Stability of Constant Solutions to the Navier–Stokes System in ${\Bbb R}^3$
Volume 28 / 2001
Applicationes Mathematicae 28 (2001), 301-310
MSC: 35B35, 35Q30, 76D05.
DOI: 10.4064/am28-3-6
Abstract
The paper examines the initial value problem for the Navier–Stokes system of viscous incompressible fluids in the three-dimensional space. We prove stability of regular solutions which tend to constant flows sufficiently fast. We show that a perturbation of a regular solution is bounded in $W^{2,1}_r({\Bbb R}^3\times [k,k+1])$ for $k\in {\Bbb N}$. The result is obtained under the assumption of smallness of the $L_2$-norm of the perturbing initial data. We do not assume smallness of the $W^{2-2/r}_r({\Bbb R}^3)$-norm of the perturbing initial data or smallness of the $L_r$-norm of the perturbing force.
