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Zbl 0878.35087
Kato, Hisako
Existence of periodic solutions of the Navier-Stokes equations. (English)
[J] J. Math. Anal. Appl. 208, No.1, 141-157 (1997). ISSN 0022-247X

The author considers the nonstationary Navier-Stokes system $$\frac{\partial u}{\partial t}-\Delta u+u\cdot\nabla u+\nabla p=f ,\quad \nabla\cdot u=0\quad x\in\Omega,\ \ t\in \bbfR$$ with boundary condition $u=0$ on $\partial\Omega$ and periodicity condition $$u(x,t+\omega)=u(x,t)\quad x\in\Omega,\ t\in \bbfR$$ Here $\Omega$ is a bounded domain in $\bbfR^n\ (n=3,4)$ with smooth boundary $\partial\Omega$. It is proved that if $f$ is a sufficiently small $\omega$-periodic function then the problem has a unique $\omega$-periodic strong solution.
[I.Sh.Mogilevskij (Ferrara)]

MSC 2000:: *35Q30 Stokes and Navier-Stokes equations
76D05 Navier-Stokes equations (fluid dynamics)
35B10 Periodic solutions of PDE

Keywords: strong periodic solution; existence; uniqueness

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Zentralblatt MATH Berlin [Germany]