Stability of Constant Solutions to the Navier–Stokes System in ${\Bbb R}^3$

Piotr Bogus/law Mucha¹

¹ Institute of Applied Mathematics and Mechanics Warsaw University Banacha 2 02-097 Warszawa, Poland

Applicationes Mathematicae, Tome 28 (2001) no. 3, pp. 301-310

Résumé

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.

Détail

DOI : 10.4064/am28-3-6

Keywords: paper examines initial value problem navier stokes system viscous incompressible fluids three dimensional space prove stability regular solutions which tend constant flows sufficiently fast perturbation regular solution bounded bbb times bbb result obtained under assumption smallness norm perturbing initial assume smallness bbb norm perturbing initial smallness r norm perturbing force