# American Institute of Mathematical Sciences

doi: 10.3934/eect.2020078

## Conditional regularity for the 3D Navier-Stokes equations in terms of the middle eigenvalue of the strain tensor

 Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China

* Corresponding author: Fan Wu

Received  April 2020 Revised  April 2020 Published  June 2020

In this paper, we consider the regularity criteria for the 3D incompressible Navier-Stokes equations involving the middle eigenvalue ($\lambda_2$) of the strain tensor. It is proved that if $\lambda^+_2$ belongs to Multiplier space or Besov space, then the weak solution remains smooth on $[0, T]$, where $\lambda^{+}_2 = \max\{\lambda_2, 0\}$. These regularity conditions allows us to improve the result obtained by Miller [7].

Citation: Fan Wu. Conditional regularity for the 3D Navier-Stokes equations in terms of the middle eigenvalue of the strain tensor. Evolution Equations & Control Theory, doi: 10.3934/eect.2020078
##### References:
 [1] D. Chae, On the spectral dynamics of the deformation tensor and new a priori estimates for the 3D Euler equations, Communications in Mathematical Physics, 263 (2005), 789-801.  doi: 10.1007/s00220-005-1465-8.  Google Scholar [2] H. B. Da Veiga, A new regularity class for the Navier-Stokes equations in $\mathbb{R}^n$, Chinese Ann. Math. Ser. B, 16 (1995), 407-412.   Google Scholar [3] B. Q. Dong and Z. M. Chen, Regularity criterion of weak solutions to the 3D Navier-Stokes equations via two velocity components, Journal of Mathematical Analysis and Applications, 338 (2008), 1-10.  doi: 10.1016/j.jmaa.2007.05.003.  Google Scholar [4] B. Q. Dong, S. Gala and Z. M. Chen, On the regularity criteria of the 3D Navier-Stokes equations in critical spaces, Acta Mathematica Scientia, 31 (2011), 591-600.  doi: 10.1016/S0252-9602(11)60259-2.  Google Scholar [5] L. Escauriaza and G. Seregin, $L_{3, \infty}$-solutions of the Navier-Stokes equations and backward uniqueness, Nonlinear Problems in Mathematical Physics & Related Topics Ⅱ, 2 (2002), 353-366.   Google Scholar [6] Z. G. Guo, P. Kucera and Z. Skalák, Regularity criterion for solutions to the Navier-Stokes equations in the whole 3D space based on two vorticity components, Journal of Mathematical Analysis and Applications, 458 (2018), 755-766.  doi: 10.1016/j.jmaa.2017.09.029.  Google Scholar [7] E. Miller, A regularity criterion for the Navier-Stokes equation involving only the middle eigenvalue of the strain tensor, Archive for Rational Mechanics and Analysis, 235 (2020), 99-139.  doi: 10.1007/s00205-019-01419-z.  Google Scholar [8] J. Neustupa and P. Penel, Regularity of a weak solution to the Navier-Stokes equation in dependence on eigenvalues and eigenvectors of the rate of deformation tensor, Trends in Partial Differential Equations of Mathematical Physics. Birkhäuser Basel, 91 (2005), 197-212.  doi: 10.1007/3-7643-7317-2_15.  Google Scholar [9] J. Neustupa and P. Penel, On regularity of a weak solution to the Navier-Stokes equation with generalized impermeability boundary conditions., Nonlinear Analysis: Theory, Methods & Applications, 66 (2007), 1753-1769.  doi: 10.1016/j.na.2006.02.043.  Google Scholar [10] J. Neustupa and P. Penel, On regularity of a weak solution to the Navier-Stokes equations with the generalized Navier Slip boundary conditions, Advances in Mathematical Physics, 2018 (2018), Art. ID 4617020, 7 pp. doi: 10.1155/2018/4617020.  Google Scholar [11] G. Prodi, Un teorema di unicita per le equazioni di Navier-Stokes, Annali di Matematica pura ed Applicata, 48 (1959), 173-182.  doi: 10.1007/BF02410664.  Google Scholar [12] P. Penel and M. Pokorny, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity, Applications of Mathematics, 49 (2004), 483-493.  doi: 10.1023/B:APOM.0000048124.64244.7e.  Google Scholar [13] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Archive for Rational Mechanics and Analysis, 9 (1962), 187-195.  doi: 10.1007/BF00253344.  Google Scholar [14] Z. Skalak, On the regularity of the solutions to the Navier-Stokes equations via the gradient of one velocity component., Nonlinear Analysis: Theory, Methods & Applications, 104 (2014), 84-89.  doi: 10.1016/j.na.2014.03.018.  Google Scholar [15] F. Wu, Blowup criterion via only the middle eigenvalue of the strain tensor in anisotropic Lebesgue spaces to the 3D double-diffusive convection equations, Journal of Mathematical Fluid Mechanics, 22 (2020), Art. 24, 9 pp. doi: 10.1007/s00021-020-0483-9.  Google Scholar [16] X. C. Zhang, A regularity criterion for the solutions of 3D Navier-Stokes equations, Journal of Mathematical Analysis and Applications, 346 (2008), 336-339.  doi: 10.1016/j.jmaa.2008.05.027.  Google Scholar [17] Z. J. Zhang, Z. A. Yao, P. Li, C. C. Guo and M. Lu, Two new regularity criteria for the 3D Navier-Stokes equations via two entries of the velocity gradient tensor, Acta Applicandae Mathematicae, 123 (2013), 43-52.  doi: 10.1007/s10440-012-9712-4.  Google Scholar [18] Z. J. Zhang, A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component, Communications on Pure & Applied Analysis, 12 (2013), 117-124.  doi: 10.3934/cpaa.2013.12.117.  Google Scholar [19] Z. F. Zhang and Q. L. Chen, Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in $\mathbb{R}^3$, Journal of Differential Equations, 216 (2015), 470-481.   Google Scholar

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##### References:
 [1] D. Chae, On the spectral dynamics of the deformation tensor and new a priori estimates for the 3D Euler equations, Communications in Mathematical Physics, 263 (2005), 789-801.  doi: 10.1007/s00220-005-1465-8.  Google Scholar [2] H. B. Da Veiga, A new regularity class for the Navier-Stokes equations in $\mathbb{R}^n$, Chinese Ann. Math. Ser. B, 16 (1995), 407-412.   Google Scholar [3] B. Q. Dong and Z. M. Chen, Regularity criterion of weak solutions to the 3D Navier-Stokes equations via two velocity components, Journal of Mathematical Analysis and Applications, 338 (2008), 1-10.  doi: 10.1016/j.jmaa.2007.05.003.  Google Scholar [4] B. Q. Dong, S. Gala and Z. M. Chen, On the regularity criteria of the 3D Navier-Stokes equations in critical spaces, Acta Mathematica Scientia, 31 (2011), 591-600.  doi: 10.1016/S0252-9602(11)60259-2.  Google Scholar [5] L. Escauriaza and G. Seregin, $L_{3, \infty}$-solutions of the Navier-Stokes equations and backward uniqueness, Nonlinear Problems in Mathematical Physics & Related Topics Ⅱ, 2 (2002), 353-366.   Google Scholar [6] Z. G. Guo, P. Kucera and Z. Skalák, Regularity criterion for solutions to the Navier-Stokes equations in the whole 3D space based on two vorticity components, Journal of Mathematical Analysis and Applications, 458 (2018), 755-766.  doi: 10.1016/j.jmaa.2017.09.029.  Google Scholar [7] E. Miller, A regularity criterion for the Navier-Stokes equation involving only the middle eigenvalue of the strain tensor, Archive for Rational Mechanics and Analysis, 235 (2020), 99-139.  doi: 10.1007/s00205-019-01419-z.  Google Scholar [8] J. Neustupa and P. Penel, Regularity of a weak solution to the Navier-Stokes equation in dependence on eigenvalues and eigenvectors of the rate of deformation tensor, Trends in Partial Differential Equations of Mathematical Physics. Birkhäuser Basel, 91 (2005), 197-212.  doi: 10.1007/3-7643-7317-2_15.  Google Scholar [9] J. Neustupa and P. Penel, On regularity of a weak solution to the Navier-Stokes equation with generalized impermeability boundary conditions., Nonlinear Analysis: Theory, Methods & Applications, 66 (2007), 1753-1769.  doi: 10.1016/j.na.2006.02.043.  Google Scholar [10] J. Neustupa and P. Penel, On regularity of a weak solution to the Navier-Stokes equations with the generalized Navier Slip boundary conditions, Advances in Mathematical Physics, 2018 (2018), Art. ID 4617020, 7 pp. doi: 10.1155/2018/4617020.  Google Scholar [11] G. Prodi, Un teorema di unicita per le equazioni di Navier-Stokes, Annali di Matematica pura ed Applicata, 48 (1959), 173-182.  doi: 10.1007/BF02410664.  Google Scholar [12] P. Penel and M. Pokorny, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity, Applications of Mathematics, 49 (2004), 483-493.  doi: 10.1023/B:APOM.0000048124.64244.7e.  Google Scholar [13] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Archive for Rational Mechanics and Analysis, 9 (1962), 187-195.  doi: 10.1007/BF00253344.  Google Scholar [14] Z. Skalak, On the regularity of the solutions to the Navier-Stokes equations via the gradient of one velocity component., Nonlinear Analysis: Theory, Methods & Applications, 104 (2014), 84-89.  doi: 10.1016/j.na.2014.03.018.  Google Scholar [15] F. Wu, Blowup criterion via only the middle eigenvalue of the strain tensor in anisotropic Lebesgue spaces to the 3D double-diffusive convection equations, Journal of Mathematical Fluid Mechanics, 22 (2020), Art. 24, 9 pp. doi: 10.1007/s00021-020-0483-9.  Google Scholar [16] X. C. Zhang, A regularity criterion for the solutions of 3D Navier-Stokes equations, Journal of Mathematical Analysis and Applications, 346 (2008), 336-339.  doi: 10.1016/j.jmaa.2008.05.027.  Google Scholar [17] Z. J. Zhang, Z. A. Yao, P. Li, C. C. Guo and M. Lu, Two new regularity criteria for the 3D Navier-Stokes equations via two entries of the velocity gradient tensor, Acta Applicandae Mathematicae, 123 (2013), 43-52.  doi: 10.1007/s10440-012-9712-4.  Google Scholar [18] Z. J. Zhang, A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component, Communications on Pure & Applied Analysis, 12 (2013), 117-124.  doi: 10.3934/cpaa.2013.12.117.  Google Scholar [19] Z. F. Zhang and Q. L. Chen, Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in $\mathbb{R}^3$, Journal of Differential Equations, 216 (2015), 470-481.   Google Scholar
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