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Global Solutions to the 3-D Incompressible Inhomogeneous Navier–Stokes System with Rough Density

Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 84)

Abstract

In this paper, we prove the global well-posedness of the 3-D incompressible inhomogeneous Navier–Stokes equations with initial data \(a_{0} \in \mathcal{M}(B_{p,1}^{-1+ \frac{3} {p} }({\mathbb{R}}^{3})),\) \(u_{0} = (u_{0}^{h},u_{0}^{3}) \in B_{p,1}^{-1+ \frac{3} {p} }({\mathbb{R}}^{3}),\) which satisfies
$$(\mu \|a_{0}\|_{ \mathcal{M}(B_{p,1}^{-1+ \frac{3} {p} })} +\| u_{0}^{h}\|_{ B_{p,1}^{-1+ \frac{3} {p} }})\exp (C_{0}\|u_{0}^{3}\|_{ B_{p,1}^{-1+ \frac{3} {p} }}^{2}\ /{\mu }^{2}) \leq c_{ 0}\mu $$
for some positive constants c 0, C 0 and \(\frac{3} {2} < p < 6.\) The novelty of this paper is to replace \(\|a_{0}\|_{ B_{q,1}^{ \frac{3} {q} }}\) in the smallness condition of [20] by the rough norm in the multiplier space \(\|a_{0}\|_{ \mathcal{M}(B_{p,1}^{-1+ \frac{3} {p} })}\) here.

Key words

Inhomogeneous Navier–Stokes equations Littlewood–Paley theory Wellposedness 

Notes

Acknowledgements

Part of this work was done when Marius Paicu was visiting Morningside Center of the Chinese Academy of Sciences in the Spring of 2012. We appreciate the hospitality of MCM and the financial support from the Chinese Academy of Sciences. P. Zhang is partially supported by NSF of China under Grant 10421101 and 10931007, the one hundred talents’ plan from Chinese Academy of Sciences under Grant GJHZ200829 and innovation grant from National Center for Mathematics and Interdisciplinary Sciences.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Academy of Mathematics & Systems ScienceChinese Academy of SciencesBeijingP. R. China
  2. 2.Institut de Mathématiques de BordeauxUniversité Bordeaux 1Talence CedexFrance
  3. 3.Academy of Mathematics & Systems Science and Hua Loo-Keng Key Laboratory of MathematicsThe Chinese Academy of SciencesBeijingP. R. China

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