In this paper, we first consider the robustness of strong solutions for 3D generalized Navier-Stokes equations, i.e., we show the set of all initial conditions $ u_0\in H^1(\mathbb{R}^3) $ with $ \nabla\cdot u_0 = 0 $ that give rise to a strong solution of the generalized Navier-Stokes equations on the time interval $ [0, T] $ is open in $ H^1(\mathbb{R}^3) $. Moreover, we prove that the Galerkin approximations of a strong solution of the 3D generalized Navier-Stokes equations converge strongly to $ u $ in $ L^\infty(0, T;H^1_{0, div}(\mathbb{T}^3)) $ and $ L^2(0, T;H^{\kappa+1}(\mathbb{T}^3)\bigcap H^1_{0, div}(\mathbb{T}^3)) $.
Citation: Ning Duan, Hao Pan, Xiaopeng Zhao. Robustness of regularity and convergence of Galerkin approximations for 3D generalized incompressible Navier-Stokes equations[J]. Communications in Analysis and Mechanics, 2026, 18(2): 406-418. doi: 10.3934/cam.2026016
In this paper, we first consider the robustness of strong solutions for 3D generalized Navier-Stokes equations, i.e., we show the set of all initial conditions $ u_0\in H^1(\mathbb{R}^3) $ with $ \nabla\cdot u_0 = 0 $ that give rise to a strong solution of the generalized Navier-Stokes equations on the time interval $ [0, T] $ is open in $ H^1(\mathbb{R}^3) $. Moreover, we prove that the Galerkin approximations of a strong solution of the 3D generalized Navier-Stokes equations converge strongly to $ u $ in $ L^\infty(0, T;H^1_{0, div}(\mathbb{T}^3)) $ and $ L^2(0, T;H^{\kappa+1}(\mathbb{T}^3)\bigcap H^1_{0, div}(\mathbb{T}^3)) $.
| [1] | E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. https://doi.org/10.1112/blms/5.1.121 |
| [2] |
M. Ainsworth, Z. Mao, Analysis and approximation of a fractional Cahn-Hilliard equation, SIAM J. Numer. Anal., 55 (2017), 1689–1718. https://doi.org/10.1137/16M1075302 doi: 10.1137/16M1075302
|
| [3] |
X. Zhao, Long time behavior of solutions to 3D generalized MHD equations, Forum Math., 32 (2020), 977–993. https://doi.org/10.1515/forum-2019-0155 doi: 10.1515/forum-2019-0155
|
| [4] |
H. Fujita, T. Kato, On the Navier-Stokes initial value problem I, Arch. Ration. Mech. Anal., 16 (1964), 269–315. https://doi.org/10.1007/BF00276188 doi: 10.1007/BF00276188
|
| [5] |
T. Kato, Strong $L^{p}$-solutions of the Navier-Stokes equations in $\mathbb{R}^{m}$ with applications to weak solutions, Math. Z., 187 (1984), 471–480. https://doi.org/10.1007/BF01174182 doi: 10.1007/BF01174182
|
| [6] | M. Cannone, Ondelettes, Paraproduits et Navier-Stokes, Arts et Sciences, Diderot editeur, 1995. |
| [7] |
F. Planchon, Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in $\mathbb{R}^{3}$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 319–336. https://doi.org/10.1016/S0294-1449(16)30107-X doi: 10.1016/S0294-1449(16)30107-X
|
| [8] |
H. Koch, D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22–35. https://doi.org/10.1006/aima.2000.1937 doi: 10.1006/aima.2000.1937
|
| [9] |
Z. Lei, F. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 1297–1304. https://doi.org/10.1002/cpa.20361 doi: 10.1002/cpa.20361
|
| [10] | R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland Publishing Company, Amsterdam, 1979. https://doi.org/10.1115/1.3424338 |
| [11] |
J.-Y. Chemin, I. Gallagher, Well-posedness and stability results for the Navier-Stokes equations in $\mathbb{R}^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 599–624. https://doi.org/10.1016/j.anihpc.2007.05.008 doi: 10.1016/j.anihpc.2007.05.008
|
| [12] | G. Seregin, Lecture Notes on Regularity Theory for The Navier-Stokes Equation, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. https://doi.org/10.1142/9314 |
| [13] |
D. Song, X. Zhao, Large-time behavior of cylindrically symmetric Navier-Stokes equations with temperature-dependent viscosity and heat conductivity, Commun. Anal. Mech., 16 (2024), 599–632. https://doi.org/10.3934/cam.2024028 doi: 10.3934/cam.2024028
|
| [14] | J. C. Robinson, J. L. Rodrigo, W. Sadowski, The Three-Dimensional Navier-Stokes Equations, In: Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2016. https://doi.org/10.1017/CBO9781139095143 |
| [15] |
S. Wang, Global well-posedness and viscosity vanishing limit of a new initial-boundary value problem on two/three-dimensional incompressible Navier-Stokes equations and/or Boussinesq equations, Commun. Anal. Mech., 17 (2025), 582–605. https://doi.org/10.3934/cam.2025023 doi: 10.3934/cam.2025023
|
| [16] | J.-L. Lions, uelques méthodes de résolution des problémes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969. |
| [17] |
J. Wu, Generalized MHD equations, J. Differential Equations, 195 (2003), 284–312. https://doi.org/10.1016/j.jde.2003.07.007 doi: 10.1016/j.jde.2003.07.007
|
| [18] |
Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491–505. https://doi.org/10.1016/j.anihpc.2006.03.014 doi: 10.1016/j.anihpc.2006.03.014
|
| [19] |
J. Fan, Y. Fukumoto, Y. Zhou, Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations, Kinet. Relat. Models, 6 (2013), 545–556. https://doi.org/10.3934/krm.2013.6.545 doi: 10.3934/krm.2013.6.545
|
| [20] |
J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285–306. https://doi.org/10.1080/03605300701382530 doi: 10.1080/03605300701382530
|
| [21] |
N. Duan, Well-posedness and decay of solutions for three-dimensional generalized Navier-Stokes equations, Comput. Math. Appl., 76 (2018), 1026–1033. https://doi.org/10.1016/j.camwa.2018.05.038 doi: 10.1016/j.camwa.2018.05.038
|
| [22] |
X. Zhao, Y. Zhou, Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations, Discret. Cont. Dyn. Syst. Ser. B, 26 (2021), 795–813. https://doi.org/10.3934/dcdsb.2020142 doi: 10.3934/dcdsb.2020142
|
| [23] |
Q. Liu, J. Zhao, S. Cui, Existence and regularizing rate estimates of solutions to a generalized magneto-hydrodynamic system in pseudomeasure spaces, Ann. Mat. Pura Appl., 191 (2012), 293–309. https://doi.org/10.1007/s10231-010-0184-8 doi: 10.1007/s10231-010-0184-8
|
| [24] |
Z. Ye, Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations, Ann. Mat. Pura Appl., 195 (2016), 1111–1121. https://doi.org/10.1007/s10231-015-0507-x doi: 10.1007/s10231-015-0507-x
|
| [25] |
Q. Jiu, H. Yu, Decay of solutions to the three-dimensional generalized Navier-Stokes equations, Asymptotic Anal., 94 (2015), 105–124. https://doi.org/10.3233/ASY-151307 doi: 10.3233/ASY-151307
|
| [26] |
N. T. Le, L. T. Tinh, On the three dimensional generalized Navier-Stokes equations with damping, Frac. Calc. Appl. Anal., 28 (2025), 1923–1967. https://doi.org/10.1007/s13540-025-00421-5 doi: 10.1007/s13540-025-00421-5
|
| [27] |
Q. Liu, Z. Zuo, Partially regular weak solutions and Liouville-type theorem to the stationary fractional Navier-Stokes equations in dimensions four and five, J. Differential Equations, 421 (2025), 291–335. https://doi.org/10.1016/j.jde.2024.11.057 doi: 10.1016/j.jde.2024.11.057
|
| [28] |
D. Chamorro, M. Mansais, Some general external forces and critical mild solutions for the fractional Navier-Stokes equations, J. Elliptic Parabolic Equ., 11 (2025), 1071–1100. https://doi.org/10.1007/s41808-025-00365-0 doi: 10.1007/s41808-025-00365-0
|
| [29] |
F. Wu, A note on energy equality for the fractional Navier-Stokes equations, Proc. Royal Soc. Edinburg. Sec. A Math., 154 (2024), 201–208. https://doi.org/ 10.1017/prm.2023.3 doi: 10.1017/prm.2023.3
|
| [30] |
O. Jarrin, Asymptotic behavior in time of a generalized Navier-Stokes-alpha model, Disc. Cont. Dyn. Syst. Ser. B, 30 (2025), 1669–1709. https://doi.org/10.3934/dcdsb.2024155 doi: 10.3934/dcdsb.2024155
|
| [31] |
J. L. Wu, Regularity criteria for the 3D generalized Navier-Stokes equations with nonlinear damping term, AIMS Math., 9 (2025), 16250–16259. https://doi.org/10.3934/math.2024786 doi: 10.3934/math.2024786
|
| [32] |
X. Xi, Y. Zhou, M. Hou, Well-posedness of mild solutions for the fractional Navier-Stokes equations in Besov spaces, Qualitative Theory Dyn. Syst., 23 (2024), 15. https://doi.org/10.1007/s12346-023-00867-z doi: 10.1007/s12346-023-00867-z
|
| [33] |
D. W. Boutros, J. D. Gibbon, Phase transitions in the fractional three-dimensional Navier-Stokes equations, Nonlinearity, 37 (2024), 045010. https://doi.org/10.1088/1361-6544/ad25be doi: 10.1088/1361-6544/ad25be
|
| [34] |
K. W. Hajduk, J. C. Robinsion, W. Sadowski, Robustness of regularity for the 3D convective Brinkman-Forchheimer equations, J. Math. Anal. Appl., 500 (2021), 125058. https://doi.org/10.1016/j.jmaa.2021.125058 doi: 10.1016/j.jmaa.2021.125058
|
| [35] | H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach, Birkhäuser Advanced Texts, Springer, Basel, 2012. |
| [36] | L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa CI. Sci., 13 (1959), 115–162. |
| [37] |
S. I. Chernysehnko, P. Constantin, J. C. Robinson, E. S. Titi, A posteriori regularity of the three-dimensional Navier-Stokes equations from numerical computations, J. Math. Phys., 48 (2006), 1–15. https://doi.org/10.1063/1.2372512 doi: 10.1063/1.2372512
|
Ning Duan, Hao Pan, Xiaopeng Zhao. Robustness of regularity and convergence of Galerkin approximations for 3D generalized incompressible Navier-Stokes equations[J]. Communications in Analysis and Mechanics, 2026, 18(2): 406-418. doi: 10.3934/cam.2026016
DownLoad: