _0, p _1< \infty , 0 < q _0, q _1 \le \infty \). If f is a measurable function satisfying
then \(f \in L ^{p, q}\), where
Proof
It is easy to check from the definition of decreasing rearrangement that if \(h \le f + g\), then \(h ^* (2 \lambda ) \le (f + g) ^* (2 \lambda ) \le f ^* (\lambda ) + g ^* (\lambda )\). Thus (9) implies
Set \(\theta = \frac{1}{1+\nu }\), \(\delta = f ^* _0 (\lambda )^{-\theta } f ^* _1 (\lambda )^{\theta }\), then
Therefore,
where \(C = 2 ^\frac{1}{p}\). \(\square \)
We would also like to mention that Riesz transform is bounded on Lorentz space. The proof can be found in [3]. See [15] for general Lorentz spaces.
Lemma 2
For \(1< p < \infty \), \(1 \le q \le \infty \), \(R _{ij} = \partial _i \partial _j \varDelta ^{-1}\) is a bounded linear operator from \(L ^{p,q} ({\mathbb {R}}^n)\) to itself. As a spatial operator, it is also bounded in time-space from \(L ^{p,q} ((0, T) \times {\mathbb {R}}^n)\) to itself.
2.3 Helmholtz Decomposition
First, recall two vector calculus identities:
For operators A and B, denote \([A, B] = AB - BA\) to be their commutator. Define \({\mathbb {P}} _{{\text {curl}}}= -{\text {curl}}{\text {curl}}\varDelta ^{-1}\) and \({\mathbb {P}} _{\nabla }= \nabla \varDelta ^{-1}{\text {div}}= {\text {Id}}- {\mathbb {P}} _{{\text {curl}}}\) to be the Helmholtz decomposition. Then we compute the following commutators:
The first two are straightforward. The third uses
and the last one is because
so we can expand \({\text {curl}}(\nabla \varphi \times \varDelta ^{-1}u)\) by (11).
Lemma 3
\(\partial _i [\varphi , {\mathbb {P}} _{{\text {curl}}}]\) and \([\varphi , {\mathbb {P}} _{{\text {curl}}}] \partial _i\) are both bounded linear operator from \(L ^p\) to \(L ^p\) for any \(1< p < \infty \), i.e.
Proof
First, we observe that by Jacobi identity \([\varphi , {\mathbb {P}} _{{\text {curl}}}] \partial _i\) and \(\partial _i [\varphi , {\mathbb {P}} _{{\text {curl}}}]\) differ by
which is bounded from \(L ^p\) to \(L ^p\) for any p, because both \({\mathbb {P}} _{{\text {curl}}}\) and multiplication by \(\partial _i \varphi \) are bounded from \(L ^p\) to \(L ^p\), so we can complete the proof by duality. For \(1< p < 3\), set \(\frac{1}{p ^*} = \frac{1}{p} - \frac{1}{3}\), from (15) we can see that
For \(\frac{3}{2}< p < \infty \), set \(1 - \frac{1}{p} = \frac{1}{q} = \frac{1}{q ^*} + \frac{1}{3}\), then \(1< p, q, q^* < \infty \). Take any \(u \in L ^p ({\mathbb {R}}^3)\) and any vector field \(v \in L ^q ({\mathbb {R}}^3)\) to get
\(\square \)
Corollary 2
\(\partial _i [\varphi , {\mathbb {P}} _{\nabla }]\) and \([\varphi , {\mathbb {P}} _{\nabla }] \partial _i\) are both bounded linear operator from \(L ^p\) to \(L ^p\) for any \(1< p < \infty \):
Proof
\({\text {Id}}= {\mathbb {P}} _{\nabla }+ {\mathbb {P}} _{{\text {curl}}}\) commutes with \(\varphi \), so \([\varphi , {\mathbb {P}} _{\nabla }] = -[\varphi , {\mathbb {P}} _{{\text {curl}}}]\). \(\square \)
Because of the smoothing effect of the Laplace potential, we have the following.
Lemma 4
Let \(\varphi \in C _c ^\infty ({\mathbb {R}}^3)\) be supported away from some openset \({\varOmega } \subset {\mathbb {R}}^3\), that is, \({\text {dist}} ({\text {supp}}\varphi , {\varOmega }) = d > 0\). Then for any \(f \in L ^1 _{\mathrm {loc}} ({\mathbb {R}}^3)\), \(k > 0\),
We also have
3 Proof of the Main Results
In this section, we show that the Local Theorem 2 leads to the main results. First, we show the pivot quantity is indeed enough to bound \(\nabla ^n \omega \).
Lemma 5
There exists \(\eta _2 > 0\) such that the following holds. Let \(\frac{11}{6}< p < 2\), \(\frac{2-p}{p-1} < \nu \le \frac{7p-12}{6-p}\). If u is a suitable solution to the Navier–Stokes equations in \((-9, 0) \times {\mathbb {R}}^3\) satisfying the condition
for some \(\delta \le \eta _2\),
then we have for any \(n \ge 0\),
Here \(C _n\) is the same constant in Theorem 2.
Proof
First, we claim that
Formally, we can take the dot product of both sides of the vorticity equation (1) with \(\omega ^0 :{=} \frac{\omega }{|\omega |}\), and recalling the convexity inequality \(\omega ^0 \cdot \varDelta \omega \le \varDelta |\omega |\), we have
Let \(\psi \in C _c ^\infty ((-9, 0] \times {\mathbb {R}}^3)\) be a cut-off function such that \(\mathbf{1} _{Q _2} \le \psi \le \mathbf{1} _{Q _3}\). Multiply (20) by \(\psi \) and then integrate in space to get
for some large universal constant \(C > 1\). The last step uses Poincaré’s inequaliting and (16). Integrate in time, we obtain
This proves the claim. A more rigorous proof can be obtained by difference quotient same as in Constantin [6] or Lions [13] Theorem 3.6, so we omit the details.
Now we interpolate between (17) and (19). Let \(\theta = \frac{1}{1+\nu }\),
where we choose \(\eta _2 = \frac{\eta _1}{4C + 1} \le \frac{1}{2} \eta _1\) from Theorem 2, and \(p _2, q _2\) are determined by
Combining the above with (17) we have
By the choice of \(\theta \) and the range of \(\nu \),
One can also easily check that \(p < 2\) implies \(q _2 < 2\), and thus by (16) and (21) the requirements of the Local Theorem 2 are satisfied with \(p _1 = q _1 = p\), and it completes the proof of the lemma. \(\square \)
Now we transform this lemma into the global coordinate. Recall that \(Q _\varepsilon (t, x)\) is defined by (7).
Corollary 3
There exists \(\eta _3 > 0\) such that, if for some \(\delta \le \eta _2\),
then
Proof
Define \({{\tilde{u}}}\) by (3). Then (22) implies
Moreover, (16) is satisfied by \({{\tilde{u}}}\). Therefore, if we choose \(\eta _3\) such that
then by Lemma 5, \({{\tilde{\omega }}} :{=} {\text {curl}}{{\tilde{u}}}\) has bounded derivatives at (0, 0), and thus finish the proof of the corollary by scaling. \(\square \)
Then we use the maximal function to go from the local bound to a global bound.
Proof of Theorem 1
First, we fix \(\frac{11}{6}< p < 2\), \(\frac{2-p}{p-1} < \nu \le \frac{7p-12}{6-p}\). Let \(\eta<< 1\) be a small constant to be specified later. Finally we fix a \(0< \delta < \infty \). For \((t, x) \in (0, T) \times {\mathbb {R}}^3\), define
If (t, x) is both a \({\mathcal {Q}}\)-Lebesgue point of \(|{\mathcal {M}}(\nabla u)| ^p\) and of \(|{\mathcal {M}}(\nabla u)| ^2\), then we claim that there exists a positive \(\varepsilon = \varepsilon _{(t, x)}\) such that one of two cases is true:
-
Case 1. \({3 \varepsilon _{(t, x)} < t ^\frac{1}{2}}\), and \({I (\varepsilon _{(t, x)}) = \eta }\).
-
Case 2. \({3 \varepsilon _{(t, x)} = t ^\frac{1}{2}}\), and \({I (\varepsilon _{(t, x)}) \le \eta }\).
This is because, by Theorem 4,
and \(I (\varepsilon )\) is a continuous function of \(\varepsilon \).
On the one hand, in both cases we have \(I (\varepsilon ) \le \eta \), which implies that
If we set \(\eta < \eta _0 ^2\), then depending on \(\delta \ge 1\) or \(\delta \le 1\), one of the two would imply admissibility condition (8) by Jensen’s inequality. Therefore \(Q _\varepsilon (t, x)\) is admissible and
so we can combine two cases and conclude
On the other hand, if we set \(\eta < \eta _3\), then in both cases \(I (\varepsilon ) \le \eta _3\). If \(\delta \le \eta _2\) one would have
by Corollary 3. If \(\delta > \eta _2\), notice that by Jensen’s inequality,
so
If we require \(\eta < (1 - \eta _2) \eta _2 ^{2 \nu } \eta _3\), then
Again by Corollary 3, we would still have (24). In conclusion, we choose
Then for any \(0< \delta < \infty \) one would have
by putting (24) and (23) together. Denote \(f = |\nabla ^n \omega | ^\frac{4}{n+2}\), and we denote \(f _1 = {\mathcal {M}}_{\mathcal {Q}}({\mathcal {M}}(|\nabla u|) ^p) ^\frac{2}{p}\), \(f _2 = {\mathcal {M}}_{\mathcal {Q}}({\mathcal {M}}(|\nabla u|) ^2)\). Then we have almost everywhere
By Theorem 3,
Finally, by the interpolation between Lorentz spaces Lemma 1,
This proves the theorem for \(q \ge 1 + 2 \nu \). Recall that p can be arbitrarily chosen between \(\frac{11}{6}\) and 2, and \(\nu \) can be chosen between \(\frac{2-p}{p-1}\) and \(\frac{7p-12}{6-p}\), so \(\nu \) can be arbitrarily small, therefore we prove the theorem for any \(q > 1\). \(\square \)
Estimates on \(\nabla ^2 u\) can be obtained by a Riesz transform of \(\varDelta u = -{\text {curl}}\omega \).
Proof of Corollary 1
We can put \(K \subset (t _0, T) \times B _R\) for some \(t _0, T, R > 0\). Denote \(Q = (t _0, T) \times B _{2R}\). Let \(\rho \in C _c ^\infty ({\mathbb {R}}^3)\) be a smooth spatial cut-off function between \(\mathbf{1} _{B _R} \le \rho \le \mathbf{1} _{B _{2R}}\). Then
Since \(\varDelta u = -{\text {curl}}\omega \), the case \(n = 1\) of Theorem 1 gives
so
As for lower order terms,
For Leray-Hopf solution, \(\Vert \nabla u\Vert _{L ^\infty _t L ^2 _x \cap L ^2 _t \dot{H} ^1 _x ((0, T) \times {\mathbb {R}}^3)} \le \Vert u _0\Vert _{L ^2}\), so
Because Riesz transform is bounded from \(L ^{\frac{4}{3}, q} ((t _0, T) \times {\mathbb {R}}^3)\) to itself by Lemma 2,
\(\square \)
Remark 1
For smooth solutions to the Navier–Stokes equation, we have \(L ^{1, q}\) estimate for the third derivatives for any \(q > 1\),
4 Local Study: Part One, Initial Energy
The next three sections are dedicated to the proof of the Local Theorem 2. In [22], the proof of the local theorem consists of the following three parts:
-
Step 1. Show the velocity u is locally small in the energy space \({\mathcal {E}} = L ^{\infty } _t L ^2 _x \cap L ^2 _t H ^1 _x\).
-
Step 2. Use De Giorgi iteration and the truncation method developed in [23] to show u is locally bounded in \(L ^\infty \).
-
Step 3. Bootstrap to higher regularity by differentiating the original equation.
In our case, directly working with u is difficult due to the lack of control on the pressure, which is nonlocal. Therefore, we would like to work on vorticity, whose evolution is governed by (6) and only involves local quantities. Since \(\omega \) is one derivative of u, we have less integrability to do any parabolic regularization, and we don’t have the local energy inequality to perform De Giorgi iteration. This motivates us to work on minus one derivative of \(\omega \), but instead of \(\omega \) we use a localization of \(\omega \). Similar as [4], we introduce a new local quantity
where \(\varphi \) and \(\varphi ^\sharp \) are a pair of fixed smooth spatial cut-off functions, which are defined between \(\mathbf{1} _{B _\frac{6}{5}} \le \varphi \le \mathbf{1} _{B _\frac{5}{4}}\), \(\mathbf{1} _{B _\frac{4}{3}} \le \varphi ^\sharp \le \mathbf{1} _{B _\frac{3}{2}}\). This v is divergence free and compactly supported. It will help us get rid of the pressure P, while staying in the same space as u: it scales the same as u, has the same regularity, inherit a local energy inequality from u, and its evolution only depends on local information. We will follow the same three steps above, but we will work on v instead of u.
For convenience, from now on we will use \(\eta \) to denote a small universal constant depending only on the smallness of \(\eta _1\), such that \(\lim _{\eta _1 \rightarrow 0} \eta = 0\). Similar as the constant C, the value of \(\eta \) may change from line to line. The purpose of this section is to obtain the smallness of v in the energy space \({\mathcal {E}}\), which is the following proposition:
Proposition 1
Under the same assumptions of the Local Theorem 2, we have
For convenience, define \(q _3\), \(q _4\), \(q _5\) by
4.1 Equations of v
We use (10) in (1) to rewrite the equation of u, then take the curl to rewrite the equation of \(\omega \), finally apply \(-{\text {curl}}\varphi ^\sharp \varDelta ^{-1}\varphi \) on the vorticity equation to obtain the equation of v:
The second term of (26) is
where \({\mathbf {B}}\) denotes the quadratic commutator
Here we used (12). The right hand side of (26) is
where \({\mathbf {L}}\) denotes the linear commutator
Here we used (13). Therefore we have the equation for v as the following:
We observe the following localization decomposition:
Lemma 6
We can decompose
where w and \(\varpi \) are harmonic inside \(B _1\).
Proof
We can compute v by
using \({\text {div}}u = 0\). We denote
which implies the first decomposition \(\varphi u = v + w\). By taking the curl,
We denote
which implies the second decomposition \(\varphi \omega = {\text {curl}}v + \varpi \). We can easily see that \(\varDelta w\) and \(\varDelta \varpi \) are both the sum of a smooth function supported outside \(B _\frac{3}{2}\) and the Newtonian potential of something supported inside \({\text {supp}} (\nabla \varphi ) \subset B _{\frac{5}{4}} \setminus B _{\frac{6}{5}}\), so they are harmonic inside \(B _1\). \(\square \)
Using this decomposition, we can continue to expand
where \({\mathbf {W}}\) denotes the remainders involving w and \(\varpi \),
By subtracting (11) from (10), for divergence free u, v we have
so
For convenience, denote the Riesz operator
Finally, we have the equation of v as
We now check the spatial integrability of these new terms.
Lemma 7
For any \(1< p < \infty \),
If we denote \(q = (\frac{1}{p} - \frac{1}{3}) _+ ^{-1}\), then
Proof
\(v, w, \varpi \) are all supported inside \(B _2\), so
Here we used Lemma 4 since \(\varphi \) and \(1 - \varphi ^\sharp \) are supported away from each other, and we also used the boundedness of Riesz transform by Lemma 2. Their derivatives are bounded by
The proof for \({\mathbf {B}}, {\mathbf {L}}, {\mathbf {W}}\) are similar, so we omit it here. \(\square \)
Since \(u \in {\mathcal {E}}\) and \(\omega \in L ^\infty _t L ^1 _x\), it can be seen from the above lemma that \(v, \nabla w, \varpi \in {\mathcal {E}}\), thus
therefore \({\mathbf {B}}, {\mathbf {L}}, {\mathbf {W}}\in L ^1 _t L ^3 _{\mathrm {loc}, x} + L ^2 _x L ^\frac{3}{2} _{\mathrm {loc},x}\). In the appendix we prove the suitability for v: it satisfies the following local energy inequality:
4.2 Energy Estimate
Multipling (29) by \(\varphi ^4\) then integrating over \({\mathbb {R}}^3\) yields
Let us discuss these terms. For the first four terms on the right hand side,
Here we use Hölder’s inequality, and \(\varphi \) is compactly supported in \(B _2\) and \( \frac{1}{q_2} + \frac{1}{q _3} + \frac{1}{6} \le 1 \). For the \({\mathbf {W}}\) term,
For the first one, we break it as
Using (10),
we have
The remaining is of lower order:
For the second one,
where
Thus \(I _{\mathbf {W}}\) can be bounded by
In summary, we conclude that, for \(-4 \le t \le 0\),
with good estimates on each of the term on the right.
4.3 Proof of Proposition 1
First we check the integrability of each terms.
Lemma 8
(Integrability) Given conditions (4) and (5), we have
Proof
Integrability of u is obtained by Sobolev embedding and that \(\varphi u\) has average 0. Integrability of \(\varphi \omega \) is given. The remaining are consequences of Lemma 7 and Sobolev embedding. \(\square \)
Proof of Proposition 1
We prove Proposition 1 using a Grönwall argument. Multiply (35) by an increasing smooth function \(\psi _1 (t)\) with \(\psi _1 (t) = 0\) for \(t \le -2\), \(\psi _1 (t) = 1\) for \(t \ge -1\), we have
Formally, we can integrate from \(-4\) to \(t < 0\) and to get
This integration is justified since v satisfies the local energy inequality (29) in distribution, and \(\psi _1 (t) \varphi ^4 (x) \in C _c ^\infty ((-4, 0] \times B _2)\). Because of (30), (31), (32), (33), (34), and
we can conclude that
where
Here we used interpolation for \( \Vert v \Vert _{L ^2} ^2 \le \Vert v \Vert _{L ^{q _3} _x} \Vert v \Vert _{L ^{q _4} _x}. \) Therefore
By Grönwall’s lemma, we conclude that for every \(-4 \le t \le 0\),
Therefore by taking the sup over \(-1 \le t \le 0\) and \(t = 0\) respectively, we conclude that
\(\square \)
5 Local Study: Part Two, De Giorgi Iteration
In this section, we derive the boundedness of v in \(Q _\frac{1}{2}\) which is the following.
Proposition 2
Let v solves (28). If (25) holds for sufficiently small \(\eta \), and we have integrability bounds in Lemma 8, then we have
The proof uses De Giorgi technique and the truncation method. First, we set a dyadically shrinking radius
Then we define dyadically shrinking cylinder \(Q _k\)’s
We also introduce positive smooth space-time cut-off functions \(\rho _k\) and \(\rho _k^\sharp \) with
Then, let \(c _k\) denote a sequence of rising energy level
We define analogous of vector derivative \(d _k\) and energy quantity \(U_k\) to get
We have the following truncation estimates:
Lemma 9
Proof
The first estimate follows from the definition. By Lemma 4 in [23], we have \(|\nabla v _k| \le d _k\) and \(|\nabla (\beta _kv)| \le 3 d _k\). Moreover, since \(|\nabla |v|| \le |\nabla v| ^2\), we see \(d _k \le d _{k - 1}\), as \(v _k\) and \(\beta _k\) are monotonously decreasing, so
Moreover, the truncation gives \(|\beta _kv| + 2 ^{-k} \mathbf{1} _{k} = v _k + 2 ^{-k} \mathbf{1} _{k} = \mathbf{1} _{k} v _{k - 1}\), so
\(\square \)
Corollary 4
(Nonlinearization) If \(f \in L ^p _t L ^q _x (Q _{k - 1})\), with
for some \(0 \le \theta \le 1\), \(0 < \sigma \le \gamma \), then, uniformly in \(\sigma \),
Proof
By interpolation,
where
Therefore, using Hölder’s inequality,
\(\square \)
First, we recall the following identities from [23]:
Since \(\alpha _kv\) is bounded, we can multiply equation (28) by \(\alpha _kv\) and obtain
using (38) and (39). Denote \({\mathbf {C}}_v= {\mathbf {B}}+ {\mathbf {L}}+ {\mathbf {W}}\). Subtracting (40) from (29), we have
Multipling by \(\rho _k\), then integrating in space and from \(\sigma \) to \(\tau \) in time,
Taking the sup over \(\tau > -T _k ^\flat \), and set \(\sigma < -T _{k - 1} ^\flat \), we obtain
Using Corollary 4, the first one is bounded by
Now let us deal with the last few terms. For simplicity, we use \(\iint {\mathrm {d}}x {\mathrm {d}}t\) to denote \(\int _{-T _k ^\natural } ^\tau \int _{{\mathbb {R}}^3} {\mathrm {d}}x {\mathrm {d}}t\) in the rest of this section.
5.1 Highest Order Nonlinear Term
Define three trilinear forms:
There are symmetric on \(v _2\), \(v _3\) positions. When we have enough integrability, that is, when
we have Leibniz is rule
The goal is to estimate the first two double integrals in (41),
We first separate \(w \otimes v\) from \(u \otimes v\), and we have
Denote \(-\nabla {\mathbf {R}}(w \otimes v) {=}: {\mathbf {W}}_2\) and we will deal with it later. The remaining \((u - w) \otimes v\) can be separated into interior part and exterior part,
The exterior part is bounded and smooth in space over the support of \(\rho _k\):
Here, we denote
Therefore we can use Leibniz is rule similarly as to as w and
by nonlinearization Corollary 4. The interior part is
Notice that the boundedness of \(\alpha _kv\) guarantees enough integrability to switch between trilinear forms. Then
In conclusion,
5.2 Lower Order Terms
For the bilinear and linear term, recall that inside \(B _1\),
Therefore,
Thus
5.3 W Terms
Finally, let us deal with
Here \(\nabla {\mathbf {R}}= \frac{1}{2} \nabla {\text {tr}}- {\mathbb {P}} _{\nabla }{\text {div}}\), so
Hence
Again, we separate \({\mathbf {W}}+ {\mathbf {W}}_2\) into exterior and interior part, with
where
Similarly as to the bilinear terms, \(\rho _k{\mathbf {W}}_{\mathrm {ext}}\) is small in \(L ^{p _3} _t L ^\infty _x\). Among the three terms in \({\mathbf {W}}_{\mathrm {int}}\), \(\rho _k^\sharp \varpi \times w\) is bounded in \(L ^{p _3} _t L ^\infty _x\), and \(\rho _k^\sharp \varpi \) is in \(L ^{p _2} _t L ^\infty _x\). Finally, for the first term,
Every term is a product of v and \(\nabla \rho _k^\sharp w\) (possibly with a Riesz transform) except \(\rho _k^\sharp w \cdot \nabla v\). Because in \({\varOmega } _k\), \(\nabla |v| = \nabla v _k\) are the same, we have
Therefore, every term of \({\mathbb {P}} _{{\text {curl}}}({\text {curl}}v \times \rho _k^\sharp w)\) is a product of v and \(\nabla \rho _kw\) or \(\nabla \rho _k^\sharp w\). Inside \(B _1\), \(w \in L ^{p _1} _t C ^\infty _x\). In conclusion,
so the sum is bounded in
provided that \(U _{k - 1} < 1\).
5.4 Proof of Proposition 2
Proof of Proposition 2
Coming back to (41), by estimates (42) on the first term, (43) on the trilinear terms, (44), (45) on the \({\mathbf {B}}, {\mathbf {L}}\) terms and (46) on the \({\mathbf {W}}\) terms, we conclude that
provided that \(U _{k - 1} < 1\). Here \(p _3 > 1\) ensures the index is strictly greater than 1. Since
by Proposition 1, we know that if \(\eta \) is small enough, \(U _k \rightarrow 0\) as \(k \rightarrow \infty \). Thus in \(Q _\frac{1}{2}\), \(|v| \le 1\) a.e.. This finishes the proof of Proposition 2. \(\square \)
6 Local Study: Part Three, More Regularity
In this section, we will show that the vorticity \(\omega \) is smooth in space. We will only work with the vorticity equation from now on. After the previous two steps, in \(B _\frac{1}{2}\) we should always decompose \(u = v + w\), because v is bounded and w is harmonic.
For convenience, given a vector \(\omega \), we denote
Let \(\partial _\bullet \) be the partial derivative in any space direction or time. Then we have
6.1 Bound Vorticity in the Energy Space
We will first show \(\omega \) is bounded in the energy space.
Proposition 3
If \(u = v + w\) in \(Q _\frac{1}{2}\), where v, w are bounded in
\(\omega = {\text {curl}}u\) solves the vorticity equation (6), then
-
(a)
\(\Vert \omega ^\frac{3}{4} \Vert _{{\mathcal {E}} (Q _\frac{1}{4})} \le C\),
-
(b)
\(\Vert \omega \Vert _{{\mathcal {E}} (Q _\frac{1}{8})} \le C\),
Proof of Proposition 3 (a)
We fix a pair of smooth space-time cut-off functions \(\varrho \) and \(\varsigma \) which satisfy
Take the dot product of the vorticity equation (6) with \(\frac{3}{2} \omega ^\frac{1}{2}\):
Therefore,
Multiply by \(\varrho ^6\) then integrate over space to get
For the left hand side, we can integrate by parts to get
where the latter can be controlled by
For the right hand side, using \(u = v + w\) over the support of \(\varrho \), we can separate
The \(\nabla v\) term can be controlled by
where
Here the second to the last inequality is due to \(\partial _i |\omega | ^\frac{3}{4} = \partial _i \omega ^\frac{3}{4} \cdot \omega ^0\). Since \(|v| \le 1\) over the support of \(\varrho \),
By using (50)–(54) in (49), we conclude that
By Hölder’s inequality,
Therefore we can write
where
since \(u = w + v\), and \(|v| \le 1\) inside \(B _\frac{1}{2}\). By (47),
Thus, by Grönwall’s inequality,
\(\square \)
Proof of Proposition 3(b)
From Proposition 3(a) and Sobolev embedding,
this interpolates the space
Multiply the vorticity equation (6) by \(\varsigma ^2 \omega \) then integrate over \({\mathbb {R}}^3\) to get
The first integral is \(L ^1\) in time because \(\omega \in L ^4 _t L ^2 _x\). For the second,
the latter is bounded \(L ^1\) in time, by \(u \in L ^\frac{4}{3} _t L ^\infty _x\) and \(\omega \in L ^4 _t L ^2 _x\). For the third integral,
w is bounded in \(L ^\frac{4}{3} _t \mathrm {Lip}_x\), and for v,
The former is \(L ^1\) in time, while the latter can be bounded by Cauchy-Schwartz,
In conclusion,
where
whose integral is bounded using (47),
By a Grönwall is argument, we have
\(\square \)
6.2 Bound Higher Derivatives in the Energy Space
Now we bootstrap to higher regularity of \(\omega \) using similar ideas as in the proof of Proposition 3.
Proposition 4
For any \(n \ge 1\), if \(u = v + w\) in \(Q _{8 ^{-n}}\), where v, w are bounded in
for some constant \(c _n\), \(\omega = {\text {curl}}u\) solves the vorticity equation (6), and is bounded in
then for any multiindex \(\alpha \) with \(|\alpha | = n\),
-
(a)
\(\Vert \nabla ^\alpha \omega ^\frac{3}{4} \Vert _{{\mathcal {E}} (Q _{8 ^{-n} / 4})} \le C _n\)
-
(b)
\(\Vert \nabla ^\alpha \omega \Vert _{{\mathcal {E}} (Q _{8 ^{-n-1}})} \le C _n\)
for some \(C _n\) depending on \(c _n\) and n.
Proof of Proposition 4(a)
Similarly, we fix smooth cut-off functions \(\varrho _n\) and \(\varsigma _n\) which satisfy
Differentiate (6) by \(\nabla ^\alpha \) to get
where
Multiply (58) by \(\frac{3}{2}\varrho _n^6 (\nabla ^\alpha \omega ) ^\frac{1}{2}\) then integrate in space to get
the same as in the proof of Proposition 3(a). Therefore,
Terms other than \({\mathbf {P}}_\alpha \) are dealt with in the same way as in Proposition 3:
The induction condition (57) ensures that \(\Vert \nabla ^\alpha \omega \Vert _{L ^2 (Q _{8 ^{-n}})} \le c _n\). Therefore
Now let’s focus on \({\mathbf {P}}_\alpha \):
We denote
First we estimate \({\mathbf {P}}_{v, k}\). By (55) and (57), when \(k = 0\),
and when \(0 < k \le n\),
Next we estimate \({\mathbf {P}}_{w, k}\). When \(0 \le k < n\),
Finally, when \(k = n\),
Therefore,
In conclusion, we have shown that
where
with integral
Taking the sum over all multi-index \(\alpha \) with size \(|\alpha | = n\), we have
Finally, Grönwall inequality gives
\(\square \)
Proof of Proposition 4(b)
Now we multiply (58) by \(\varsigma _n^2 \nabla ^\alpha \omega \) then integrate over \({\mathbb {R}}^3\) to get that
For the same reason, the only term that we need to take care of is \({\mathbf {P}}_\alpha \) term, and the others are dealt the same as with Proposition 3(b):
The last term can be absorbed into the left, and we will use Grönwall on the remaining terms.
Now we shall focus on the \({\mathbf {P}}_\alpha \) term. From Proposition 4(a), we have
Again, by interpolation,
First we estimate \({\mathbf {P}}_{w, k}\). In this case, for any \(0 \le k \le n\),
Then we estimate \({\mathbf {P}}_{v, k}\). When \(0 < k \le n\),
For the case \(k = 0\) of the \(v\) term, we put the curl on \(\nabla ^\alpha \omega \) to obtain
where \(|\nabla \nabla ^\alpha \omega |\) term can be absorbed to the left. By (59) and Sobolev embedding,
Therefore
In conclusion,
where
has the integral \(\int _{-8 ^{-2n} / 16} ^0 {\varPhi } (t) {\mathrm {d}}t \le C _n\). Finally, Grönwall is inequality gives
\(\square \)
6.3 Proof of the Local Theorem
Proof of the Local Theorem 2
First, Proposition 1 gives
where \(\eta \) can be chosen arbitrarily small if we pick \(\eta _1\) small. Next, by Proposition 2, we know that
These two steps implies (47). As for (48), \({\text {curl}}w = \varpi \) in \(B _1\), so we use interpolation in (37) to get that
w is harmonic inside \(B _1\), therefore
due to (36) and \(p _1 \ge \frac{4}{3}\). Therefore, we can use Proposition 3 to obtain
The next step is to use Proposition 4 iteratively. Suppose that for \(n \ge 1\) we know that
which is equivalent to (57). Let \(\varphi _n\) and \(\varphi ^\sharp _n\) be a pair of smooth spatial cut-off functions, with
and set
On the one hand, \(\nabla v _n\) is a Riesz transform of \(\varphi _n \omega \) up to lower order terms, so by the boundedness of Riesz transform we know that
On the other hand, we have similar boundedness estimates to those of Proposition 2 as before, so
\(w _n\) is harmonic in \(B _{\frac{1}{8^n + 4}}\), so we also have
Therefore, by Proposition 4,
By induction, we have
for any n. By Sobolev embedding, this implies, for any n, that
\(\square \)
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Acknowledgements
The first author was partially supported by the National Science Foundation grant: DMS 1907981, and the second author was partially supported by the National Science Fundation grant: DMS RTG 1840314.
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Suitability of Solutions
Suitability of Solutions
Theorem 5
Let u be a suitable weak solution to the Navier–Stokes equation in \({\mathbb {R}}^3\). That is, \(u \in L ^\infty _t L ^2 _x \cap L ^2 _t \dot{H} ^1 _x\) solves the following equation in the sense of distribution:
where P is the pressure, and u satisfies the following local energy inequality in the sense of distribution:
Suppose \(v \in L ^\infty _t L ^2 _x \cap L ^2 _t \dot{H} ^1 _x\) is compactly supported in space and solves the following equation,
where \(\omega = {\text {curl}}u\) is the vorticity, \({\mathbf {C}}_v\in L ^1 _t L ^2 _{\mathrm {loc},x} + L ^2 _t L ^\frac{6}{5} _{\mathrm {loc},x}\) is a force term, and
is a symmetric Riesz operator. Moreover, suppose v differs from \(\varphi u\) by
for some fixed \(\varphi \in C _c ^\infty ({\mathbb {R}}^3)\). Then v satisfies the following local energy inequality:
Proof
It is well-known that the pressure P can be recovered from u by
Since
the Navier–Stokes equation (60) can be rewritten as
and local energy inequality (61) can be rewritten as
First, multiply (64) by \(\varphi \), to get
Denote
for these commutator terms. Subtracting the equation of v from this equation of \(\varphi u\), we will have the equation for w. In summary,
Recall from [22] that \(\varDelta u \in L ^{\frac{4}{3}-\varepsilon } _{\mathrm {loc}(t, x)}\). Since \(\varDelta w \in L ^2 _{t, x}\), we have \(\varDelta v \in L ^{\frac{4}{3}-\varepsilon } _{\mathrm {loc}(t, x)}\). Moreover, \({\mathbf {C}}_u, {\mathbf {C}}_v\in L ^1 _t L ^2 _{\mathrm {loc},x} + L ^2 _t L ^\frac{6}{5} _{\mathrm {loc},x}\), and \(\varphi u, v \in {\mathcal {E}}\) are compactly supported. Therefore, we can multiply (66) and (67) by w, and (68) by \(\varphi u\) and v, to get
Now take the sum of (69)–(72). The \(\partial _t\) terms are
The \(\omega \times \) terms are
The \(\nabla {\mathbf {R}}\) terms are
Here we use \({\text {div}}v = 0, {\text {div}}(\varphi u) = {\text {div}}w = u \cdot \nabla \varphi \). The \(\varDelta \) terms are
The commutator terms are
In summary, half the sum of these four identities (69)–(72) gives
Next, multiply local energy inequality of u (65) by \(\varphi ^2\). Then
The quadratic commutator terms in (74) are
The cubic commutator terms in (74) are
Therefore, local energy inequality for \(\varphi u\) can be simplified as
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Vasseur, A., Yang, J. Second Derivatives Estimate of Suitable Solutions to the 3D Navier–Stokes Equations. Arch Rational Mech Anal 241, 683–727 (2021). https://doi.org/10.1007/s00205-021-01661-4
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DOI: https://doi.org/10.1007/s00205-021-01661-4