Abstract:
In this paper, we investigate the global well-posedness of three-dimensional Navier–Stokes system with the axisymmetric initial data and with various types of smallness c...Show MoreMetadata
Abstract:
In this paper, we investigate the global well-posedness of three-dimensional Navier–Stokes system with the axisymmetric initial data and with various types of smallness conditions on the swirl component u^{\theta}_0 of the initial velocity field. In particular, we proved that if u^{\theta}_01_{r\leq M} is sufficiently small (compared with ∥u0∥L2, \parallel \omega^{\theta}_0\parallel_{L^2}, and \parallel \frac {\omega^{\theta}_0}{r}\parallel_{L^2}) for M large enough, then (1.1) has a unique global smooth solution. Furthermore, we prove that as long as the local axisymmetric smooth solution u satisfying \parallel u{\theta}1_{r\leq \delta}\parallel_{L^{\alpha}((0,T);L^{\beta})}\lt+\infty for δ>0, \frac {2}{\alpha}+\frac {3}{\beta}\lt 1 β>6, or (α,β)=(4,6), and any T\lt \infty u will not blow-up in finite time.
Published in: International Mathematics Research Notices ( Volume: 2014, Issue: 3, 2014)
DOI: 10.1093/imrn/rns232
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- Index Terms
- Global Solution ,
- Three-dimensional System ,
- Axisymmetric Solution ,
- Navier Stokes System ,
- Three-dimensional Navier-Stokes ,
- Proof Of Theorem ,
- Unique Solution ,
- Positive Constant ,
- Local Solution ,
- Vertical Component ,
- Horizontal Components ,
- Organization Of The Paper ,
- Proof Of The Lemma ,
- Integration By Parts ,
- Cylindrical Coordinate System ,
- Gronwall’s Inequality ,
- Embedding Theorem ,
- Lebesgue Dominated Convergence ,
- Notations In This Paper ,
- Small Assumption
Keywords assist with retrieval of results and provide a means to discovering other relevant content. Learn more.
- Index Terms
- Global Solution ,
- Three-dimensional System ,
- Axisymmetric Solution ,
- Navier Stokes System ,
- Three-dimensional Navier-Stokes ,
- Proof Of Theorem ,
- Unique Solution ,
- Positive Constant ,
- Local Solution ,
- Vertical Component ,
- Horizontal Components ,
- Organization Of The Paper ,
- Proof Of The Lemma ,
- Integration By Parts ,
- Cylindrical Coordinate System ,
- Gronwall’s Inequality ,
- Embedding Theorem ,
- Lebesgue Dominated Convergence ,
- Notations In This Paper ,
- Small Assumption