# 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

show all references

##### 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
 [1] Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241 [2] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [3] Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463 [4] Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 [5] Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443 [6] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [7] Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267 [8] Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467 [9] Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 [10] Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 [11] Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073 [12] Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297 [13] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [14] Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 [15] Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050 [16] Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323 [17] Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076 [18] Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249 [19] Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377 [20] Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

2019 Impact Factor: 0.953

Article outline