Global well-posedness of the Navier-Stokes-omega equations. (English) Zbl 1408.76129
Summary: First, we prove that the local solution to the Navier-Stokes-omega equations is unique when the spatial dimension \(n\) satisfies \(3\leq n\leq 6\). Then, a regularity criterion is established for any \(n\geq 3\). As a corollary, it is proved that the smooth solution exists globally when \(3\leq n\leq 6\).
MSC:
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35Q35 | PDEs in connection with fluid mechanics |
References:
| [1] | W. Layton, I. Stanculescu, C. Trenchea, Theory of the \(\operatorname{NS} - \overline{\omega}\operatorname{NS} - \alpha \); W. Layton, I. Stanculescu, C. Trenchea, Theory of the \(\operatorname{NS} - \overline{\omega}\operatorname{NS} - \alpha \) · Zbl 1229.76031 |
| [2] | Kato, T.; Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41, 7, 891-907 (1988) · Zbl 0671.35066 |
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