Well-posedness of abstract distributed-order fractional diffusion equations
CPAA Home
Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent
March 2014, 13(2): 585-603. doi: 10.3934/cpaa.2014.13.585
Regularity criterion for 3D Navier-Stokes equations in Besov spaces
Daoyuan Fang 1, and Chenyin Qian 2,
Department of Mathematics, Zhejiang University, Hangzhou 310027
Department of Mathematics, Zhejiang University, Hangzhou, 310027, China
Received October 2012 Revised July 2013 Published October 2013
Several regularity criterions of Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes equations are obtained. The results show that a weak solution $u$ becomes regular if the gradient of velocity component $\nabla_{h}{u}$ (or $ \nabla{u_3}$) satisfies the additional conditions in the class of $L^{q}(0,T; \dot{B}_{p,r}^{s}(\mathbb{R}^{3}))$, where $\nabla_{h}=(\partial_{x_{1}},\partial_{x_{2}})$ is the horizontal gradient operator. Besides, we also consider the anisotropic regularity criterion for the weak solution of Navier-Stokes equations in $\mathbb{R}^3$. Finally, we also get a further regularity criterion, when give the sufficient condition on $\partial_3u_3$.
Keywords: 3D Navier-Stokes equations, Leray-Hopf weak solution, Regularity criterion..
Mathematics Subject Classification: 35Q30, 76D0.
Citation: Daoyuan Fang, Chenyin Qian. Regularity criterion for 3D Navier-Stokes equations in Besov spaces. Communications on Pure & Applied Analysis, 2014, 13 (2) : 585-603. doi: 10.3934/cpaa.2014.13.585
H. Bahouri, R. Danchin and J. Y. Chemin, "Fourier Analysis and Nonlinear Partial Differential Equations, A Series of Comprehensive Studies in Mathematics,'' Springer Heidelberg Dordrecht London New York. Springer-Verlag Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. Google Scholar
H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in $\mathbbR^n$, Chinese Ann. Math. Ser. B, 16 (1995), 407-412. Google Scholar
L. C. Berselli, On a regularity criterion for the solutions to the 3D Navier-Stokes equations, Dierential Integral Equations, 15 (2002), 1129-1137. Google Scholar
C. S. Cao and E. S. Titi, Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor, Arch. Rational Mech. Anal., 202 (2011), 919-932. doi: 10.1007/s00205-011-0439-6. Google Scholar
Z. F. Zhang and Q. L. Chen, Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in $\mathbbR^3$, J. Differential Equations, 216 (2005), 470-481. doi: 10.1016/j.jde.2005.06.001. Google Scholar
A. Cheskidov and R. Shvydkoy, On the regularity of weak solutions of the 3D Navier-Stokes equations in $B^{-1}_{\infty,\infty}$,, \arXiv{0708.3067v2 [math.AP]}., (). Google Scholar
G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations,'' Vol. I, II. Springer, New York, 1994. doi: 10.1007/978-0-387-09620-9. Google Scholar
S. Gala, A remark on the blow-up criterion of strong solutions to the Navier-Stokes equations, Applied Mathematics and Computation, 217 (2011), 9488-9491. doi: 10.1016/j.amc.2011.03.156. Google Scholar
E. Hopf, Über die anfang swetaufgabe für die hydrodynamischer grundgleichungan, Math. Nach., 4 (1951), 213-231. Google Scholar
H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194. doi: 10.1007/s002090000130. Google Scholar
H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z., 242 (2002), 251-278. doi: 10.1007/s002090100332. Google Scholar
H. Kozono and N. Yatsu, Extension criterion via two-components of vorticity on strong solution to the 3D Navier-Stokes equations, Math. Z., 246 (2003), 55-68. doi: 10.1007/s00209-003-0576-1. Google Scholar
I. Kukavica and M. Ziane, Navier-Stokes equations with regularity in one direction, J. Math Phys., 48 (2007), 065203. doi: 10.1063/1.2395919. Google Scholar
I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 453-469. doi: 10.1088/0951-7715/19/2/012. Google Scholar
O. A. Ladyzhenskaya, "The Boundary Value Problems of Mathematical Physics," Springer, Berlin, 1985. Google Scholar
J. Leray, Sur le mouvement d'um liquide visqieux emlissant l'space, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354. Google Scholar
J. Neustupa and P. Penel, Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier-Stokes equations, In "11 Mathematical Fluid Mechanics: Recent Results and Open Problems"(J. Neustupa and P. Penel eds.), Advances in Mathematical Fluid Mechanics, pp. 239-267. Birkhäuser, Basel, 2001. doi: 10.1007/978-3-0348-8243-9_10. Google Scholar
P. Penel and M. Pokorný, On anisotropic regularity criteria for the Solutions to 3D Navier-Stokes equations, J. Math. Fluid Mech., 13 (2011), 341-353. doi: 10.1007/s00021-010-0038-6. Google Scholar
P. Penel and M. Pokorný, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity, Appl. Math., 49 (2004), 483-493. doi: 10.1023/B:APOM.0000048124.64244.7e. Google Scholar
M. Pokorný, On the result of He concerning the smoothness of solutions to the Navier-Stokes equations, Electron. J. Differ. Equ., 11 (2003), 1-8. Google Scholar
G. Prodi, Un teorema di unicità per el equazioni di Navier-Stokes, Ann. Mat. Pura Appl. IV, 48 (1959), 173-82. Google Scholar
J. Serrin, "The Initial Value Problems for the Navier-Stokes Equations, in Nonlinear Problems," edited by R. E. Langer, University of Wisconsin Press, Madison, WI, 1963. Google Scholar
H. Sohr, "The Navier-Stokes Equations, An Elementary Functional Analytic Approach," Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-0551-3. Google Scholar
B. Q. Yuan and B. Zhang, Blow-up criterion of strong solutions to the Navier-Stokes equations in Besov spaces with negative indices, J. Differential Equations, 242 (2007), 1-10. doi: 0.1016/j.jde.2007.07.009. Google Scholar
Y. Zhou and M. Pokorný, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107. doi: 10.1088/0951-7715/23/5/004. Google Scholar
Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299
Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141
Hui Chen, Daoyuan Fang, Ting Zhang. Regularity of 3D axisymmetric Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1923-1939. doi: 10.3934/dcds.2017081
Alessio Falocchi, Filippo Gazzola. Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021151
Vladimir V. Chepyzhov, E. S. Titi, Mark I. Vishik. On the convergence of solutions of the Leray-$\alpha $ model to the trajectory attractor of the 3D Navier-Stokes system. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 481-500. doi: 10.3934/dcds.2007.17.481
Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101
Zoran Grujić. Regularity of forward-in-time self-similar solutions to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 837-843. doi: 10.3934/dcds.2006.14.837
Fan Wu. Conditional regularity for the 3D Navier-Stokes equations in terms of the middle eigenvalue of the strain tensor. Evolution Equations & Control Theory, 2021, 10 (3) : 511-518. doi: 10.3934/eect.2020078
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
Anthony Suen. Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1775-1798. doi: 10.3934/dcds.2020093
Teng Wang, Yi Wang. Large-time behaviors of the solution to 3D compressible Navier-Stokes equations in half space with Navier boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2811-2838. doi: 10.3934/cpaa.2021080
Donatella Donatelli, Nóra Juhász. The primitive equations of the polluted atmosphere as a weak and strong limit of the 3D Navier-Stokes equations in downwind-matching coordinates. Discrete & Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022002
Jingrui Wang, Keyan Wang. Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 5003-5019. doi: 10.3934/dcds.2017215
Sadek Gala. A new regularity criterion for the 3D MHD equations in $R^3$. Communications on Pure & Applied Analysis, 2012, 11 (3) : 973-980. doi: 10.3934/cpaa.2012.11.973
Aseel Farhat, M. S Jolly, Evelyn Lunasin. Bounds on energy and enstrophy for the 3D Navier-Stokes-$\alpha$ and Leray-$\alpha$ models. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2127-2140. doi: 10.3934/cpaa.2014.13.2127
Ahmad Mohammad Alghamdi, Sadek Gala, Chenyin Qian, Maria Alessandra Ragusa. The anisotropic integrability logarithmic regularity criterion for the 3D MHD equations. Electronic Research Archive, 2020, 28 (1) : 183-193. doi: 10.3934/era.2020012
Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637
Shijin Ding, Zhilin Lin, Dongjuan Niu. Boundary layer for 3D plane parallel channel flows of nonhomogeneous incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4579-4596. doi: 10.3934/dcds.2020193
Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142
Hakima Bessaih, María J. Garrido-Atienza. Longtime behavior for 3D Navier-Stokes equations with constant delays. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1931-1948. doi: 10.3934/cpaa.2020085
Daoyuan Fang Chenyin Qian