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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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