On the global stability of large Fourier mode for 3-D Navier-Stokes equations. (English) Zbl 1532.35350
Summary: In this paper, we first prove the global existence of strong solutions to 3-D incompressible Navier-Stokes equations with solenoidal initial data, which writes in the cylindrical coordinates is of the form: \( \operatorname{A}(r, z) \cos N \theta + \operatorname{B}(r, z) \sin N \theta \), provided that \(N\) is large enough. In particular, we prove that the corresponding solution has almost the same frequency \(N\) for any positive time. The main idea of the proof is first to write the solution in trigonometrical series in \(\theta\) variable and estimate the coefficients separately in some scale-invariant spaces, then we handle a sort of weighted sum of these norms of the coefficients in order to close the a priori estimate of the solution. Furthermore, we shall extend the above well-posedness result for initial data which is a linear combination of axisymmetric data without swirl and infinitely many large mode trigonometric series in the angular variable.
MSC:
| 35Q30 | Navier-Stokes equations |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35D35 | Strong solutions to PDEs |
| 35B45 | A priori estimates in context of PDEs |
| 35B07 | Axially symmetric solutions to PDEs |
| 42A32 | Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.) |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Keywords:
Navier-Stokes equations; global well-posedness; cylindrical coordinates; large Fourier modeReferences:
| [1] | Albritton, D.; Brué, E.; Colombo, M., Non-uniqueness of Leray solutions of the forced Navier-Stokes equations. Ann. Math. (2), 415-455 (2022) · Zbl 1497.35337 |
| [2] | Buckmaster, T.; Vicol, V., Nonuniqueness of weak solutions to the Navier-Stokes equation. Ann. Math. (2), 1, 101-144 (2019) · Zbl 1412.35215 |
| [3] | Chae, D.; Lee, J., On the regularity of the axisymmetric solutions of the Navier-Stokes equations. Math. Z., 645-671 (2002) · Zbl 0992.35068 |
| [4] | Chemin, J.-Y.; Gallagher, I., Large, global solutions to the Navier-Stokes equations, slowly varying in one direction. Trans. Am. Math. Soc., 2859-2873 (2010) · Zbl 1189.35220 |
| [5] | Chemin, J.-Y.; Gallagher, I.; Zhang, P., Sums of large global solutions to the incompressible Navier-Stokes equations. J. Reine Angew. Math., 65-82 (2013) · Zbl 1300.35068 |
| [6] | Chemin, J.-Y.; Zhang, P., Remarks on the global solutions of 3-D Navier-Stokes system with one slow variable. Commun. Partial Differ. Equ., 878-896 (2015) · Zbl 1323.35128 |
| [7] | Chen, C. C.; Strain, R.; Yau, H. T.; Tsai, T. P., Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations. Int. Math. Res. Not. IMRN, 9 (2008) · Zbl 1154.35068 |
| [8] | Chen, C. C.; Strain, R.; Tsai, T. P.; Yau, H. T., Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations. II. Commun. Partial Differ. Equ., 203-232 (2009) · Zbl 1173.35095 |
| [9] | Chen, J.; Hou, T., Finite time blowup of 2D Boussinesq and 3D Euler equations with \(C^{1 , \alpha}\) velocity and boundary. Commun. Math. Phys., 3, 1559-1667 (2021) · Zbl 1485.35071 |
| [10] | Constantin, P.; Foias, C., Navier-Stokes Equations. Chicago Lectures in Mathematics (1988), University of Chicago Press: University of Chicago Press Chicago, IL · Zbl 0687.35071 |
| [11] | Elgindi, T., Finite-time singularity formation for \(C^{1 , \alpha}\) solutions to the incompressible Euler equations on \(\mathbb{R}^3\). Ann. Math. (2), 3, 647-727 (2021) · Zbl 1492.35199 |
| [12] | Elgindi, T.; Jeong, I.-J., Finite-time singularity formation for strong solutions to the axi-symmetric 3D Euler equations. Ann. PDE, 2 (2019) · Zbl 1436.35055 |
| [13] | Fefferman, C. L., Existence and smoothness of the Navier-Stokes equation, 57-67 · Zbl 1194.35002 |
| [14] | Gallagher, I.; Iftimie, D.; Planchon, F., Asymptotics and stability for global solutions to the Navier-Stokes equations. Ann. Inst. Fourier (Grenoble), 1387-1424 (2003) · Zbl 1038.35054 |
| [15] | Gallay, T.; Šverák, V., Uniqueness of axisymmetric viscous flows originating from circular vortex filaments. Ann. Sci. Éc. Norm. Supér. (4), 1025-1071 (2019) · Zbl 1461.35182 |
| [16] | Hopf, E., Über die Anfangswertaufgabe für die hydrodynamischen Grundgle-ichungen. Math. Nachr., 213-231 (1951) · Zbl 0042.10604 |
| [17] | Iskauriaza, L.; Seregin, G. A.; Šverák, V., \( L_{3 , \infty}\) -solutions of Navier-Stokes equations and backward uniqueness. Usp. Mat. Nauk. Russ. Math. Surv., 2(350), 211-250 (2003), (in Russian); translation in · Zbl 1064.35134 |
| [18] | Jia, H.; Šverák, V., Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions. Invent. Math., 1, 233-265 (2014) · Zbl 1301.35089 |
| [19] | Jia, H.; Šverák, V., Are the incompressible 3d Navier-Stokes equations locally ill-posed in the natural energy space?. J. Funct. Anal., 12, 3734-3766 (2015) · Zbl 1317.35176 |
| [20] | Koch, G.; Nadirashvili, N.; Seregin, G. A.; Šverák, V., Liouville theorems for the Navier-Stokes equations and applications. Acta Math., 83-105 (2009) · Zbl 1208.35104 |
| [21] | Ladyzhenskaja, O. A., Uniqueness and smoothness of generalized solutions of Navier-Stokes equations. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 169-185 (1967) · Zbl 0194.12805 |
| [22] | Ladyzhenskaja, O. A., Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 155-177 (1968), (in Russian) |
| [23] | Leonardi, S.; Málek, J.; Nečas, J.; Pokorny, M., On axially symmetric flows in \(\mathbb{R}^3\). Z. Anal. Anwend., 639-649 (1999) · Zbl 0943.35066 |
| [24] | Leray, J., Essai sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math., 193-248 (1933), (in French) · JFM 59.0763.02 |
| [25] | Liu, Y., Solving the axisymmetric Navier-Stokes equations in critical spaces (I): the case with small swirl component. J. Differ. Equ., 287-315 (2022) · Zbl 1508.35040 |
| [26] | Liu, Y.; Zhang, P., On the global well-posedness of 3-D axisymmetric Navier-Stokes system with small swirl component. Calc. Var. Partial Differ. Equ., 1 (2018) · Zbl 1384.35065 |
| [27] | Liu, Y.; Zhang, P., Global solutions of 3-D Navier-Stokes system with small unidirectional derivative. Arch. Ration. Mech. Anal., 1405-1444 (2020) · Zbl 1435.35280 |
| [28] | Liu, Y.; Xu, L., On the existence and structures of almost axisymmetric solutions to 3-D Navier-Stokes equations. SIAM J. Math. Anal., 1, 458-485 (2023) · Zbl 1512.35445 |
| [29] | Shen, Z., \( L^p\) estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier (Grenoble), 513-546 (1995) · Zbl 0818.35021 |
| [30] | Stein, E. M., Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series (1970), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0207.13501 |
| [31] | Tao, T., Localisation and compactness properties of the Navier-Stokes global regularity problem. Anal. PDE, 1, 25-107 (2013) · Zbl 1287.35058 |
| [32] | Ukhovskii, M. R.; Yudovich, V. I., Axially symmetric flows of ideal and viscous fluids filling the whole space. J. Appl. Math. Mech., 52-61 (1968) |
| [33] | Wei, D., Regularity criterion to the axially symmetric Navier-Stokes equations. J. Math. Anal. Appl., 402-413 (2016) · Zbl 1330.35298 |
| [34] | Zhong, J., Harmonic analysis for some Schroedinger type operators (1993), Princeton University, 84 pp |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
