Strong solutions of the steady nonlinear Navier-Stokes system in domains with exits to infinity
The solvability of the stationary Navier-Stokes equations in weighted Sobolev and Hölder spaces is studied. The class of domains \(\Omega\subset\mathbb{R}^n\), \(n= 2,3\), having \(m>1\) exits to infinity \(\Omega_i\) of the form \[ \Omega_i= \{x\in\mathbb{R}^n;\;|x'|< g_i(x_n),\;x_n>0\} \] and the flow of viscous incompressible fluid in such domain is considered. First, the problem in the case of zero fluxes in the space of exponentially vanishing at infinity functions is studied. These results are based on the Banach contraction principle and refer to dimensions \(n=2,3\). Then, for arbitrarily fluxes and in three-dimensional domains \(\Omega\) the solvability of this problem for arbitrarily large data is proved. The case of two-dimensional domains \(\Omega\) is more complicated and for such domains the solvability only for small data is proved.
scientific article; zbMATH DE number 1089780
scientific article