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Second Derivatives Estimate of Suitable Solutions to the 3D Navier–Stokes Equations

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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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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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Correspondence to Jincheng Yang.

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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:

$$\begin{aligned} \partial _tu + u \cdot \nabla u + \nabla P&= \varDelta u, \qquad {\text {div}}u = 0, \end{aligned}$$
(60)

where P is the pressure, and u satisfies the following local energy inequality in the sense of distribution:

$$\begin{aligned} \partial _t\frac{|u|^2}{2}+ {\text {div}}\left( u \left( \frac{|u|^2}{2}+ P \right) \right) + |\nabla u| ^2 \le \varDelta \frac{|u|^2}{2}. \end{aligned}$$
(61)

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,

$$\begin{aligned} \partial _tv + \omega \times v + \nabla {\mathbf {R}}(u \otimes v)&= \varDelta v + {\mathbf {C}}_v, \qquad {\text {div}}v = 0 \end{aligned}$$
(62)

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

$$\begin{aligned} {\mathbf {R}}= \frac{1}{2} {\text {tr}}-\varDelta ^{-1}{\text {div}}{\text {div}}\end{aligned}$$

is a symmetric Riesz operator. Moreover, suppose v differs from \(\varphi u\) by

$$\begin{aligned} \varphi u - v = w \in L ^\infty _t H ^1 _x \cap L ^2 _t H ^2 _x \end{aligned}$$

for some fixed \(\varphi \in C _c ^\infty ({\mathbb {R}}^3)\). Then v satisfies the following local energy inequality:

$$\begin{aligned} \partial _t\frac{|v|^2}{2}+ {\text {div}}\left( v {\mathbf {R}}(u \otimes v) \right) + |\nabla v| ^2&\le \varDelta \frac{|v|^2}{2}+ v \cdot {\mathbf {C}}_v. \end{aligned}$$
(63)

Proof

It is well-known that the pressure P can be recovered from u by

$$\begin{aligned} P = -\varDelta ^{-1}{\text {div}}{\text {div}}(u \otimes u). \end{aligned}$$

Since

$$\begin{aligned} u \cdot \nabla u + \nabla P&= \nabla \frac{|u|^2}{2}+ \omega \times u - \nabla \varDelta ^{-1}{\text {div}}{\text {div}}(u \otimes u) \\&= \omega \times u + \nabla {\mathbf {R}}(u \otimes u), \end{aligned}$$

the Navier–Stokes equation (60) can be rewritten as

$$\begin{aligned} \partial _tu + \omega \times u + \nabla {\mathbf {R}}(u \otimes u)&= \varDelta u, \end{aligned}$$
(64)

and local energy inequality (61) can be rewritten as

$$\begin{aligned} \partial _t\frac{|u|^2}{2}+ {\text {div}}\left( u {\mathbf {R}}(u \otimes u) \right) + |\nabla u| ^2 \le \varDelta \frac{|u|^2}{2}. \end{aligned}$$
(65)

First, multiply (64) by \(\varphi \), to get

$$\begin{aligned} \partial _t\varphi u + \omega \times \varphi u + \nabla {\mathbf {R}}(u \otimes \varphi u) = \varDelta (\varphi u) + [\nabla {\mathbf {R}}, \varphi ] (u \otimes u) + [\varphi , \varDelta ] u. \end{aligned}$$

Denote

$$\begin{aligned} {\mathbf {C}}_u= [\nabla {\mathbf {R}}, \varphi ] (u \otimes u) + [\varphi , \varDelta ] u \end{aligned}$$

for these commutator terms. Subtracting the equation of v from this equation of \(\varphi u\), we will have the equation for w. In summary,

$$\begin{aligned} \partial _t\varphi u + \omega \times \varphi u + \nabla {\mathbf {R}}(u \otimes \varphi u)&= \varDelta (\varphi u) + {\mathbf {C}}_u, \end{aligned}$$
(66)
$$\begin{aligned} \partial _tv + \omega \times v + \nabla {\mathbf {R}}(u \otimes v)&= \varDelta v + {\mathbf {C}}_v, \end{aligned}$$
(67)
$$\begin{aligned} \partial _tw + \omega \times w + \nabla {\mathbf {R}}(u \otimes w)&= \varDelta w + {\mathbf {C}}_u- {\mathbf {C}}_v. \end{aligned}$$
(68)

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

$$\begin{aligned} w \cdot \partial _t(\varphi u) + w \cdot \omega \times \varphi u + w \cdot \nabla {\mathbf {R}}(u \otimes \varphi u)&= w \cdot \varDelta (\varphi u) + w \cdot {\mathbf {C}}_u, \end{aligned}$$
(69)
$$\begin{aligned} w \cdot \partial _tv + w \cdot \omega \times v + w \cdot \nabla {\mathbf {R}}(u \otimes v)&= w \cdot \varDelta v + w \cdot {\mathbf {C}}_v\end{aligned}$$
(70)
$$\begin{aligned} \varphi u \cdot \partial _tw + \varphi u \cdot \omega \times w + \varphi u \cdot \nabla {\mathbf {R}}(u \otimes w)&= \varphi u \cdot \varDelta w + \varphi u \cdot ({\mathbf {C}}_u- {\mathbf {C}}_v). \end{aligned}$$
(71)
$$\begin{aligned} v \cdot \partial _tw + v \cdot \omega \times w + v \cdot \nabla {\mathbf {R}}(u \otimes w)&= v \cdot \varDelta w + v \cdot ({\mathbf {C}}_u- {\mathbf {C}}_v). \end{aligned}$$
(72)

Now take the sum of (69)–(72). The \(\partial _t\) terms are

$$\begin{aligned}&\varphi u \cdot \partial _tw + w \cdot \partial _t(\varphi u) + v \cdot \partial _tw + w \cdot \partial _tv \\&\quad = \partial _t(\varphi u \cdot w) + \partial _t(w \cdot v) \\&\quad = \partial _t(|\varphi u| ^2 - |v| ^2). \end{aligned}$$

The \(\omega \times \) terms are

$$\begin{aligned} w \cdot \omega \times \varphi u + \varphi u \cdot \omega \times w + w \cdot \omega \times v + v \cdot \omega \times w = 0. \end{aligned}$$

The \(\nabla {\mathbf {R}}\) terms are

$$\begin{aligned}&w \cdot \nabla {\mathbf {R}}(u \otimes \varphi u) + v \cdot \nabla {\mathbf {R}}(u \otimes w) + \varphi u \cdot \nabla {\mathbf {R}}(u \otimes w) + w \cdot \nabla {\mathbf {R}}(u \otimes v) \\&\quad = {\text {div}}(w {\mathbf {R}}(u \otimes \varphi u)) + {\text {div}}(v {\mathbf {R}}(u \otimes w)) \\&\qquad + {\text {div}}(\varphi u {\mathbf {R}}(u \otimes w)) + {\text {div}}(w {\mathbf {R}}(u \otimes v)) \\&\qquad - {\text {div}}(w) \nabla {\mathbf {R}}(u \otimes \varphi u) - {\text {div}}(v) \nabla {\mathbf {R}}(u \otimes w) \\&\qquad - {\text {div}}(\varphi u) \nabla {\mathbf {R}}(u \otimes w) - {\text {div}}(\varphi ) \nabla {\mathbf {R}}(u \otimes v) \\&\quad = 2 {\text {div}}(\varphi u {\mathbf {R}}(u \otimes \varphi u) - v {\mathbf {R}}(u \otimes v)) \\&\qquad - (u \cdot \nabla \varphi ) \left( \nabla {\mathbf {R}}(u \otimes \varphi u) + \nabla {\mathbf {R}}(u \otimes w) + \nabla {\mathbf {R}}(u \otimes v) \right) \\&\quad = 2 {\text {div}}(\varphi u {\mathbf {R}}(u \otimes \varphi u) - v {\mathbf {R}}(u \otimes v)) - 2 (u \cdot \nabla \varphi ) {\mathbf {R}}(u \otimes \varphi u). \end{aligned}$$

Here we use \({\text {div}}v = 0, {\text {div}}(\varphi u) = {\text {div}}w = u \cdot \nabla \varphi \). The \(\varDelta \) terms are

$$\begin{aligned}&\varphi u \cdot \varDelta w + w \cdot \varDelta (\varphi u) + v \cdot \varDelta w + w \cdot \varDelta v \\&\quad = \varDelta (u \cdot w) - 2 \nabla (\varphi u) : \nabla w + \varDelta (v \cdot w) - 2 \nabla v : \nabla w \\&\quad = \varDelta (|\varphi u| ^2 - |v| ^2) - 2 (|\nabla (\varphi u)| ^2 - |\nabla v| ^2). \end{aligned}$$

The commutator terms are

$$\begin{aligned} w \cdot {\mathbf {C}}_u+ \varphi u \cdot ({\mathbf {C}}_u- {\mathbf {C}}_v) + w \cdot {\mathbf {C}}_v+ v \cdot ({\mathbf {C}}_u- {\mathbf {C}}_v) = 2 \varphi u \cdot {\mathbf {C}}_u- 2 v \cdot {\mathbf {C}}_v. \end{aligned}$$

In summary, half the sum of these four identities (69)–(72) gives

$$\begin{aligned}&\partial _t\frac{|\varphi u| ^2 - |v| ^2}{2} + {\text {div}}(\varphi u {\mathbf {R}}(u \otimes \varphi u) - v {\mathbf {R}}(u \otimes v)) + |\nabla (\varphi u)| ^2 - |\nabla v| ^2 \nonumber \\&\quad = \varDelta \frac{|\varphi u| ^2 - |v| ^2}{2} + \varphi u \cdot {\mathbf {C}}_u- v \cdot {\mathbf {C}}_v+ (u \cdot \nabla \varphi ) {\mathbf {R}}(\varphi u \otimes u). \end{aligned}$$
(73)

Next, multiply local energy inequality of u (65) by \(\varphi ^2\). Then

$$\begin{aligned}&\partial _t\frac{|\varphi u| ^2}{2} + |\varphi \nabla u| ^2 + {\text {div}}\left( \varphi ^2 u {\mathbf {R}}(u \otimes u) \right) \nonumber \\&\quad \le \varDelta \frac{|\varphi u| ^2}{2} + [\varphi ^2, \varDelta ] \frac{|u|^2}{2}+ [{\text {div}}, \varphi ^2] \left( u {\mathbf {R}}(u \otimes u) \right) , \nonumber \\&\partial _t\frac{|\varphi u| ^2}{2} + |\nabla (\varphi u)| ^2 + {\text {div}}\left( \varphi u {\mathbf {R}}(u \otimes \varphi u) \right) \nonumber \\&\quad \le \varDelta \frac{|\varphi u| ^2}{2} + [\varphi ^2, \varDelta ] \frac{|u|^2}{2}+ |u \otimes \nabla \varphi | ^2 + 2 (u \otimes \nabla \varphi ) : (\varphi \nabla u) \nonumber \\&\qquad + [{\text {div}}, \varphi ^2] \left( u {\mathbf {R}}(u \otimes u) \right) + {\text {div}}(\varphi u [{\mathbf {R}}, \varphi ] (u \otimes u)). \end{aligned}$$
(74)

The quadratic commutator terms in (74) are

$$\begin{aligned}&[\varphi ^2, \varDelta ] \frac{|u|^2}{2}+ |u \otimes \nabla \varphi | ^2 + 2 (u \otimes \nabla \varphi ) : (\varphi \nabla u) \\&\quad = [\varphi ^2, \varDelta ] \frac{|u|^2}{2}+ |u| ^2 |\nabla \varphi | ^2 + 2 \nabla \varphi \cdot \varphi \nabla u \cdot u \\&\quad = -2 \nabla (\varphi ^2) \cdot \nabla \frac{|u|^2}{2}- \varDelta (\varphi ^2) \frac{|u|^2}{2}+ |u| ^2 |\nabla \varphi | ^2 + 2 \nabla \varphi \cdot \nabla u \cdot \varphi u \\&\quad = -4 \varphi \nabla \varphi \cdot \nabla \frac{|u|^2}{2}- \frac{1}{2} \varDelta (\varphi ^2) |u| ^2 + |u| ^2 |\nabla \varphi | ^2 + 2 \nabla \varphi \cdot \nabla u \cdot \varphi u \\&\quad = -2 \varphi \nabla \varphi \cdot \nabla u \cdot u- \varphi \varDelta \varphi |u| ^2 \\&\quad = \varphi u \cdot (-2 \nabla \varphi \cdot \nabla u - (\varDelta \varphi ) u) \\&\quad = \varphi u \cdot [\varphi , \varDelta ] u. \end{aligned}$$

The cubic commutator terms in (74) are

$$\begin{aligned}&[{\text {div}}, \varphi ^2] \left( u {\mathbf {R}}(u \otimes u) \right) + {\text {div}}(\varphi u [{\mathbf {R}}, \varphi ] (u \otimes u)) \\&\quad = 2 \varphi \nabla \varphi \cdot u {\mathbf {R}}(u \otimes u) + \varphi u \cdot \nabla [{\mathbf {R}}, \varphi ] (u \otimes u) + {\text {div}}(\varphi u) [{\mathbf {R}}, \varphi ] (u \otimes u) \\&\quad = 2 \varphi (u \cdot \nabla \varphi ) {\mathbf {R}}(u \otimes u) + \varphi u \cdot \nabla [{\mathbf {R}}, \varphi ] (u \otimes u) + (u \cdot \nabla \varphi ) [{\mathbf {R}}, \varphi ] (u \otimes u) \\&\quad = 2 \varphi (u \cdot \nabla \varphi ) {\mathbf {R}}(u \otimes u) + \varphi u \cdot \nabla [{\mathbf {R}}, \varphi ] (u \otimes u) \\&\qquad + (u \cdot \nabla \varphi ) {\mathbf {R}}(\varphi u \otimes u) - (u \cdot \nabla \varphi ) \varphi {\mathbf {R}}( u \otimes u) \\&\quad = \varphi u \cdot \nabla \varphi {\mathbf {R}}(u \otimes u) + \varphi u \cdot \nabla [{\mathbf {R}}, \varphi ] (u \otimes u) + (u \cdot \nabla \varphi ) {\mathbf {R}}(\varphi u \otimes u) \\&\quad = \varphi u \cdot [\nabla , \varphi ] {\mathbf {R}}(u \otimes u) + \varphi u \cdot \nabla [{\mathbf {R}}, \varphi ] (u \otimes u) + (u \cdot \nabla \varphi ) {\mathbf {R}}(\varphi u \otimes u) \\&\quad = \varphi u \cdot \left( [\nabla , \varphi ] {\mathbf {R}}- \nabla [\varphi , {\mathbf {R}}] \right) (u \otimes u) + (u \cdot \nabla \varphi ) {\mathbf {R}}(\varphi u \otimes u) \\&\quad = \varphi u \cdot [\nabla {\mathbf {R}}, \varphi ] (u \otimes u) + (u \cdot \nabla \varphi ) {\mathbf {R}}(\varphi u \otimes u). \end{aligned}$$

Therefore, local energy inequality for \(\varphi u\) can be simplified as

$$\begin{aligned}&\partial _t\frac{|\varphi u| ^2}{2} + |\nabla (\varphi u)| ^2 + {\text {div}}\left( \varphi u {\mathbf {R}}(u \otimes \varphi u) \right) \\&\quad \le \varDelta \frac{|\varphi u| ^2}{2} + \varphi u \cdot {\mathbf {C}}_u+ (u \cdot \nabla \varphi ) {\mathbf {R}}(\varphi u \otimes u). \end{aligned}$$

Subtracting (73) from this, we obtain (63). \(\square \)

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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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