Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-29T07:50:11.008Z Has data issue: false hasContentIssue false
  • English
  • Français

Axisymmetric flows in the exterior of a cylinder

Part of: Partial differential equations Parabolic equations and systems

Published online by Cambridge University Press:  01 February 2019

K. Abe  and
G. Seregin
K. Abe
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka City University, Sugimoto 3-3-138, Sumiyoshi-ku Osaka558-8585, Japan ([email protected])
G. Seregin
Affiliation:
Mathematical Institute, Oxford University, Oxford, 24-29 St Giles', OX1 3LB, UK and Voronezh State University, Voronezh, Russia ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We study an initial-boundary value problem of the three-dimensional Navier-Stokes equations in the exterior of a cylinder $\Pi =\{x=(x_{h}, x_3)\ \vert \vert x_{h} \vert \gt 1\}$, subject to the slip boundary condition. We construct unique global solutions for axisymmetric initial data $u_0\in L^{3}\cap L^{2}(\Pi )$ satisfying the decay condition of the swirl component $ru^{\theta }_{0}\in L^{\infty }(\Pi )$.

Keywords

Navier-Stokes equationsaxisymmetric solutionsexterior of a cylinder

MSC classification

Primary: 35B07: Axially symmetric solutions
Secondary: 35K20: Initial-boundary value problems for second-order parabolic equations 35K90: Abstract parabolic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 4 , August 2020 , pp. 1671 - 1698
Check for updates
Copyright
Copyright © Royal Society of Edinburgh 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Abidi, H.. Résultats de régularité de solutions axisymétriques pour le système de Navier-Stokes. Bull. Sci. Math. 132 (2008), 592–624.CrossRefGoogle Scholar
2Agmon, S., Douglis, A. and Nirenberg, L.. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12 (1959), 623–727.CrossRefGoogle Scholar
3Bae, H.-O. and Jin, B.. Regularity for the Navier-Stokes equations with slip boundary condition. Proc. Amer. Math. Soc. 136 (2008), 2439–2443.CrossRefGoogle Scholar
4Beirão da Veiga, H.. Vorticity and regularity for flows under the Navier boundary condition. Commun. Pure Appl. Anal. 5 (2006), 907–918.CrossRefGoogle Scholar
5Beirão da Veiga, H.. Vorticity and regularity for viscous incompressible flows under the Dirichlet boundary condition. Results and related open problems. J. Math. Fluid Mech. 9 (2007), 506–516.CrossRefGoogle Scholar
6Caffarelli, L., Kohn, R. and Nirenberg, L.. Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982), 771–831.CrossRefGoogle Scholar
7Calderón, C. P.. Existence of weak solutions for the Navier-Stokes equations with initial data in L p. Trans. Amer. Math. Soc. 318 (1990), 179–200.Google Scholar
8Chae, D. and Lee, J.. On the regularity of the axisymmetric solutions of the Navier-Stokes equations. Math. Z. 239 (2002), 645–671.CrossRefGoogle Scholar
9Chen, C.-C., Strain, R. M., Tsai, T.-P. and Yau, H.-T.. Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations. Int. Math. Res. Not. IMRN (2008), 31, Art. ID rnn016.Google Scholar
10Chen, C.-C., Strain, R. M., Tsai, T.-P. and Yau, H.-T.. Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations. II. Comm. Partial Differential Equations 34 (2009), 203–232.CrossRefGoogle Scholar
11Choi, K., Kiselev, A. and Yao, Y.. Finite time blow up for a 1D model of 2D Boussinesq system. Comm. Math. Phys. 334 (2015), 1667–1679.CrossRefGoogle Scholar
12Choi, K., Hou, T. Y., Kiselev, A., Luo, G., Sverak, V. and Yao, Y.. On the finite-time blowup of a one-dimensional model for the three-dimensional axisymmetric Euler equations. Comm. Pure Appl. Math. 70 (2017), 2218–2243.CrossRefGoogle Scholar
13Constantin, P. and Fefferman, C.. Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ. Math. J. 42 (1993), 775–789.CrossRefGoogle Scholar
14Farwig, R. and Rosteck, V.. Resolvent estimates of the Stokes system with Navier boundary conditions in general unbounded domains. Adv. Differential Equations 21 (2016), 401–428.Google Scholar
15Farwig, R., Kozono, H. and Sohr, H.. An L q-approach to Stokes and Navier-Stokes equations in general domains. Acta Math. 195 (2005), 21–53.CrossRefGoogle Scholar
16Farwig, R., Kozono, H. and Sohr, H.. On the Helmholtz decomposition in general unbounded domains. Arch. Math. (Basel) 88 (2007), 239–248.CrossRefGoogle Scholar
17Farwig, R., Kozono, H. and Sohr, H.. On the Stokes operator in general unbounded domains. Hokkaido Math. J. 38 (2009), 111–136.CrossRefGoogle Scholar
18Gustafson, S., Kang, K. and Tsai, T.-P.. Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary. J. Differential Equations 226 (2006), 594–618.CrossRefGoogle Scholar
19Hopf, E.. über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4 (1951), 213–231.CrossRefGoogle Scholar
20Hou, T. and Luo, G.. On the finite-time blow up of a 1d model for the 3d incompressible euler equations, arxiv:1311.2613.Google Scholar
21Jiu, Q. and Xin, Z.. Some regularity criteria on suitable weak solutions of the 3-D incompressible axisymmetric Navier-Stokes equations. In Lectures on partial differential equations. New Stud. Adv. Math., pp. 119–139 (Somerville, MA: Int. Press, 2003).Google Scholar
22Kato, T.. Strong L p-solutions of the Navier-Stokes equation in R m, with applications to weak solutions. Math. Z. 187 (1984), 471–480.CrossRefGoogle Scholar
23Koch, G., Nadirashvili, N., Seregin, G. and Šverák, V.. Liouville theorems for the Navier-Stokes equations and applications. Acta Math. 203 (2009), 83–105.CrossRefGoogle Scholar
24Ladyženskaya, 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) 7 (1968), 155–177.Google Scholar
25Ladyžhenskaya, O. A., Solonnikov, V. A. and Uraltseva, N. N.. Linear and Quasilinear Equations of Parabolic Type. volume 23 of Translations of Mathematical Monographs (Providence, RI: American Mathematical Society, 1968).CrossRefGoogle Scholar
26Lei, Z., Navas, E. A. and Qi Zhang, S.. A priori bound on the velocity in axially symmetric Navier-Stokes equations. Comm. Math. Phys. 341 (2016), 289–307.CrossRefGoogle Scholar
27Lemarié-Rieusset, P. G.. Recent developments in the Navier-Stokes problem, vol. 431 (Boca Raton, FL: Chapman & Hall/CRC, 2002).CrossRefGoogle Scholar
28Leonardi, S., Málek, J., Nečas, J. and Pokorný, M.. On axially symmetric flows in R3. Z. Anal. Anwendungen 18 (1999), 639–649.CrossRefGoogle Scholar
29Leray, J.. Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63 (1934), 193–248.CrossRefGoogle Scholar
30Lieberman, G. M.. Second order parabolic differential equations (River Edge, NJ: World Scientific Publishing Co., Inc., 1996).CrossRefGoogle Scholar
31Loftus, J. B. and Qi Zhang, S.. A priori bounds for the vorticity of axially symmetric solutions to the Navier-Stokes equations. Adv. Differential Equations 15 (2010), 531–560.Google Scholar
32Lunardi, A.. Analytic semigroups and optimal regularity in parabolic problems. Progress in Nonlinear Differential Equations and their Applicationsvol. 16 (Basel: Birkhäuser Verlag, 1995).Google Scholar
33Luo, G. and Hou, T.. Potentially singular solutions of the 3d incompressible euler equations, arxiv:1310.0497.Google Scholar
34Majda, A. J. and Bertozzi, A. L.. Vorticity and incompressible flow, volume 27 of Cambridge Texts in Applied Mathematics (Cambridge: Cambridge University Press, 2002).Google Scholar
35Neustupa, J. and Pokorný, M.. An interior regularity criterion for an axially symmetric suitable weak solution to the Navier-Stokes equations. J. Math. Fluid Mech. 2 (2000), 381–399.CrossRefGoogle Scholar
36Neustupa, J. and Pokorný, M.. Axisymmetric flow of Navier-Stokes fluid in the whole space with non-zero angular velocity component. In Proceedings of Partial Differential Equations and Applications (Olomouc, 1999), vol. 126, pp. 469–481 (2001).Google Scholar
37Seregin, G.. Local regularity of suitable weak solutions to the Navier-Stokes equations near the boundary. J. Math. Fluid Mech. 4 (2002), 1–29.CrossRefGoogle Scholar
38Seregin, G.. Lecture notes on regularity theory for the Navier-Stokes equations. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., 2015.Google Scholar
39Seregin, G. and Šverák, V.. On type I singularities of the local axi-symmetric solutions of the Navier-Stokes equations. Comm. Partial Differential Equations 34 (2009), 171–201.CrossRefGoogle Scholar
40Seregin, G. and Šverák, V.. On global weak solutions to the Cauchy problem for the Navier-Stokes equations with large L 3-initial data. Nonlinear Anal. 154 (2017), 269–296.CrossRefGoogle Scholar
41Stein, E. M.. Singular integrals and differentiability properties of functions. Princeton Mathematical Seriesvol. 30 (Princeton, N.J.: Princeton University Press, 1970).Google Scholar
42Ukhovskii, M. R. and Iudovich, V. I.. Axially symmetric flows of ideal and viscous fluids filling the whole space. J. Appl. Math. Mech. 32 (1968), 52–61.CrossRefGoogle Scholar
43Zaja̧czkowski, W. M.. Global existence of axially symmetric solutions to Navier-Stokes equations with large angular component of velocity. Colloq. Math. 100 (2004), 243–263.CrossRefGoogle Scholar
44Zaja̧czkowski, W. M.. Global axially symmetric solutions with large swirl to the Navier-Stokes equations. Topol. Methods Nonlinear Anal. 29 (2007), 295–331.Google Scholar