Abstract
We consider instationary Navier–Stokes equations with prescribed time-dependent fluxes in a cylindrical domain \(\Omega \subset {\mathbb R}^n,n\ge 3,\) with several exits to infinity. First, we prove existence and uniqueness of time-dependent Poiseuille flow in \(L^q\)-spaces on infinite straight cylinders over time interval \((0,T)\), \(0<T\le \infty \), by obtaining an estimate of the pressure gradient using heat semigroups and techniques of Fourier multipliers. Then, based on sharp estimates in Besov spaces of the nonlinear term in the Navier–Stokes equations, we show the existence, uniqueness of a strong solution with prescribed time-dependent flux in each exit of \(\Omega \) in \(L^p(0,T;L^q_\beta (\Omega )), p>2, q\in (\frac{n-1}{2},\infty )\), with an exponential weight along the axial directions of \(\Omega \).
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Acknowledgments
Part of this work was done during the stay of the author in the Institute of Mathematics, AMSS, CAS, China by the support of 2012 CAS-TWAS Postdoctoral Fellowship, grant No. 3240267229. The author is indebted much to Professor Reinhard Farwig for his careful reading the manuscript of the paper, correction of errors and suggestion for concise description for some parts of proofs.
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Ri, MH. Incompressible Navier–Stokes flows with time-dependent prescribed fluxes in domains with cylindrical outlets to infinity. Ann Univ Ferrara 61, 119–148 (2015). https://doi.org/10.1007/s11565-014-0217-0
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DOI: https://doi.org/10.1007/s11565-014-0217-0
Keywords
- Navier–Stokes equations
- Infinite cylindrical domains
- Nonstationary Poiseuille flow
- Time-dependent fluxes
- Exponential weight