Global Well-Posedness of the 3D Generalized Navier-Stokes Equations with Fractional Partial Dissipation

Abstract

The existence of a global solution to the 3D incompressible Navier-Stokes equations is an outstanding open problem. However, the 3D incompressible Navier-Stokes equations with hyperdissipation \((-\Delta )^{\alpha }u\) always possesses a global smooth solution if \(\alpha \geq \frac{5}{4}\). Yang-Jiu-Wu (The 3D incompressible Navier-Stokes equations with partial hyperdissipation, Math. Nachr. 292(8):1823–1836, 2019) reduced the hyperdissipation \((-\Delta )^{\alpha }u\) to \(((\Lambda _{1}^{\frac{5}{2}}, \Lambda _{2}^{\frac{5}{2}})u_{1}, (\Lambda _{2}^{ \frac{5}{2}}, \Lambda _{3}^{\frac{5}{2}})u_{2}, ( \Lambda _{3}^{\frac{5}{2}}, \Lambda _{1}^{\frac{5}{2}})u_{3})^{\top }\) and obtained the global existence and uniqueness in \(H^{1}\) space. However, the higher-order derivative estimate of the solution is a nontrivial question. In this paper, we prove that this solution is a global solution in \(H^{s}\) with \(s>\frac{5}{2}\) applying a single directional commutator estimate.

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Acknowledgements

The authors would like to thank the referee for the valuable comments and suggestions which helped to improve the paper greatly. The research of B Yuan was partially supported by the National Natural Science Foundation of China (No. 11471103).

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Appendix A

Appendix A

In this section, we present the Littlewood-Paley theory, the definition of Besov space and the proof of the Proposition 2.1.

Choose two nonnegative radial functions \(\chi \), \(\varphi \in C_{c}^{\infty }(\mathbb{R}^{3})\) supported in \(\mathcal{B} =\{\xi \in \mathbb{R}^{3}, |\xi |\leq \frac{4}{3}\}\) and \(\mathcal{C}=\{\xi \in \mathbb{R}^{3}, \frac{3}{4}\leq |\xi |\leq \frac{8}{3}\}\) respectively, such that

$$\begin{aligned} \chi (\xi )+\sum _{j\geq 0}\varphi (2^{-j}\xi )=1, \quad \xi \in \mathbb{R}^{3}, \\ \sum _{j\in \mathbb{Z}}\varphi (2^{-j}\xi )=1, \quad \xi \in \mathbb{R}^{3}\setminus \{0\}. \end{aligned}$$

Let \(h=\mathcal{F}^{-1}\varphi \) and \(\widetilde{h}=\mathcal{F}^{-1}\chi \), the frequency localization operators are defined by

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

Formally, \(\Delta _{j}\) is a frequency projection to the annulus \(\{|\xi | \approx 2^{j}\}\), \(S_{j}\) is a frequency projection to the ball \(\{|\xi | \leq 2^{j}\}\). Observe that \(\Delta _{j}=S_{j+1}-S_{j}\).

Let \(s\in \mathbb{R}\), \(1\leq p\), \(q\leq \infty \), the inhomogeneous Besov space \({B}^{s}_{p,q}\) is defined by

$$\begin{aligned} {B}^{s}_{p,q}=\{f\in \mathcal{S}'(\mathbb{R}^{3})\big| \|f\|_{{B}^{s}_{p,q}}< \infty \}, \end{aligned}$$

where

$$\begin{aligned} \|f\|_{{B}^{s}_{p,q}}= \textstyle\begin{cases} \bigg(\sum _{j\geq -1}2^{jsq}\|\triangle _{j}f\|_{p}^{q}\bigg)^{ \frac{1}{q}}, \ if\ q< \infty , \\ \mathrm{sup}_{j\geq -1} 2^{js}\|\triangle _{j}f\|_{L^{p}}, \ if \ q= \infty . \end{cases}\displaystyle \end{aligned}$$

In particular, \(\|f\|_{B^{s}_{2,2}}\thickapprox \|f\|_{H^{s}}\).

Now, we prove the Proposition 2.1. Taking the inner product with \(u\) in \(\mathbb{R}^{3}\times (0, T)\) on (1.3)1-(1.3)3, respectively, using integration by part and divergence free condition of \(u\), one has

$$\begin{aligned} &\|u\|^{2}_{2}+2\int _{0}^{T} \bigg(\| (\Lambda _{1} ^{\frac{5}{ 4}}, \Lambda _{2}^{\frac{5}{ 4}}) u_{1} \|^{2}_{2} +\| ( \Lambda _{2}^{ \frac{5}{ 4}}, \Lambda _{3}^{\frac{5}{ 4}})u_{2}\|^{2}_{2} +\| ( \Lambda _{3}^{\frac{5}{ 4}} , \Lambda _{1}^{\frac{5}{ 4}} )u_{3}\|^{2}_{2} \bigg)\text{d}t\le \|u_{0}\|^{2}_{2}. \end{aligned}$$
(4.1)

In the following, we establish the \(\dot{H}^{1}\)-estimate. Putting the operator \(\nabla \) to the equations (1.3)1-(1.3)3 and taking the \(L^{2}\) inner product with \(\nabla u_{i}\) for \(i=1,2,3\), respectively, we have

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \|\nabla u\|^{2}_{2}+ \| (\Lambda _{1} ^{ \frac{5}{ 4}}, \Lambda _{2}^{\frac{5}{ 4}}) \nabla u_{1} \|^{2}_{2} + \| ( \Lambda _{2}^{\frac{5}{ 4}}, \Lambda _{3}^{\frac{5}{ 4}})\nabla u_{2}\|^{2}_{2} +\| (\Lambda _{3} ^{\frac{5}{ 4}} , \Lambda _{1} ^{\frac{5}{ 4}} )\nabla u_{3}\|^{2}_{2} \\ =&-\sum _{i,j,k=1}^{3}\int _{\mathbb{R}^{3}}\partial _{j} u_{k} \partial _{k} u_{i} \partial _{j} u_{i} dx\triangleq I. \end{aligned}$$
(4.2)

In the following, we bound the term \(I\).

$$\begin{aligned} I =&-\sum _{i,k=1}^{3}\int _{\mathbb{R}^{3}}\partial _{1} u_{k} \partial _{k} u_{i} \partial _{1} u_{i} \text{d}x -\sum _{i,k=1}^{3} \int _{\mathbb{R}^{3}}\partial _{2} u_{k} \partial _{k} u_{i} \partial _{2} u_{i} \text{d}x -\sum _{i,k=1}^{3}\int _{\mathbb{R}^{3}} \partial _{3} u_{k} \partial _{k} u_{i} \partial _{3} u_{i} \text{d}x \\ \triangleq &I_{1}+I_{2}+I_{3}. \end{aligned}$$

We estimate \(I_{1}\) in detail.

$$\begin{aligned} I_{1} =& -\int _{\mathbb{R}^{3}}\partial _{1} u_{1} \partial _{1}u_{1} \partial _{1} u_{1} \text{d}x-\int _{\mathbb{R}^{3}}\partial _{1} u_{1} \partial _{1} u_{2} \partial _{1} u_{2} \text{d}x-\int _{\mathbb{R}^{3}} \partial _{1} u_{1} \partial _{1} u_{3} \partial _{1} u_{3} \text{d}x \\ -&\int _{\mathbb{R}^{3}}\partial _{1} u_{2} \partial _{2} u_{1} \partial _{1} u_{1} \text{d}x-\int _{\mathbb{R}^{3}}\partial _{1} u_{2} \partial _{2} u_{2} \partial _{1} u_{2} \text{d}x-\int _{\mathbb{R}^{3}} \partial _{1} u_{2} \partial _{2} u_{3} \partial _{1} u_{3} \text{d}x \\ -&\int _{\mathbb{R}^{3}}\partial _{1} u_{3} \partial _{3} u_{1} \partial _{1} u_{1} \text{d}x-\int _{\mathbb{R}^{3}}\partial _{1} u_{3} \partial _{3} u_{2} \partial _{1} u_{2}\text{d}x-\int _{\mathbb{R}^{3}} \partial _{1} u_{3} \partial _{3}u_{3} \partial _{1} u_{3} \text{d}x\triangleq \sum _{k=1}^{9}I_{1k}. \end{aligned}$$
(4.3)

\(I_{1}\) can be divided into two classes \(I_{11}\), \(I_{13}\), \(I_{14}\), \(I_{17}\), \(I_{18 }\), \(I_{19}\) and \(I_{12}\), \(I_{15}\), \(I_{16}\). Using Hölder, Gagliardo-Nirenberg, Minkowski inequalities, Lemma 2.1, Lemma 2.2 and Young inequality, we have

$$\begin{aligned} I_{14} \le &\|\partial _{1} u_{2}\|_{L^{2}}\|\partial _{2} u_{1}\|_{L^{4}_{x_{1}}L^{2}_{x_{2}}L^{ \infty }_{x_{3}}} \|\partial _{1} u_{1}\|_{L^{4}_{x_{1}}L^{\infty }_{x_{2}} L^{2}_{x_{3}}} \\ \le & \|\partial _{1} u_{2}\|_{L^{2}}\|\Lambda _{1}^{\frac{1}{4}} \partial _{2} u_{1}\|_{L^{2}_{x_{1}}L^{2}_{x_{2}}L^{\infty }_{x_{3}}} \|\Lambda _{1}^{\frac{1}{4}}\partial _{1} u_{1}\|_{L^{2}_{x_{1}}L^{ \infty }_{x_{2}} L^{2}_{x_{3}}} \\ \le & \|\partial _{1} u_{2}\|_{L^{2}}\|\Lambda _{1}^{\frac{1}{4}} \partial _{2} u_{1}\|_{L^{\infty }_{x_{3}}L^{2}_{x_{1}}L^{2}_{x_{2}}} \|\Lambda _{1}^{\frac{1}{4}}\partial _{1} u_{1}\|_{L^{\infty }_{x_{2}}L^{2}_{x_{1}} L^{2}_{x_{3}}} \\ \le & \|\partial _{1} u_{2}\|_{L^{2}}\|\Lambda _{1}^{\frac{1}{4}} \partial _{2} u_{1}\|^{\frac{1}{2}}_{L^{2}}\|\Lambda _{1}^{\frac{1}{4}} \partial _{3}\partial _{2} u_{1}\|^{\frac{1}{2}}_{L^{2}} \|\Lambda _{1}^{ \frac{1}{4}}\partial _{1} u_{1}\|^{\frac{1}{2}}_{L^{2}}\|\Lambda _{1}^{ \frac{1}{4}}\partial _{2}\partial _{1} u_{1}\|^{\frac{1}{2}}_{L^{2}} \\ \le & \|\partial _{1} u_{2}\|_{L^{2}}\|(\Lambda _{1}^{\frac{5}{4}}, \Lambda _{2}^{\frac{5}{4}}) u_{1}\|^{\frac{1}{2}}_{L^{2}}\|(\Lambda _{1}^{ \frac{5}{4}},\Lambda _{2}^{\frac{5}{4}}) \partial _{3} u_{1}\|^{\frac{1}{2}}_{L^{2}} \|\Lambda _{1}^{\frac{5}{4}} u_{1}\|^{\frac{1}{2}}_{L^{2}}\|\Lambda _{1}^{ \frac{5}{4}}\partial _{2} u_{1}\|^{\frac{1}{2}}_{L^{2}} \\ \le & \frac{1}{16}\bigg(\|(\Lambda _{1}^{\frac{5}{4}},\Lambda _{2}^{ \frac{5}{4}}) \partial _{3} u_{1}\|^{2}_{L^{2}}+\|\Lambda _{1}^{\frac{5}{4}} \partial _{2} u_{1}\|^{2}_{L^{2}}\bigg) \\ & +C \|\partial _{1} u_{2}\|^{2}_{L^{2}} \bigg(\|(\Lambda _{1}^{\frac{5}{4}},\Lambda _{2}^{\frac{5}{4}}) u_{1}\|^{2}_{L^{2}}+ \|\Lambda _{1}^{\frac{5}{4}} u_{1}\|^{2}_{L^{2}}\bigg). \end{aligned}$$
(4.4)

Arguing similarly as the above argument, we obtain

$$\begin{aligned} &I_{11}+I_{13}+I_{17}+I_{18}+I_{19} \\ \le &\frac{1}{8}\bigg(\|(\Lambda _{1}^{\frac{5}{4}},\Lambda _{2}^{ \frac{5}{4}}) \nabla u_{1}\|^{2}_{L^{2}} +\|(\Lambda _{2}^{\frac{5}{4}}, \Lambda _{3}^{\frac{5}{4}}) \nabla u_{2}\|^{2}_{L^{2}}+\|(\Lambda _{1}^{ \frac{5}{4}}, \Lambda _{3}^{\frac{5}{4}})\nabla u_{3}\|^{2}_{L^{2}}\bigg) \\ & +C \|\nabla u\|^{2}_{2}\bigg(\|(\Lambda _{1}^{\frac{5}{4}},\Lambda _{2}^{ \frac{5}{4}}) u_{1}\|^{2}_{L^{2}} +\|(\Lambda _{2}^{\frac{5}{4}},\Lambda _{3}^{ \frac{5}{4}}) u_{2}\|^{2}_{L^{2}}+\|(\Lambda _{1}^{\frac{5}{4}}, \Lambda _{3}^{ \frac{5}{4}}) u_{3}\|^{2}_{L^{2}}\bigg). \end{aligned}$$
(4.5)

Because the form of \(I_{12}\), \(I_{15}\), \(I_{16}\) is different from the others in \(I_{1}\), we use a different method to estimate the three terms. By divergence free condition of \(u\), integrating by part, and Hölder, Gagliardo-Nirenberg, Minkowski inequalities, Lemma 2.1, Lemma 2.2 and Young inequality, one has

$$\begin{aligned} &I_{12}+I_{15}=\int _{\mathbb{R}^{3}}\partial _{3}u_{3} \partial _{1} u_{2} \partial _{1} u_{2} \text{d}x =-2\int _{\mathbb{R}^{3}}u_{3} \partial _{1} u_{2} \partial _{13} u_{2} \text{d}x \\ \le & C\|u_{3}\|_{L^{2}_{x_{3}}L^{2}_{x_{2}}L^{\infty }_{x_{1}}} \| \partial _{1} u_{2}\|_{L^{4}_{x_{3}}L^{\infty }_{x_{2}}L^{2}_{x_{1}}} \|\partial _{13} u_{2}\|_{L^{4}_{x_{3}}L^{2}_{x_{2}}L^{2}_{x_{1}}} \\ \le & C\|u_{3}\|^{\frac{1}{2}}_{L^{2}}\|\partial _{1} u_{3}\|^{\frac{1}{2}}_{L^{2}} \|\Lambda _{3}^{\frac{1}{4}}\partial _{1} u_{2}\|_{L^{2}}^{\frac{1}{2}} \| \Lambda _{3}^{\frac{1}{4}}\partial _{12} u_{2}\|_{L^{2}}^{\frac{1}{2}} \| \Lambda ^{\frac{5}{4}}_{3}\partial _{1} u_{2}\|_{L^{2}} \\ \le & C\|u_{3}\|^{\frac{3}{5}}_{L^{2}}\|\Lambda ^{\frac{5}{4}}_{1} u_{3}\|^{ \frac{2}{5}}_{L^{2}} \|\partial _{1} u_{2}\|_{L^{2}}^{\frac{2}{5}} \| \Lambda _{3}^{\frac{5}{4}}\partial _{1} u_{2}\|_{L^{2}}^{\frac{1}{10}} \|( \Lambda _{2}^{\frac{5}{4}}, \Lambda ^{\frac{5}{4}}_{3})\partial _{1} u_{2}\|^{ \frac{3}{2}}_{L^{2}} \\ \le & \frac{1}{8} \|(\Lambda _{2}^{\frac{5}{4}}, \Lambda ^{\frac{5}{4}}_{3}) \partial _{1} u_{2}\|^{2}_{L^{2}} +C\|\partial _{1} u_{2}\|_{L^{2}}^{2} \|\Lambda ^{\frac{5}{4}}_{1} u_{3}\|^{2}_{L^{2}} \|u_{3}\|^{3}_{L^{2}}. \end{aligned}$$
(4.6)

Similarly,

$$\begin{aligned} &I_{16}=\int _{\mathbb{R}^{3}}u_{2}\partial _{12}u_{3} \partial _{1} u_{3} \text{d}x+\int _{\mathbb{R}^{3}}u_{2} \partial _{2} u_{3} \partial _{11}u_{3} \text{d}x \\ \le & \|u_{2}\|_{L^{2}_{x_{3}}L^{2}_{x_{1}}L^{\infty }_{x_{2}}}\| \partial _{1}u_{3}\|_{L^{4}_{x_{3}}L^{\infty }_{x_{1}}L^{2}_{x_{2}}} \| \partial _{12} u_{3}\|_{L^{4}_{x_{3}}L^{2}_{x_{1}}L^{2}_{x_{2}}} \\ & +\|u_{2}\|_{L^{2}_{x_{3}}L^{2}_{x_{1}}L^{\infty }_{x_{2}}} \| \partial _{2} u_{3}\|_{L^{4}_{x_{3}}L^{\infty }_{x_{1}}L^{2}_{x_{2}}} \|\partial _{11}u_{3}\|_{L^{4}_{x_{3}}L^{2}_{x_{1}}L^{2}_{x_{2}}} \\ \le & \frac{1}{8}\|(\Lambda _{1}^{\frac{5}{4}}, \Lambda _{3}^{\frac{5}{4}}) \nabla u_{3}\|^{2}_{L^{2}}+C\|\partial _{2} u_{2}\|_{L^{2}}^{2}\|( \Lambda _{3}^{\frac{5}{4}}, \Lambda _{1}^{\frac{5}{4}})u_{3}\|^{2}_{L^{2}} \|u_{2} \|_{L^{2}}^{2} \\ &+C \|\partial _{2}u_{3}\|^{ 2}_{L^{2}} \|\Lambda _{2}^{\frac{5}{4}} u_{2} \|_{L^{2}}^{2} \|u_{2}\|_{L^{2}}^{ 3}. \end{aligned}$$
(4.7)

Adding (4.4)-(4.7), we get

$$\begin{aligned} I_{1} \le & \frac{1}{4}\bigg(\|(\Lambda _{1}^{\frac{5}{4}},\Lambda _{2}^{ \frac{5}{4}}) \nabla u_{1}\|^{2}_{L^{2}}+\|(\Lambda _{2}^{\frac{5}{4}}, \Lambda _{3}^{\frac{5}{4}})\nabla u_{2}\|^{2}_{L^{2}}+\|(\Lambda _{3}^{ \frac{5}{4}},\Lambda _{1}^{\frac{5}{4}})\nabla u_{3}\|^{2}_{L^{2}}\bigg) \\ & +C \|\nabla u\|^{2}_{2}\bigg(\|(\Lambda _{1}^{\frac{5}{4}},\Lambda _{2}^{ \frac{5}{4}}) u_{1}\|^{2}_{L^{2}} +\|(\Lambda _{2}^{\frac{5}{4}},\Lambda _{3}^{ \frac{5}{4}}) u_{2}\|^{2}_{L^{2}}+\|(\Lambda _{1}^{\frac{5}{4}}, \Lambda _{3}^{ \frac{5}{4}}) u_{3}\|^{2}_{L^{2}}\bigg) \\ &\times \bigg(1+ \|u\|^{3}_{L^{2}}+ \|u\|_{L^{2}}^{2}\bigg). \end{aligned}$$
(4.8)

By the same way as in deriving (4.8), one has

$$\begin{aligned} I_{2}+I_{3} \le & \frac{1}{4}\bigg(\|(\Lambda _{1}^{\frac{5}{4}}, \Lambda _{2}^{\frac{5}{4}}) \nabla u_{1}\|^{2}_{L^{2}}+\|(\Lambda _{2}^{ \frac{5}{4}},\Lambda _{3}^{\frac{5}{4}})\nabla u_{2}\|^{2}_{L^{2}}+\|( \Lambda _{3}^{\frac{5}{4}},\Lambda _{1}^{\frac{5}{4}})\nabla u_{3}\|^{2}_{L^{2}} \bigg) \\ & +C \|\nabla u\|^{2}_{2}\bigg(\|(\Lambda _{1}^{\frac{5}{4}},\Lambda _{2}^{ \frac{5}{4}}) u_{1}\|^{2}_{L^{2}} +\|(\Lambda _{2}^{\frac{5}{4}},\Lambda _{3}^{ \frac{5}{4}}) u_{2}\|^{2}_{L^{2}}+\|(\Lambda _{1}^{\frac{5}{4}}, \Lambda _{3}^{ \frac{5}{4}}) u_{3}\|^{2}_{L^{2}}\bigg) \\ &\times \bigg(1+ \|u\|^{3}_{L^{2}}+ \|u\|_{L^{2}}^{2}\bigg). \end{aligned}$$
(4.9)

Inserting (4.8), (4.9) to (4.2), we have

$$\begin{aligned} &\frac{d}{dt} \|\nabla u\|^{2}_{2} + \| (\Lambda _{1} ^{ \frac{5}{ 4}}, \Lambda _{2}^{\frac{5}{ 4}}) \nabla u_{1} \|^{2}_{2} + \| ( \Lambda _{2}^{\frac{5}{ 4}}, \Lambda _{3}^{\frac{5}{ 4}})\nabla u_{2}\|^{2}_{2} +\| (\Lambda _{3} ^{\frac{5}{ 4}} , \Lambda _{1} ^{\frac{5}{ 4}} )\nabla u_{3}\|^{2}_{2} \\ \le &C\|\nabla u\|^{2}_{L^{2}}\bigg(\|(\Lambda _{1}^{\frac{5}{4}}, \Lambda _{2}^{\frac{5}{4}}) u_{1}\|_{L^{2}}^{2}+\|(\Lambda _{2}^{ \frac{5}{4}},\Lambda _{3}^{\frac{5}{4}}) u_{2}\|_{L^{2}}^{2}+\|(\Lambda _{1}^{ \frac{5}{4}}, \Lambda _{3}^{\frac{5}{4}}) u_{3}\|_{L^{2}}^{2}\bigg) \\ &\times \bigg(1+\|u\|^{2}_{L^{2}}+\|u\|^{3}_{L^{2}}\bigg). \end{aligned}$$
(4.10)

By Gronwall’s inequality, we obtain

$$\begin{aligned} &\|\nabla u\|^{2}_{2} + \int _{0}^{T}\bigg(\| (\Lambda _{1} ^{ \frac{5}{ 4}}, \Lambda _{2}^{\frac{5}{ 4}}) \nabla u_{1} \|^{2}_{2} + \| ( \Lambda _{2}^{\frac{5}{ 4}}, \Lambda _{3}^{\frac{5}{ 4}})\nabla u_{2}\|^{2}_{2} +\| (\Lambda _{3} ^{\frac{5}{ 4}} , \Lambda _{1} ^{\frac{5}{ 4}} )\nabla u_{3}\|^{2}_{2} \bigg)\text{d}t \\ \le & \|\nabla u_{0}\|^{2}_{L^{2}}e^{C\int ^{T}_{0}\big(\|(\Lambda _{1}^{ \frac{5}{4}}, \ \Lambda _{2}^{\frac{5}{4}}) u_{1}\|_{L^{2}}^{2}+\|(\Lambda _{2}^{ \frac{5}{4}}, \ \Lambda _{3}^{\frac{5}{4}}) u_{2}\|_{L^{2}}^{2}+\|(\Lambda _{3}^{ \frac{5}{4}}, \ \Lambda _{1}^{\frac{5}{4}}) u_{3}\|_{L^{2}}^{2}\big) \big(1+ \|u\|^{3}_{L^{2}}\big)\text{d}t} \\ \le & \|\nabla u_{0}\|^{2}_{L^{2}}e^{ C\big(1+\|u_{0}\|_{L^{2}}^{5} \big)}. \end{aligned}$$
(4.11)

Noting the \(L^{2}\)-estimate (4.1) of \(u\), the proof of the Proposition 2.1 is completed.

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Li, F., Yuan, B. Global Well-Posedness of the 3D Generalized Navier-Stokes Equations with Fractional Partial Dissipation. Acta Appl Math 171, 20 (2021). https://doi.org/10.1007/s10440-021-00388-4

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Mathematics Subject Classification

  • 35Q30
  • 76D03

Keywords

  • Navier-Stokes equations
  • Fractional partial dissipation
  • Global solution