Abstract
In this paper, we investigate the global well-posedness for the three dimensional inhomogeneous incompressible Navier–Stokes system with axisymmetric initial data. We obtain the global existence and uniqueness of the axisymmetric solution provided that
Furthermore, if \({u_0 \in L^{1}}\) and \({ru^{\theta}_{0}\in L^{1} \cap L^{2}}\) , we have the decay estimate
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abidi H., Gui G.L., Zhang P.: On the wellposedness of three-dimensional inhomogeneous Navier–Stokes equations in the critical spaces. Arch. Rational Mech. Anal. 204(1), 189–230 (2012). doi:10.1007/s00205-011-0473-4
Abidi H., Zhang P.: Global smooth axisymmetric solutions of 3-D inhomogeneous incompressible Navier–Stokes system. Calc. Var. Partial Differ. Equ. 54(3), 3251–3276 (2015). doi:10.1007/s00526-015-0902-6
Chae D., Lee J.: On the regularity of the axisymmetric solutions of the Navier–Stokes equations. Math. Z. 239(4), 645–671 (2002). doi:10.1007/s002090100317
Chen H., Fang D.Y., Zhang T.: Regularity of 3D axisymmetric Navier–Stokes equations. arXiv:1505.00905
Danchin R.: Local and global well-posedness results for flows of inhomogeneous viscous fluids. Adv. Differential Equations 9(3–4), 353–386 (2004)
Danchin R.: The inviscid limit for density-dependent incompressible fluids. Ann. Fac. Sci. Toulouse Math. (6) 15(4), 637–688 (2006). doi:10.5802/afst.1133
Danchin R.: A Lagrangian approach for the incompressible Navier–Stokes equations with variable density. Commun. Pure Appl. Math. 65(10), 1458–1480 (2012). doi:10.1002/cpa.21409
Kim H.: A blow-up criterion for the nonhomogeneous incompressible Navier–Stokes equations. SIAM J. Math. Anal. 37(5), 1417–1434 (2006). doi:10.1137/S0036141004442197
Ladyzenskaja 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. 7, 155–177 (1968)
Ladyzenskaja O.A., Solonnikov V.A.: The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids. Boundary value problems of mathematical physics, and related questions of the theory of functions, 8. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 52, 52–109, 218–219 (1975).
Lei Z., Zhang Qi S.: Criticality of the axially symmetric Navier–Stokes equations. arXiv:1505.02628
Leonardi S., Málek J., Nečas J., Pokorný M.: On axially symmetric flows in \({{\mathbb R}^{3}}\). Z. Anal. Anwendungen 18(3), 639–649 (1999). doi:10.4171/ZAA/903
Lions P.L.: Mathematical topics in fluid mechanics. Incompressible models, vol. 1. The Clarendon Press, Oxford University Press, New York (1996)
Liu J.G., Wang W.C.: Energy and helicity preserving schemes for hydro- and magnetohydro-dynamics flows with symmetry. J. Comput. Phys. 200(1), 8–33 (2004). doi:10.1016/j.jcp.2004.03.005
Liu J.G., Wang W.C.: Convergence analysis of the energy and helicity preserving scheme for axisymmetric flows. SIAM J. Numer. Anal. 44(6), 2456–2480 (2006). doi:10.1137/050639314
Liu J.G., Wang W.C.: Characterization and regularity for axisymmetric solenoidal vector fields with application to Navier–Stokes equation. SIAM J. Math. Anal. 41(5), 1825–1850 (2009). doi:10.1137/080739744
Simon J.: Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure. SIAM J. Math. Anal. 21(5), 1093–1117 (1990). doi:10.1137/0521061
Paicu M., Zhang P., Zhang Z.F.: Global unique solvability of inhomogeneous Navier–Stokes equations with bounded density. Comm. Partial Differential Equations 38(7), 1208–1234 (2013). doi:10.1080/03605302.2013.780079
Ukhovskii M.R., Iudovich V.I.: Axially symmetric flows of ideal and viscous fluids filling the whole space. J. Appl. Math. Mech. 32(1), 52–61 (1968). doi:10.1016/0021-8928(68)90147-0
Wei D.Y.: Regularity criterion to the axially symmetric Navier–Stokes equations. J. Math. Anal. Appl. 435(1), 402–413 (2016). doi:10.1016/j.jmaa.2015.09.088
Zhang P., Zhang T.: Global axisymmetric solutions to three-dimensional Navier–Stokes system. Int. Math. Res. Not. 2014(3), 610–642 (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F. Lin
Rights and permissions
About this article
Cite this article
Chen, H., Fang, D. & Zhang, T. Global Axisymmetric Solutions of Three Dimensional Inhomogeneous Incompressible Navier–Stokes System with Nonzero Swirl. Arch Rational Mech Anal 223, 817–843 (2017). https://doi.org/10.1007/s00205-016-1046-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-016-1046-3