Long time existence of regular solutions to Navier–Stokes equations in cylindrical domains under boundary slip conditions

Volume 169 / 2005

W. M. Zajączkowski Studia Mathematica 169 (2005), 243-285 MSC: 35Q35, 76D03, 76D05. DOI: 10.4064/sm169-3-3

Abstract

Long time existence of solutions to the Navier–Stokes equations in cylindrical domains under boundary slip conditions is proved. Moreover, the existence of solutions with no restrictions on the magnitude of the initial velocity and the external force is shown. However, we have to assume that the quantity $$ I=\sum_{i=1}^2(\|\partial_{x_3}^iv(0)\|_{L_2({\mit\Omega})}+ \|\partial_{x_3}^if\|_{L_2({\mit\Omega}\times(0,T))}) $$ is sufficiently small, where $x_3$ is the coordinate along the axis parallel to the cylinder. The time of existence is inversely proportional to $I$. Existence of solutions is proved by the Leray–Schauder fixed point theorem applied to problems for $h^{(i)}=\partial_{x_3}^iv$, $q^{(i)}=\partial_{x_3}^ip$, $i=1,2$, which follow from the Navier–Stokes equations and corresponding boundary conditions. Existence is proved in Sobolev–Slobodetskiĭ spaces: $h^{(i)}\in W_\delta^{2+\beta,1+\beta/2}({\mit\Omega}\times(0,T))$, where $i=1,2$, $\beta\in(0,1)$, $\delta\in(1,2)$, $5/\delta<3+\beta$, $3/\delta<2+\beta$.

Authors

  • W. M. ZajączkowskiInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-956 Warszawa, Poland
    and
    Institute of Mathematics and Cryptology
    Military University of Technology
    Kaliskiego 2
    00-908 Warszawa, Poland
    e-mail

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