On the stationary and nonstationary Navier-Stokes equations in \(R^ n\) (Q1824110)

The author solves the stationary and nonstationary Navier-Stokes equations: \[ -\nu \Delta v+(v\cdot \nabla)v+\nabla p=f,\quad \nabla \cdot v=0\quad in\quad R^ n,\quad \lim_{| x| \to \infty}v(x)=0,\quad \lim_{| x| \to \infty}p(x)=0 \] and \[ u'- \nu \Delta u+(u\cdot \nabla)u+\nabla q=f,\quad \nabla \cdot u=0\quad in\quad (0,T)\times R^ n, \] \[ u(0,x)=u_ 0(x),\quad \lim_{| x| \to \infty}u(t,x)=0,\quad \lim_{| x| \to \infty}q(t,x)=0. \] The existence and uniqueness results are derived. The asymptotic behaviour of a stationary solution u is established too. The cases \(n\geq 3\), \(n=3\) and \(n\geq 4\) are considered.