Archive for Rational Mechanics and Analysis

, Volume 223, Issue 2, pp 817–843 | Cite as

Global Axisymmetric Solutions of Three Dimensional Inhomogeneous Incompressible Navier–Stokes System with Nonzero Swirl

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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
$$\left\|\frac{a_{0}}{r}\right\|_{\infty} {\rm and}\|u_{0}^{\theta}\|_{3} {\rm are sufficiently small}.$$
Furthermore, if \({u_0 \in L^{1}}\) and \({ru^{\theta}_{0}\in L^{1} \cap L^{2}}\) , we have the decay estimate
$$\begin{aligned} \|u^{\theta}(t)\|_{2}^{2} + \langle t \rangle \|\nabla(u^{\theta}e_{\theta})(t)\|_{2}^{2} + t\langle t \rangle(\|u_{t}^{\theta}(t)\|_{2}^{2} + \|\Delta(u^{\theta}e_{\theta})(t)\|_{2}^{2}) \leqq C \langle t\rangle^{-\frac{5}{2}}, \\ \quad \forall t > 0. \end{aligned}$$
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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang UniversityHangzhouChina

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